GEOMETRIC STRUCTURES RESEARCH SEMINAR - PAST TALKS
co-organised by the research groups of
Andrei Agrachev, Antonio Lerario and Luca Rizzi
SISSA, Trieste, Italy
Title: Nodal sets of eigenfunctions of sub-Laplacians
Title: Nodal sets of eigenfunctions of sub-Laplacians
Time: January 11th, 2024, at 14:00 a.m. (Rome Time) - both in-presence in room 134 at SISSA and streamed on Zoom
Time: January 11th, 2024, at 14:00 a.m. (Rome Time) - both in-presence in room 134 at SISSA and streamed on Zoom
Abstract: Nodal sets of eigenfunctions of elliptic operators on compact manifolds have been studied extensively over the past decades. In a recent work, we initiated the study of nodal sets of eigenfunctions of hypoelliptic operators on compact manifolds, focusing on sub-Laplacians (e.g. on compact quotients of the Heisenberg group). Our results show that nodal sets behave in an anisotropic way which can be analyzed with standard tools from sub-Riemannian geometry such as sub-Riemannian dilations, nilpotent approximation and desingularization at singular points. This is a joint work with S. Eswarathasan.
Abstract: Nodal sets of eigenfunctions of elliptic operators on compact manifolds have been studied extensively over the past decades. In a recent work, we initiated the study of nodal sets of eigenfunctions of hypoelliptic operators on compact manifolds, focusing on sub-Laplacians (e.g. on compact quotients of the Heisenberg group). Our results show that nodal sets behave in an anisotropic way which can be analyzed with standard tools from sub-Riemannian geometry such as sub-Riemannian dilations, nilpotent approximation and desingularization at singular points. This is a joint work with S. Eswarathasan.
Title: Exceptional real Lie algebras $f_4$ and $e_6$ via contactifications
Title: Exceptional real Lie algebras $f_4$ and $e_6$ via contactifications
Time: September 4th, 2023, at 11:00 a.m. (Rome Time) - both in-presence in room 134 at SISSA and streamed on Zoom
Time: September 4th, 2023, at 11:00 a.m. (Rome Time) - both in-presence in room 134 at SISSA and streamed on Zoom
Abstract: In Cartan's PhD thesis, there is a formula defining a certain rank 8 vector distribution in dimension 15, whose algebra of authomorphism is the split real form of the simple exceptional complex Lie algebra $f_4$. Cartan's formula is written in the standard Cartesian coordinates in $\mathbb{R}^{15}$. In the talk I will explain how to find analogous formula for the flat models of any bracket generating distribution $D$ whose symbol algebra $n(D)$ is constant and 2-step graded, $n(D) = n−2 \oplus n−1$. I will use the general formula to provide other distributions with symmetries being real forms of simple exceptional Lie algebras $f_4$ and $e_6$.
Abstract: In Cartan's PhD thesis, there is a formula defining a certain rank 8 vector distribution in dimension 15, whose algebra of authomorphism is the split real form of the simple exceptional complex Lie algebra $f_4$. Cartan's formula is written in the standard Cartesian coordinates in $\mathbb{R}^{15}$. In the talk I will explain how to find analogous formula for the flat models of any bracket generating distribution $D$ whose symbol algebra $n(D)$ is constant and 2-step graded, $n(D) = n−2 \oplus n−1$. I will use the general formula to provide other distributions with symmetries being real forms of simple exceptional Lie algebras $f_4$ and $e_6$.
Title: MCP and geodesic dimension of sub-Finsler Heisenberg groups
Title: MCP and geodesic dimension of sub-Finsler Heisenberg groups
Time: June 15th, 2023, at 14:00 (Rome Time) - both in-presence and streamed on Zoom
Time: June 15th, 2023, at 14:00 (Rome Time) - both in-presence and streamed on Zoom
Abstract: We will discuss the Heisenberg group equipped with an $\ell^p$-sub-Finsler metric. We will explore its geometry through the corresponding (Finsler) isoperimetric problem. Subsequently, we will analyse these spaces as metric measure spaces, considering whether the measure contraction property holds. Furthermore, we will also compute their geodesic dimension. It will become apparent how the answers to these questions are controlled by the value of p (and its Hölder conjugate q). This value determines whether the $\ell^p$-sub-Finsler metric is branching or not, whether it possesses a negligible cut locus, and whether its geodesics are sufficiently smooth or not. Joint work with K. Tashiro.
Abstract: We will discuss the Heisenberg group equipped with an $\ell^p$-sub-Finsler metric. We will explore its geometry through the corresponding (Finsler) isoperimetric problem. Subsequently, we will analyse these spaces as metric measure spaces, considering whether the measure contraction property holds. Furthermore, we will also compute their geodesic dimension. It will become apparent how the answers to these questions are controlled by the value of p (and its Hölder conjugate q). This value determines whether the $\ell^p$-sub-Finsler metric is branching or not, whether it possesses a negligible cut locus, and whether its geodesics are sufficiently smooth or not. Joint work with K. Tashiro.
Title: Lipschitz Geometry of Germs of Real Surfaces
Title: Lipschitz Geometry of Germs of Real Surfaces
Time: June 1st, 2023, at 11:00 a.m. (Rome Time) - streamed on Zoom
Time: June 1st, 2023, at 11:00 a.m. (Rome Time) - streamed on Zoom
Abstract: Lipschitz Geometry is now an intensively developed part of Singularity Theory. I am going to make an introductory talk on the subject. I am going to explain the general directions of Lipschitz geometry (inner, outer and ambient) on the example of germs of Real Semialgebraic Surfaces.
Abstract: Lipschitz Geometry is now an intensively developed part of Singularity Theory. I am going to make an introductory talk on the subject. I am going to explain the general directions of Lipschitz geometry (inner, outer and ambient) on the example of germs of Real Semialgebraic Surfaces.
Title: First-order heat content asymptotics on RCD(K,N) spaces
Title: First-order heat content asymptotics on RCD(K,N) spaces
Time: May 19, 2023, at 11:00 a.m. (Rome Time) - both in-presence and streamed on Zoom
Time: May 19, 2023, at 11:00 a.m. (Rome Time) - both in-presence and streamed on Zoom
Abstract: We study the small-time asymptotics of the heat content associated with a bounded open set when the ambient space is an RCD(K,N) metric measure space. By adapting a technique due to Savo, we establish the existence of a first-order asymptotic expansion, under a regularity condition for the boundary of the domain that we call measured interior geodesic condition. We carefully study such a condition, relating it to the properties of the disintegration associated with the signed distance function from the boundary. This is a joint work with Emanuele Caputo.
Abstract: We study the small-time asymptotics of the heat content associated with a bounded open set when the ambient space is an RCD(K,N) metric measure space. By adapting a technique due to Savo, we establish the existence of a first-order asymptotic expansion, under a regularity condition for the boundary of the domain that we call measured interior geodesic condition. We carefully study such a condition, relating it to the properties of the disintegration associated with the signed distance function from the boundary. This is a joint work with Emanuele Caputo.
Title: Gromov's h-principle for corank two distribution of odd rank with maximal first Kronecker index
Title: Gromov's h-principle for corank two distribution of odd rank with maximal first Kronecker index
Time: April 7, 2023, at 11:00 a.m. (Rome Time) - both in-presence and streamed on Zoom
Time: April 7, 2023, at 11:00 a.m. (Rome Time) - both in-presence and streamed on Zoom
Abstract: While establishing various versions of the h-principle for contact distributions (Eliashberg (1989) in dimension 3, Borman-Eliashberg-Murphy (2015) in arbitrary dimension, and even-contact contact (D. McDuff, 1987) distributions are among the most remarkable advances in differential topology in the last four decades, very little is known about analogous results for other classes of distributions, e.g. generic distributions of corank 2 or higher. The smallest dimensional nontrivial case of corank 2 distributions are Engel distributions, i.e. the maximally nonholonomic rank 2 distributions on $4$-manifolds. This case is highly nontrivial and was treated recently by Casals-Pérez-del Pino-Presas (2017) and Casals-Pérez-Presas (2017). In my talk, I will show how to use the method of contex integration in order to establish all versions of the h-principle for corank 2 distribution of arbitrary odd rank satisfying a natural generic assumption on the associated pencil of skew-symmetric forms. During the talk I will try to give all the necessary background related to the method of convex integration in principle. This is the joint work with Milan Jovanovic, Javier Martinez-Aguinaga, and Alvaro del Pino.
Abstract: While establishing various versions of the h-principle for contact distributions (Eliashberg (1989) in dimension 3, Borman-Eliashberg-Murphy (2015) in arbitrary dimension, and even-contact contact (D. McDuff, 1987) distributions are among the most remarkable advances in differential topology in the last four decades, very little is known about analogous results for other classes of distributions, e.g. generic distributions of corank 2 or higher. The smallest dimensional nontrivial case of corank 2 distributions are Engel distributions, i.e. the maximally nonholonomic rank 2 distributions on $4$-manifolds. This case is highly nontrivial and was treated recently by Casals-Pérez-del Pino-Presas (2017) and Casals-Pérez-Presas (2017). In my talk, I will show how to use the method of contex integration in order to establish all versions of the h-principle for corank 2 distribution of arbitrary odd rank satisfying a natural generic assumption on the associated pencil of skew-symmetric forms. During the talk I will try to give all the necessary background related to the method of convex integration in principle. This is the joint work with Milan Jovanovic, Javier Martinez-Aguinaga, and Alvaro del Pino.
Title: From CT Scans to four-manifold topology
Title: From CT Scans to four-manifold topology
Time: March 2, 2023, at 2:00 pm (Rome Time) - both in-presence and streamed on Zoom
Time: March 2, 2023, at 2:00 pm (Rome Time) - both in-presence and streamed on Zoom
Abstract: Integration over lines is the mathematical basis of many modern methods of tomography, including Computerized Tomography scans. In this talk, a recent geometrization using indefinite metrics of signature (2,2) is presented of the seminal work of Fritz John on the problem. The contemporary mathematical background is 4-manifold topology and the use of neutral metrics to explore co-dimension two problems.
Abstract: Integration over lines is the mathematical basis of many modern methods of tomography, including Computerized Tomography scans. In this talk, a recent geometrization using indefinite metrics of signature (2,2) is presented of the seminal work of Fritz John on the problem. The contemporary mathematical background is 4-manifold topology and the use of neutral metrics to explore co-dimension two problems.
Title: Sub-Lorentzian problem on the Heisenberg group (Yu. Sachkov, E. Sachkova)
Title: Sub-Lorentzian problem on the Heisenberg group (Yu. Sachkov, E. Sachkova)
Time: February 16, 2023, at 2:00 pm (Rome Time) - online only
Time: February 16, 2023, at 2:00 pm (Rome Time) - online only
Abstract: The sub-Riemannian problem on the Heisenberg group is well known, it is a cornerstone of sub-Riemannian geometry. It can be stated as a time-optimal problem with a planar set of control parameters, a circle.
Abstract: The sub-Riemannian problem on the Heisenberg group is well known, it is a cornerstone of sub-Riemannian geometry. It can be stated as a time-optimal problem with a planar set of control parameters, a circle.
The talk will be devoted to its natural variation, the time-optimal problem with a hyperbola as a set of control parameters.
The talk will be devoted to its natural variation, the time-optimal problem with a hyperbola as a set of control parameters.
This variation is the sub-Lorentzian problem on the Heisenberg group. For this problem we will describe the following results:
This variation is the sub-Lorentzian problem on the Heisenberg group. For this problem we will describe the following results:
1) The reachable set from the identity of the group,
1) The reachable set from the identity of the group,
2) Pontryagin maximum principle, parameterization of extremal trajectories, exponential mapping,
2) Pontryagin maximum principle, parameterization of extremal trajectories, exponential mapping,
3) Diffeomorphic property of the exponential mapping, its inverse,
3) Diffeomorphic property of the exponential mapping, its inverse,
4) Optimality of extremal trajectories, optimal synthesis,
4) Optimality of extremal trajectories, optimal synthesis,
5) Sub-Lorentzian distance,
5) Sub-Lorentzian distance,
6) Sub-Lorentzian spheres of positive and zero radii.
6) Sub-Lorentzian spheres of positive and zero radii.
Results 1), 2) were obtained by M.Grochowski (2006), the rest results are new.
Results 1), 2) were obtained by M.Grochowski (2006), the rest results are new.
Title: Systolic inequality and volume of the unit ball in Carnot groups
Title: Systolic inequality and volume of the unit ball in Carnot groups
Time: January 19, 2023, at 2:00 pm (Rome Time) - online only
Time: January 19, 2023, at 2:00 pm (Rome Time) - online only
Abstract: Roughly speaking, a systolic inequality on a length measure space asserts that the minimal length of non-contractible closed curve is controlled by the product of the (root of) total measure and a constant depending only on its topology. Such inequalities hold for (a class of) Riemannian manifolds and Alexandrov spaces with the constants depending only on the Hausdorff dimension.
Abstract: Roughly speaking, a systolic inequality on a length measure space asserts that the minimal length of non-contractible closed curve is controlled by the product of the (root of) total measure and a constant depending only on its topology. Such inequalities hold for (a class of) Riemannian manifolds and Alexandrov spaces with the constants depending only on the Hausdorff dimension.
We proved the systolic inequality on quotient spaces of Carnot groups, which is a class of closed sub-Riemannian manifolds, with the constant depending only on the Hausdorff dimension. Actually it is equivalent to give a uniform lower bound of volume of the unit ball in Carnot groups of a given Hausdorff dimension.
We proved the systolic inequality on quotient spaces of Carnot groups, which is a class of closed sub-Riemannian manifolds, with the constant depending only on the Hausdorff dimension. Actually it is equivalent to give a uniform lower bound of volume of the unit ball in Carnot groups of a given Hausdorff dimension.
Title: Surface area on sub-Riemannian measure manifolds
Title: Surface area on sub-Riemannian measure manifolds
Time: December 15, 2022 at 2:00 pm (Rome Time)
Time: December 15, 2022 at 2:00 pm (Rome Time)
Abstract: We present an area formula in equiregular sub-Riemannian measure manifolds. The perimeter measure of a smooth bounded open set is related to the spherical measure of its boundary, using the sub-Riemannian distance. To perform the intrinsic blow-up at the boundary new difficulties appear, that also involve the nilpotent approximation of the sub-Riemannian manifold. The density of the perimeter measure naturally arises as a geometric invariant that can be explicitly related to different objects, like the nilpotent approximation, the tangent Riemannian metric and the shape of the tangent unit ball. The area formula for the perimeter measure is achieved by showing that this invariant is equal to the Federer density. These results are a joint work with Sebastiano Don (Brescia University).
Abstract: We present an area formula in equiregular sub-Riemannian measure manifolds. The perimeter measure of a smooth bounded open set is related to the spherical measure of its boundary, using the sub-Riemannian distance. To perform the intrinsic blow-up at the boundary new difficulties appear, that also involve the nilpotent approximation of the sub-Riemannian manifold. The density of the perimeter measure naturally arises as a geometric invariant that can be explicitly related to different objects, like the nilpotent approximation, the tangent Riemannian metric and the shape of the tangent unit ball. The area formula for the perimeter measure is achieved by showing that this invariant is equal to the Federer density. These results are a joint work with Sebastiano Don (Brescia University).
Title: Brownian motion under the Bures-Wasserstein geometry
Title: Brownian motion under the Bures-Wasserstein geometry
Time: November 17, 2022 at 2:00 pm (Rome Time)
Time: November 17, 2022 at 2:00 pm (Rome Time)
Abstract: The Bures-Wasserstein geometry of positive definite matrices is closely related to the optimal transport of Gaussian measures and has several applications such as in optimization and physics. We present a detailed formula of the Brownian motion under such geometry, which has an extra mean curvature drift term and is reminiscent of well-studied stochastic processes such as Dyson's Brownian, suggesting broader framework of matrix geometry for these processes. Moreover, we investigate ideas utilizing the mean curvature drift term to design a control system.
Abstract: The Bures-Wasserstein geometry of positive definite matrices is closely related to the optimal transport of Gaussian measures and has several applications such as in optimization and physics. We present a detailed formula of the Brownian motion under such geometry, which has an extra mean curvature drift term and is reminiscent of well-studied stochastic processes such as Dyson's Brownian, suggesting broader framework of matrix geometry for these processes. Moreover, we investigate ideas utilizing the mean curvature drift term to design a control system.
Title: Attainable sets for step 2 free Carnot groups with non-negative controls and inequalities for independent random variables
Title: Attainable sets for step 2 free Carnot groups with non-negative controls and inequalities for independent random variables
Time: November 10, 2022 at 2:00 pm (Rome Time)
Time: November 10, 2022 at 2:00 pm (Rome Time)
Abstract: In recent works H.Abels and E.B.Vinberg considered free nilpotent Lie semigroups and suggested a probability interpretaion of such semigroups of step 2. With a help of an algebraic method they obtained an explicit description of the step 2 rank 3 free nilpotant Lie semigroup. This result implies some non trivial inequalities for a system of three independent random variables x, y, z. For example, if P(x < y) = 3/5 and P(y < z) = 3/5, then P(x < z) >= 1/3 (an obvious bound is 1/5).
Abstract: In recent works H.Abels and E.B.Vinberg considered free nilpotent Lie semigroups and suggested a probability interpretaion of such semigroups of step 2. With a help of an algebraic method they obtained an explicit description of the step 2 rank 3 free nilpotant Lie semigroup. This result implies some non trivial inequalities for a system of three independent random variables x, y, z. For example, if P(x < y) = 3/5 and P(y < z) = 3/5, then P(x < z) >= 1/3 (an obvious bound is 1/5).
We regard these free nilpotent Lie semigroups as attainable sets for some control systems. We describe the boundary of the attainable set with a help of first and second order optimality conditions. It turns out that the curved faces of the attainable set consist of the ends of optimal trajectories with the number of control switching corresponding to the face dimension. We give an explicit answer in the case of rank 3 and upper bounds for the number for control switchings in the case of rank 4.
We regard these free nilpotent Lie semigroups as attainable sets for some control systems. We describe the boundary of the attainable set with a help of first and second order optimality conditions. It turns out that the curved faces of the attainable set consist of the ends of optimal trajectories with the number of control switching corresponding to the face dimension. We give an explicit answer in the case of rank 3 and upper bounds for the number for control switchings in the case of rank 4.
Title: Absolute parallelism constructions for 2-nondegenerate CR hypersurfaces
Title: Absolute parallelism constructions for 2-nondegenerate CR hypersurfaces
Time: October 27, 2022 at 10:00 am (Rome Time) - room 134
Time: October 27, 2022 at 10:00 am (Rome Time) - room 134
Abstract: The talk will introduce and demonstrate through examples a Tanaka-theoretic general method (developed in joint work with Igor Zelenko) for solving local equivalence problems applicable to a broad class of 2-nondegenerate hypersurface-type CR manifolds, namely to all such structures that are uniquely determined by the geometry naturally induced on their associated Levi leaf space. We will apply the general method to an instructive family of CR hypersurfaces in complex 6-space, reducing their local equivalence problem to one of absolute parallelisms that we explicitly construct in local coordinates. The talk will also review further applications of the general method, applications to estimating symmetry group dimensions and to classifications of homogeneous structures.
Abstract: The talk will introduce and demonstrate through examples a Tanaka-theoretic general method (developed in joint work with Igor Zelenko) for solving local equivalence problems applicable to a broad class of 2-nondegenerate hypersurface-type CR manifolds, namely to all such structures that are uniquely determined by the geometry naturally induced on their associated Levi leaf space. We will apply the general method to an instructive family of CR hypersurfaces in complex 6-space, reducing their local equivalence problem to one of absolute parallelisms that we explicitly construct in local coordinates. The talk will also review further applications of the general method, applications to estimating symmetry group dimensions and to classifications of homogeneous structures.
Title: What is a para-CR structure of type (k,r,s) and why it describes geometry of ODEs and PDEs of finite type
Title: What is a para-CR structure of type (k,r,s) and why it describes geometry of ODEs and PDEs of finite type
Time: September 13, 2022 at 2:00 pm (Rome Time), in room 133 at SISSA.
Time: September 13, 2022 at 2:00 pm (Rome Time), in room 133 at SISSA.
Abstract: I will talk about a geometry of manifolds equipped with a pair of integrable vector distributions. Such geometry is suitable to describe a geometry of a large class of Partial Differential Equations (PDEs) considered modulo various types of changes of variables. This class of PDEs is called `the finite type', meaning that their space of solutions is finite dimensional. I will illustrate the para-CR structure of type (k,r,s) geometry with a few examples having applications in theoretical physics.
Abstract: I will talk about a geometry of manifolds equipped with a pair of integrable vector distributions. Such geometry is suitable to describe a geometry of a large class of Partial Differential Equations (PDEs) considered modulo various types of changes of variables. This class of PDEs is called `the finite type', meaning that their space of solutions is finite dimensional. I will illustrate the para-CR structure of type (k,r,s) geometry with a few examples having applications in theoretical physics.
Title: Motivic Vitushkin's invariants
Title: Motivic Vitushkin's invariants
Time: July 28, 2022 at 2:00 pm (Rome Time)
Time: July 28, 2022 at 2:00 pm (Rome Time)
Abstract: I will explain how, in a joint work with Immanuel Halupczok (Düsseldorf Univ.), we define in definable nonarchimedean geometry a sequence of invariants which is the counterpart in this context of the sequence of Vituskin's invariants in real geometry. For this we use the theory of t-stratification in its uniform version.
Abstract: I will explain how, in a joint work with Immanuel Halupczok (Düsseldorf Univ.), we define in definable nonarchimedean geometry a sequence of invariants which is the counterpart in this context of the sequence of Vituskin's invariants in real geometry. For this we use the theory of t-stratification in its uniform version.
We also define a notion of preorder on the ring of motivic constructible functions, which is compatible with motivic integration. As in the real case, our invariants are related to a notion of metric entropy.
We also define a notion of preorder on the ring of motivic constructible functions, which is compatible with motivic integration. As in the real case, our invariants are related to a notion of metric entropy.
Title: The large-data limit of the MBO scheme for data clustering
Title: The large-data limit of the MBO scheme for data clustering
Time: June 27, 2022 at 4:00 pm (Rome Time)
Time: June 27, 2022 at 4:00 pm (Rome Time)
Abstract: The MBO scheme is an efficient scheme used for data clustering, the task of partitioning a given dataset into several clusters. In this talk, I will present a rigorous analysis of the MBO scheme for data clustering in the large-data limit. Each iteration of the MBO scheme corresponds to one step of implicit gradient descent for the thresholding energy on the similarity graph of the dataset. For a subset of the nodes of the graph, the thresholding energy is the amount of heat transferred from the subset to its complement. It is then natural to think that outcomes of the MBO scheme are (local) minimizers of this energy. We prove that the algorithm is consistent, in the sense that these (local) minimizers converge to minimizers of a suitably weighted optimal partition problem. This is joint work with Jona Lelmi (U Bonn).
Abstract: The MBO scheme is an efficient scheme used for data clustering, the task of partitioning a given dataset into several clusters. In this talk, I will present a rigorous analysis of the MBO scheme for data clustering in the large-data limit. Each iteration of the MBO scheme corresponds to one step of implicit gradient descent for the thresholding energy on the similarity graph of the dataset. For a subset of the nodes of the graph, the thresholding energy is the amount of heat transferred from the subset to its complement. It is then natural to think that outcomes of the MBO scheme are (local) minimizers of this energy. We prove that the algorithm is consistent, in the sense that these (local) minimizers converge to minimizers of a suitably weighted optimal partition problem. This is joint work with Jona Lelmi (U Bonn).
Title: Morse inequalities for eigenvalue branches of generic families of self-adjoint matrices
Title: Morse inequalities for eigenvalue branches of generic families of self-adjoint matrices
Time: June 16, 2022 at 2:00 pm (Rome Time)
Time: June 16, 2022 at 2:00 pm (Rome Time)
Abstract: The eigenvalue branches of families of self-adjoint matrices are not smooth at points corresponding to repeated eigenvalues (called diabolic points or Dirac points). Generalizing the notion of critical points as points for which the homotopical type of (local) sub-level set changes after the passage through the corresponding value, in the case of the generic family we give an effective criterion for a diabolic point to be critical for those branches and compute the contribution of each such critical point to the Morse polynomial of each branch, getting the appropriate Morse inequalities as a byproduct of the theory. These contributions are expressed in terms of the homologies of Grassmannians. The motivation comes from the Floquet-Bloch theory of Schroedinger equations with periodic potential and other problems in Mathematical Physics. The talk is based on the joint work with Gregory Berkolaiko.
Abstract: The eigenvalue branches of families of self-adjoint matrices are not smooth at points corresponding to repeated eigenvalues (called diabolic points or Dirac points). Generalizing the notion of critical points as points for which the homotopical type of (local) sub-level set changes after the passage through the corresponding value, in the case of the generic family we give an effective criterion for a diabolic point to be critical for those branches and compute the contribution of each such critical point to the Morse polynomial of each branch, getting the appropriate Morse inequalities as a byproduct of the theory. These contributions are expressed in terms of the homologies of Grassmannians. The motivation comes from the Floquet-Bloch theory of Schroedinger equations with periodic potential and other problems in Mathematical Physics. The talk is based on the joint work with Gregory Berkolaiko.
Title: Facet Volumes of Polytopes
Title: Facet Volumes of Polytopes
Time: June 9, 2022 at 2:00 pm (Rome Time)
Time: June 9, 2022 at 2:00 pm (Rome Time)
Abstract: We consider what we call facet volume vectors of polytopes. Every full-dimensional polytope in R^d with n facets defines n positive real numbers: the n (d-1)-dimensional volumes of its facets. For instance, every triangle defines three lenghts; every tetrahedron defines four areas.
Abstract: We consider what we call facet volume vectors of polytopes. Every full-dimensional polytope in R^d with n facets defines n positive real numbers: the n (d-1)-dimensional volumes of its facets. For instance, every triangle defines three lenghts; every tetrahedron defines four areas.
We study the space of all such vectors. We show that for fixed integers d\geq 2 and n\geq d+1 the configuration space of all facet volume vectors of all d-polytopes in R^d with n facets is a full dimensional cone in R^n, and we describe this cone in terms of inequalities. For tetrahedra this is a cone over a regular octahedron.
We study the space of all such vectors. We show that for fixed integers d\geq 2 and n\geq d+1 the configuration space of all facet volume vectors of all d-polytopes in R^d with n facets is a full dimensional cone in R^n, and we describe this cone in terms of inequalities. For tetrahedra this is a cone over a regular octahedron.
Joint work with Pavle Blagojevic and Alexander Heaton.
Joint work with Pavle Blagojevic and Alexander Heaton.
Title: Smale’s 7th problem: an overview
Title: Smale’s 7th problem: an overview
Time: May 26, 2022 at 2:00 pm (Rome Time)
Time: May 26, 2022 at 2:00 pm (Rome Time)
Abstract: Smale’s 7th problem demands for an algorithm to find finite collections of points in the 2-sphere, in such a way that they minimize some energy that one may think of as the classical electrostatic potential. This beautiful problem (which is the computational version of a problem posed by J. J. Thomson, the discoverer of the electron) has attracted the attention of dozens of researchers and, although it is considered extremely difficult, the hope for solving it has not vanished. In this talk I will present the problem from a general perspective, showing its relations with other questions, mentioning the most important results obtained to the date and posing several open questions in the path to the total solution. The talk will be directed for a general audience.
Abstract: Smale’s 7th problem demands for an algorithm to find finite collections of points in the 2-sphere, in such a way that they minimize some energy that one may think of as the classical electrostatic potential. This beautiful problem (which is the computational version of a problem posed by J. J. Thomson, the discoverer of the electron) has attracted the attention of dozens of researchers and, although it is considered extremely difficult, the hope for solving it has not vanished. In this talk I will present the problem from a general perspective, showing its relations with other questions, mentioning the most important results obtained to the date and posing several open questions in the path to the total solution. The talk will be directed for a general audience.
Title: Real structures on tropical varieties
Title: Real structures on tropical varieties
Time: May 12, 2022 at 2:00 pm (Rome Time)
Time: May 12, 2022 at 2:00 pm (Rome Time)
Abstract: We will propose a definition of a real structure on a non-singular projective tropical variety. This definition takes its inspiration from the Viro's patchworking theorem. In the local setting, we will prove that such a structure on a matroidal fan is equivalent to an orientation on the underlying matroid. We will then generalize Viro's theorem to this setting. This is a joint work with Johannes Rau and Kris Shaw.
Abstract: We will propose a definition of a real structure on a non-singular projective tropical variety. This definition takes its inspiration from the Viro's patchworking theorem. In the local setting, we will prove that such a structure on a matroidal fan is equivalent to an orientation on the underlying matroid. We will then generalize Viro's theorem to this setting. This is a joint work with Johannes Rau and Kris Shaw.
Title: Equilibrium measures and logarithmic potential theory (Part 2 of 2)
Title: Equilibrium measures and logarithmic potential theory (Part 2 of 2)
Time: May 5, 2022 at 2:00 pm (Rome Time)
Time: May 5, 2022 at 2:00 pm (Rome Time)
Abstract: We introduce the probability measure associated to the zeroes of Hermite polynomials and discuss how a rescaling is necessary in order to get convergence (in the weak star topology). Using the tools from logarithmic potential theory from previous talk, we show convergence of this measure to the semi-circular law. We then introduce Gaussian Ensambles and the empirical and statistical eigenvalue distributions for hermitian matrices in these ensambles. We show how these ensambles fit in a more general framework and we discuss the generalized Wigner theorem, highlighting a parallelism with Laplace asymptotic method. From this we get as a corollary the convergence of the (rescaled) statistical eigenvalue distribution for GOE matrices to the semi-circular law.
Abstract: We introduce the probability measure associated to the zeroes of Hermite polynomials and discuss how a rescaling is necessary in order to get convergence (in the weak star topology). Using the tools from logarithmic potential theory from previous talk, we show convergence of this measure to the semi-circular law. We then introduce Gaussian Ensambles and the empirical and statistical eigenvalue distributions for hermitian matrices in these ensambles. We show how these ensambles fit in a more general framework and we discuss the generalized Wigner theorem, highlighting a parallelism with Laplace asymptotic method. From this we get as a corollary the convergence of the (rescaled) statistical eigenvalue distribution for GOE matrices to the semi-circular law.
Title: Equilibrium measures and logarithmic potential theory (Part 1 of 2)
Title: Equilibrium measures and logarithmic potential theory (Part 1 of 2)
Time: April 28, 2022 at 2:00 pm (Rome Time)
Time: April 28, 2022 at 2:00 pm (Rome Time)
Abstract: The aim of these talks is to show that the (rescaled) zeroes of Hermite polynomials and the (rescaled) eigenvalues of matrices in the Gaussian Orthogonal Ensemble share the same asymptotic distribution, i.e. the semi-circle law of radius \sqrt(2). We address this problem through logarithmic potential theory.
Abstract: The aim of these talks is to show that the (rescaled) zeroes of Hermite polynomials and the (rescaled) eigenvalues of matrices in the Gaussian Orthogonal Ensemble share the same asymptotic distribution, i.e. the semi-circle law of radius \sqrt(2). We address this problem through logarithmic potential theory.
We begin by showing how we can interpret the zeroes of orthogonal polynomials as equilibrium configurations for the electrostatic Stieltjes model on the real line, which serves as a motivation for their study. We then introduce logarithmic potential and energy of a measure with respect to an external potential. Under suitable hypothesis on such potential, we show the existence and uniqueness of a minimizing measure for the energy, which we call the equilibrium measure. A characterization of such equilibrium measures is also provided. We end the talk briefly discussing the equilibrium measure for a Gaussian potential.
We begin by showing how we can interpret the zeroes of orthogonal polynomials as equilibrium configurations for the electrostatic Stieltjes model on the real line, which serves as a motivation for their study. We then introduce logarithmic potential and energy of a measure with respect to an external potential. Under suitable hypothesis on such potential, we show the existence and uniqueness of a minimizing measure for the energy, which we call the equilibrium measure. A characterization of such equilibrium measures is also provided. We end the talk briefly discussing the equilibrium measure for a Gaussian potential.
The tools we developed here will be used in the second talk to address our starting problem.
The tools we developed here will be used in the second talk to address our starting problem.
Title: Bakry–Émery curvature and sub-Riemannian geometry
Title: Bakry–Émery curvature and sub-Riemannian geometry
Time: April 21, 2022 at 2:00 pm (Rome Time)
Time: April 21, 2022 at 2:00 pm (Rome Time)
Abstract: In this talk we discuss some generalization of comparison theorems involving Bakry Émery curvature in sub-Riemannian geometry. In particular we will focus on comparison theorems for distortion coefficients appearing in geometric interpolation inequalities, such as the Brunn-Minkovski inequality. The model spaces for comparison are variational problems coming from optimal control theory.
Abstract: In this talk we discuss some generalization of comparison theorems involving Bakry Émery curvature in sub-Riemannian geometry. In particular we will focus on comparison theorems for distortion coefficients appearing in geometric interpolation inequalities, such as the Brunn-Minkovski inequality. The model spaces for comparison are variational problems coming from optimal control theory.
Title: Minimal boundaries in non-smooth spaces with Ricci Curvature bounded below
Title: Minimal boundaries in non-smooth spaces with Ricci Curvature bounded below
Time: March 31, 2022 at 2:00 pm (Rome Time)
Time: March 31, 2022 at 2:00 pm (Rome Time)
Abstract: The goal of the seminar is to report on recent joint work with Daniele Semola. Motivated by a question of Gromov to establish a “synthetic regularity theory" for minimal surfaces in non-smooth ambient spaces, we address the question in the setting of non-smooth spaces satisfying Ricci curvature lower bounds in a synthetic sense via optimal transport. The talk is meant to be accessibile also to non specialists.
Abstract: The goal of the seminar is to report on recent joint work with Daniele Semola. Motivated by a question of Gromov to establish a “synthetic regularity theory" for minimal surfaces in non-smooth ambient spaces, we address the question in the setting of non-smooth spaces satisfying Ricci curvature lower bounds in a synthetic sense via optimal transport. The talk is meant to be accessibile also to non specialists.
Title: Spectral stability, spectral flow and circular relative equilibria for the Newtonian n-body problem
Title: Spectral stability, spectral flow and circular relative equilibria for the Newtonian n-body problem
Time: March 24, 2022 at 2:00 pm (Rome Time)
Time: March 24, 2022 at 2:00 pm (Rome Time)
Abstract: For the Newtonian (gravitational) $n$-body problem in the Euclidean $d$-dimensional space, $d\ge 2$, the simplest possible periodic solutions are provided by circular relative equilibria (RE), namely solutions in which each body rigidly rotates about the center of mass and the configuration of the whole system is constant in time and central configuration. A classical problem in celestial mechanics aims at relating the (in-)stability properties of a (RE) to the index properties of the central configuration generating it.
Abstract: For the Newtonian (gravitational) $n$-body problem in the Euclidean $d$-dimensional space, $d\ge 2$, the simplest possible periodic solutions are provided by circular relative equilibria (RE), namely solutions in which each body rigidly rotates about the center of mass and the configuration of the whole system is constant in time and central configuration. A classical problem in celestial mechanics aims at relating the (in-)stability properties of a (RE) to the index properties of the central configuration generating it.
In this talk, we discuss some sufficient conditions that imply the spectral instability of planar and non-planar (RE) generated by a central configuration.
In this talk, we discuss some sufficient conditions that imply the spectral instability of planar and non-planar (RE) generated by a central configuration.
The key ingredients are a new formula that allows to compute the spectral flow of a path of symmetric matrices having degenerate starting point, and a symplectic decomposition of the phase space of the linearized Hamiltonian system along a given (RE) which allows us to rule out the uninteresting part of the dynamics corresponding to the translational and (partially) to the rotational symmetry of the problem.
The key ingredients are a new formula that allows to compute the spectral flow of a path of symmetric matrices having degenerate starting point, and a symplectic decomposition of the phase space of the linearized Hamiltonian system along a given (RE) which allows us to rule out the uninteresting part of the dynamics corresponding to the translational and (partially) to the rotational symmetry of the problem.
This talk is based on a recent joint work with Prof. Dr. Luca Asselle (Ruhr Universit\"at Bochum, Germany) and Prof. Dr. Li Wu (Shandong University, Jinan, China).
This talk is based on a recent joint work with Prof. Dr. Luca Asselle (Ruhr Universit\"at Bochum, Germany) and Prof. Dr. Li Wu (Shandong University, Jinan, China).
Title: An h-principle for complements of discriminants
Title: An h-principle for complements of discriminants
Time: March 17, 2022 at 2:00 pm (Rome Time)
Time: March 17, 2022 at 2:00 pm (Rome Time)
Abstract: In classical algebraic geometry, discriminants appear naturally in various moduli spaces as the loci parametrising degenerate objects. The motivating example for this talk is the locus of singular sections of a line bundle on a smooth projective complex variety, the complement of which is a moduli space of smooth hypersurfaces.
Abstract: In classical algebraic geometry, discriminants appear naturally in various moduli spaces as the loci parametrising degenerate objects. The motivating example for this talk is the locus of singular sections of a line bundle on a smooth projective complex variety, the complement of which is a moduli space of smooth hypersurfaces.
I will present an approach to studying the homology of such moduli spaces of non-singular algebraic sections via algebro-topological tools. The main idea is to prove an "h-principle" which translates the problem into a purely homotopical one.
I will present an approach to studying the homology of such moduli spaces of non-singular algebraic sections via algebro-topological tools. The main idea is to prove an "h-principle" which translates the problem into a purely homotopical one.
I shall explain how to talk effectively about singular sections of vector bundles and what an h-principle is. To demonstrate the usefulness of homotopical methods, and using a bit of rational homotopy theory, we will prove together a homological stability result for moduli spaces of smooth hypersurfaces of increasing degree.
I shall explain how to talk effectively about singular sections of vector bundles and what an h-principle is. To demonstrate the usefulness of homotopical methods, and using a bit of rational homotopy theory, we will prove together a homological stability result for moduli spaces of smooth hypersurfaces of increasing degree.
Title: Kato limit spaces
Title: Kato limit spaces
Time: February 24, 2022 at 2:00 pm (Rome Time)
Time: February 24, 2022 at 2:00 pm (Rome Time)
Abstract: Consider a sequence of Riemannian manifolds. Assume that this sequence converges, in the measured Gromov-Hausdorff sense, to a possibly non-smooth metric measure space. What are the properties of this limit space? In a series of celebrated works from the nineties, Cheeger and Colding addressed this question under the assumption of a uniform lower bound on the Ricci curvature of the manifolds. This has led to the fruitful development of a synthetic theory of Ricci curvature lower bounds. In this talk, I will present a couple of joint works with Gilles Carron (Nantes Université) and Ilaria Mondello (Université de Créteil) where we relax the uniform Ricci lower bound assumption and work in the context of a weaker uniform Kato-type assumption, namely that the part of the lowest eigenvalue of the Ricci tensor lying under a certain threshold belongs to a given Kato class. Under this assumption which authorizes the Ricci curvature to degenerate to - infinity but in a « heat-kernel controlled » way, we show that most results of Cheeger and Colding are still true, including rectifiability on which I shall focus.
Abstract: Consider a sequence of Riemannian manifolds. Assume that this sequence converges, in the measured Gromov-Hausdorff sense, to a possibly non-smooth metric measure space. What are the properties of this limit space? In a series of celebrated works from the nineties, Cheeger and Colding addressed this question under the assumption of a uniform lower bound on the Ricci curvature of the manifolds. This has led to the fruitful development of a synthetic theory of Ricci curvature lower bounds. In this talk, I will present a couple of joint works with Gilles Carron (Nantes Université) and Ilaria Mondello (Université de Créteil) where we relax the uniform Ricci lower bound assumption and work in the context of a weaker uniform Kato-type assumption, namely that the part of the lowest eigenvalue of the Ricci tensor lying under a certain threshold belongs to a given Kato class. Under this assumption which authorizes the Ricci curvature to degenerate to - infinity but in a « heat-kernel controlled » way, we show that most results of Cheeger and Colding are still true, including rectifiability on which I shall focus.
Title: The entropy Morse-Sard Theorem.
Title: The entropy Morse-Sard Theorem.
Time:* December 16, 2:00pm (Rome Time)
Time:* December 16, 2:00pm (Rome Time)
Abstract: In these series of three seminars, we will present a proof of the classical Morse-Sard Theorem, based on results from semialgebraic geometry. It is a bit long, but it also gives a bound on the so-called entropy dimension of the set of critical values of a smooth function defined on a closed ball of R^n. This proof is due to Yomdin and Comte.
Abstract: In these series of three seminars, we will present a proof of the classical Morse-Sard Theorem, based on results from semialgebraic geometry. It is a bit long, but it also gives a bound on the so-called entropy dimension of the set of critical values of a smooth function defined on a closed ball of R^n. This proof is due to Yomdin and Comte.
*This consists of a series of three lectures: November 25; December 2; December 16.
*This consists of a series of three lectures: November 25; December 2; December 16.
Please note that the lecture initially scheduled on December 9 is rescheduled on December 16.
Please note that the lecture initially scheduled on December 9 is rescheduled on December 16.
Title: The entropy Morse-Sard Theorem.
Title: The entropy Morse-Sard Theorem.
Time:* December 2, 2:00pm (Rome Time)
Time:* December 2, 2:00pm (Rome Time)
Abstract: In these series of three seminars, we will present a proof of the classical Morse-Sard Theorem, based on results from semialgebraic geometry. It is a bit long, but it also gives a bound on the so-called entropy dimension of the set of critical values of a smooth function defined on a closed ball of R^n. This proof is due to Yomdin and Comte.
Abstract: In these series of three seminars, we will present a proof of the classical Morse-Sard Theorem, based on results from semialgebraic geometry. It is a bit long, but it also gives a bound on the so-called entropy dimension of the set of critical values of a smooth function defined on a closed ball of R^n. This proof is due to Yomdin and Comte.
*This consists of a series of three lectures: November 25; December 2; December 9.
*This consists of a series of three lectures: November 25; December 2; December 9.
Title: The entropy Morse-Sard Theorem.
Title: The entropy Morse-Sard Theorem.
Time:* November 25, 2:00pm (Rome Time)
Time:* November 25, 2:00pm (Rome Time)
Abstract: In these series of three seminars, we will present a proof of the classical Morse-Sard Theorem, based on results from semialgebraic geometry. It is a bit long, but it also gives a bound on the so-called entropy dimension of the set of critical values of a smooth function defined on a closed ball of R^n. This proof is due to Yomdin and Comte.
Abstract: In these series of three seminars, we will present a proof of the classical Morse-Sard Theorem, based on results from semialgebraic geometry. It is a bit long, but it also gives a bound on the so-called entropy dimension of the set of critical values of a smooth function defined on a closed ball of R^n. This proof is due to Yomdin and Comte.
*This consists of a series of three lectures: November 25; December 2; December 9.
*This consists of a series of three lectures: November 25; December 2; December 9.
Title: Local fields: Gaussian measures and random processes.
Title: Local fields: Gaussian measures and random processes.
Time: November 11, 2:00pm (Rome Time)
Time: November 11, 2:00pm (Rome Time)
Abstract: Gaussian measures on Banach spaces over local fields can be defined and constructed by exploiting the orthogonality structures of such spaces. We discuss these constructions and their merit by exhibiting the interesting properties of the objects they produce. Since these probabilistic objects also have a rich algebraic structure, interesting questions and problems arise.
Abstract: Gaussian measures on Banach spaces over local fields can be defined and constructed by exploiting the orthogonality structures of such spaces. We discuss these constructions and their merit by exhibiting the interesting properties of the objects they produce. Since these probabilistic objects also have a rich algebraic structure, interesting questions and problems arise.
Title: Infinite dimensional grassmannians, quantum states and optimal transport
Title: Infinite dimensional grassmannians, quantum states and optimal transport
Time: July 6, 2021, 2:00 pm (Rome Time) ---------- Please note the exceptional change of schedule. This is a joint seminar between "Analysis and Mathematical Physics" and "Geometric Structures" seminars.
Time: July 6, 2021, 2:00 pm (Rome Time) ---------- Please note the exceptional change of schedule. This is a joint seminar between "Analysis and Mathematical Physics" and "Geometric Structures" seminars.
Abstract: In this seminar we report on a recent work in collaboration with F. Cavalletti where we develop the basic theory of optimal transport for the quantum states of the C*-algebra of the compact operators on a (separable) Hilbert space.
Abstract: In this seminar we report on a recent work in collaboration with F. Cavalletti where we develop the basic theory of optimal transport for the quantum states of the C*-algebra of the compact operators on a (separable) Hilbert space.
As usual, states are interpreted as the noncommutative replacement of probability measures; via the spectral theorem applied to their density matrices, we associate to states discrete measures on the grassmannian of the finite dimensional subspaces. In this way we can treat them as ordinary probability measures and develop the theory of optimal transport.
As usual, states are interpreted as the noncommutative replacement of probability measures; via the spectral theorem applied to their density matrices, we associate to states discrete measures on the grassmannian of the finite dimensional subspaces. In this way we can treat them as ordinary probability measures and develop the theory of optimal transport.
The metric geometry of the grassmannian, as an infinite dimensional manifold plays a decisive role and part of the talk will be devoted to describing its rich structure. Notably the grassmannian is an Alexandrov space with non negative curvature.
The metric geometry of the grassmannian, as an infinite dimensional manifold plays a decisive role and part of the talk will be devoted to describing its rich structure. Notably the grassmannian is an Alexandrov space with non negative curvature.
Finally we will interpret pure normal states of the tensor product $H\otimes H$ as families of transport maps. This idea leads to the possibility of giving a definition of the Wasserstein cost for such objects.
Finally we will interpret pure normal states of the tensor product $H\otimes H$ as families of transport maps. This idea leads to the possibility of giving a definition of the Wasserstein cost for such objects.
Title: Estimating high order derivatives of a function through geometry and topology of its zero set
Title: Estimating high order derivatives of a function through geometry and topology of its zero set
Time: June 29, 2021, 4:15 pm (Rome Time)
Time: June 29, 2021, 4:15 pm (Rome Time)
Abstract: An order d rigidity inequality for a smooth function f is an explicit lower bound for the (d+1)-st derivatives of f, which holds, if f exhibits certain patterns, forbidden for polynomials of degree d.
Abstract: An order d rigidity inequality for a smooth function f is an explicit lower bound for the (d+1)-st derivatives of f, which holds, if f exhibits certain patterns, forbidden for polynomials of degree d.
We discuss some recent results in this direction, which use as an input the ``density'' of the zero set Z of f, or, in contrast, its topology. In particular, we interpret in terms of rigidity inequalities some recent results of Lerario and Stecconi, comparing topology of smooth transversal singularities, and of their polynomial approximations.
We discuss some recent results in this direction, which use as an input the ``density'' of the zero set Z of f, or, in contrast, its topology. In particular, we interpret in terms of rigidity inequalities some recent results of Lerario and Stecconi, comparing topology of smooth transversal singularities, and of their polynomial approximations.
Title: Minimal entropy of geometric 3-manifolds
Title: Minimal entropy of geometric 3-manifolds
Time: June 22, 2021, 4:15 pm (Rome Time)
Time: June 22, 2021, 4:15 pm (Rome Time)
Abstract: For a closed smooth manifold M it's natural to ask whether there exists a Riemannian metric which has minimal topological entropy. In this seminar we will investigate this question in dimension 3. First we will recall the results of A.Katok in dimension 2, then we will describe Anderson-Paternain results: which geometric 3-manifolds have metrics of minimal entropy? It turns out the answer is positive except when M is modelled on H^2*R, Sol or the universal cover of PSL(2,R), A crucial point is that only manifolds modelled on H^3 have positive minimal entropy which reveals in fact a chaotic aspect of negative curvature!
Abstract: For a closed smooth manifold M it's natural to ask whether there exists a Riemannian metric which has minimal topological entropy. In this seminar we will investigate this question in dimension 3. First we will recall the results of A.Katok in dimension 2, then we will describe Anderson-Paternain results: which geometric 3-manifolds have metrics of minimal entropy? It turns out the answer is positive except when M is modelled on H^2*R, Sol or the universal cover of PSL(2,R), A crucial point is that only manifolds modelled on H^3 have positive minimal entropy which reveals in fact a chaotic aspect of negative curvature!
Title: Computing efficiently the non-properness set of polynomial maps on the plane
Title: Computing efficiently the non-properness set of polynomial maps on the plane
Time: June 15, 2021, 4:15 pm (Rome Time)
Time: June 15, 2021, 4:15 pm (Rome Time)
Abstract: I will present new mathematical and computational tools to develop a complete and efficient algorithm for computing the set of non-properness of polynomial maps in the complex (and real) plane. In particular, this is a subset of the plane where a dominant polynomial map as above is not proper. The algorithm takes into account the sparsity of polynomials, and the genericness of the coefficients as it depends on their Newton polytopes. As a byproduct it provides a finer representation of the set of non-properness as a union of algebraic or semi-algebraic sets, that correspond to edges of the Newton polytopes, which is of independent interest. This is a joint work with Elias Tsigaridas.
Abstract: I will present new mathematical and computational tools to develop a complete and efficient algorithm for computing the set of non-properness of polynomial maps in the complex (and real) plane. In particular, this is a subset of the plane where a dominant polynomial map as above is not proper. The algorithm takes into account the sparsity of polynomials, and the genericness of the coefficients as it depends on their Newton polytopes. As a byproduct it provides a finer representation of the set of non-properness as a union of algebraic or semi-algebraic sets, that correspond to edges of the Newton polytopes, which is of independent interest. This is a joint work with Elias Tsigaridas.
Title: Configurations spaces of particles: homological solid, liquid, and gas
Title: Configurations spaces of particles: homological solid, liquid, and gas
Time: June 8, 2021, 4:15 pm (Rome Time)
Time: June 8, 2021, 4:15 pm (Rome Time)
Abstract: Configuration spaces of points in the plane are well studied and the topology of such spaces is well understood. But what if you replace points by particles with some positive thickness, and put them in a container with boundaries? It seems like not much is known. To mathematicians, this is a natural generalization of the configuration space of points, perhaps interesting for its own sake. But is also important from the point of view of physics––physicists might call such a space the "phase space" or "energy landscape" for a hard-spheres system. Since hard-spheres systems are observed experimentally to undergo phase transitions (analogous to water changing into ice), it would be quite interesting to understand topological underpinnings of such transitions.
Abstract: Configuration spaces of points in the plane are well studied and the topology of such spaces is well understood. But what if you replace points by particles with some positive thickness, and put them in a container with boundaries? It seems like not much is known. To mathematicians, this is a natural generalization of the configuration space of points, perhaps interesting for its own sake. But is also important from the point of view of physics––physicists might call such a space the "phase space" or "energy landscape" for a hard-spheres system. Since hard-spheres systems are observed experimentally to undergo phase transitions (analogous to water changing into ice), it would be quite interesting to understand topological underpinnings of such transitions.
We have just started to understand the homology of these configuration spaces, and based on our results so far we suggest working definitions of "homological solid, liquid, and gas". This is joint work with a number of collaborators, including Hannah Alpert, Ulrich Bauer, Kelly Spendlove, and Robert MacPherson.
We have just started to understand the homology of these configuration spaces, and based on our results so far we suggest working definitions of "homological solid, liquid, and gas". This is joint work with a number of collaborators, including Hannah Alpert, Ulrich Bauer, Kelly Spendlove, and Robert MacPherson.
Title: The Witten deformation on singular spaces
Title: The Witten deformation on singular spaces
Time: June 1st, 2021, 4:15 pm (Rome Time)
Time: June 1st, 2021, 4:15 pm (Rome Time)
Abstract: In his seminal paper “Supersymmetry and Morse theory” (Journal Diff. Geom. 1982) Witten, inspired by ideas from quantum field theory, gave a new analytic proof of the famous Morse inequalities. The Witten deformation plays an important role in the generalisation by Bismut and Zhang of the comparison theorem between analytic and topological torsion of a smooth compact manifold, aka Cheeger-Mu ̈ller theorem.
Abstract: In his seminal paper “Supersymmetry and Morse theory” (Journal Diff. Geom. 1982) Witten, inspired by ideas from quantum field theory, gave a new analytic proof of the famous Morse inequalities. The Witten deformation plays an important role in the generalisation by Bismut and Zhang of the comparison theorem between analytic and topological torsion of a smooth compact manifold, aka Cheeger-Mu ̈ller theorem.
The aim of this talk is to explain the generalisation of the Witten deformation to certain singular spaces. We will explain the case of singular spaces with conical singularities equipped with a radial Morse function as well as the case of singular algebraic complex curves equipped with a stratified Morse function in the sense of Goresky and MacPherson. A first result in both situations is the proof of the Morse inequalities for the L2-cohomology (or equivalently the intersection cohomology). A much stronger result is the generalisation of the comparison between the so called Witten complex and an appropriate singular Morse-Thom-Smale complex.
The aim of this talk is to explain the generalisation of the Witten deformation to certain singular spaces. We will explain the case of singular spaces with conical singularities equipped with a radial Morse function as well as the case of singular algebraic complex curves equipped with a stratified Morse function in the sense of Goresky and MacPherson. A first result in both situations is the proof of the Morse inequalities for the L2-cohomology (or equivalently the intersection cohomology). A much stronger result is the generalisation of the comparison between the so called Witten complex and an appropriate singular Morse-Thom-Smale complex.
In the first part of this talk, I will give a gentle introduction to the Witten deformation for a smooth compact manifold.
In the first part of this talk, I will give a gentle introduction to the Witten deformation for a smooth compact manifold.
Title: Asymptotic topology of random excursion sets
Title: Asymptotic topology of random excursion sets
Time: May 25, 2021, 4:15 pm (Rome Time)
Time: May 25, 2021, 4:15 pm (Rome Time)
Abstract: Let f be a smooth random Gaussian field over the unit ball of R^n. It is very natural
Abstract: Let f be a smooth random Gaussian field over the unit ball of R^n. It is very natural
to imagine that for a high level u, {f>u} is mainly composed of small components homeomorphic to n-balls. I will explain that in average, this intuition is true. After recalling the historical background of this subject, and will present the ideas of the proof, which holds on (deterministic) Morse theory and a control of random critical points of given index.
to imagine that for a high level u, {f>u} is mainly composed of small components homeomorphic to n-balls. I will explain that in average, this intuition is true. After recalling the historical background of this subject, and will present the ideas of the proof, which holds on (deterministic) Morse theory and a control of random critical points of given index.
Title: Anti-concentration and the geometry of polynomials
Title: Anti-concentration and the geometry of polynomials
Time: May 11, 2021, 4:15 pm (Rome Time)
Time: May 11, 2021, 4:15 pm (Rome Time)
Abstract: Let X be a random variable taking values in {0,...,n} with standard deviation sigma and let f_X be its probability generating function. Pemantle conjectured that if sigma is large and f_X has no roots close to 1 in the complex plane then X must approximate a normal distribution. In this talk, I will discuss the resolution of Pemantle's conjecture and its application to prove a conjecture of Ghosh, Liggett and Pemantle by proving a multivariate central limit theorem for, so called, strong Rayleigh distributions. I will also touch on some more recent work connecting anti-concentration for random variables with the zeros of their probability generating functions.
Abstract: Let X be a random variable taking values in {0,...,n} with standard deviation sigma and let f_X be its probability generating function. Pemantle conjectured that if sigma is large and f_X has no roots close to 1 in the complex plane then X must approximate a normal distribution. In this talk, I will discuss the resolution of Pemantle's conjecture and its application to prove a conjecture of Ghosh, Liggett and Pemantle by proving a multivariate central limit theorem for, so called, strong Rayleigh distributions. I will also touch on some more recent work connecting anti-concentration for random variables with the zeros of their probability generating functions.
This talk is based on joint work with Marcus Michelen.
This talk is based on joint work with Marcus Michelen.
Title: Semicontinuity of Betti numbers: A little surgery cannot kill homology
Title: Semicontinuity of Betti numbers: A little surgery cannot kill homology
Time: May 4, 2021, 4:15 pm (Rome Time)
Time: May 4, 2021, 4:15 pm (Rome Time)
Abstract: A consequence of Thom Isotopy Lemma is that the set of solutions of a regular smooth equation is stable under C^1-small perturbations (it remains isotopic to the original one), but what happens if the perturbation is just C^0-small? In this case, the topology of the set of solution may change. However, it turns out that the Homology groups cannot "decrease". In this talk I will present such result and some related examples and applications. This theorem is useful in those contexts where the price to pay to approximate something in C^1 is higher than in C^0. For instance in the search for quantitative bounds (here the price can be the degree of an algebraic approximation) or in combination with Eliashberg's and Mishachev's holonomic approximation Theorem (which is C^0 at most).
Abstract: A consequence of Thom Isotopy Lemma is that the set of solutions of a regular smooth equation is stable under C^1-small perturbations (it remains isotopic to the original one), but what happens if the perturbation is just C^0-small? In this case, the topology of the set of solution may change. However, it turns out that the Homology groups cannot "decrease". In this talk I will present such result and some related examples and applications. This theorem is useful in those contexts where the price to pay to approximate something in C^1 is higher than in C^0. For instance in the search for quantitative bounds (here the price can be the degree of an algebraic approximation) or in combination with Eliashberg's and Mishachev's holonomic approximation Theorem (which is C^0 at most).
Title: Infinite-dimensional geometry with symmetry
Title: Infinite-dimensional geometry with symmetry
Time: April 27, 2021, 4:30 pm (Rome Time)
Time: April 27, 2021, 4:30 pm (Rome Time)
Abstract: Most theorems in finite-dimensional algebraic geometry break down in infinite dimensions---for instance, the polynomial ring C[x_1,x_2,...] is not Noetherian. However, it turns out that some results do survive when a sufficiently large symmetry group is imposed; e.g., ideals in C[x_1,x_2,...] that are preserved under all variable permutations do satisfy the ascending chain condition.
Abstract: Most theorems in finite-dimensional algebraic geometry break down in infinite dimensions---for instance, the polynomial ring C[x_1,x_2,...] is not Noetherian. However, it turns out that some results do survive when a sufficiently large symmetry group is imposed; e.g., ideals in C[x_1,x_2,...] that are preserved under all variable permutations do satisfy the ascending chain condition.
This phenomenon is relevant in pure and applied mathematics, since many algebraic models come in infinite families with highly symmetric infinite-dimensional limits. Here the symmetry is typically captured by either the infinite symmetric group or the infinite general linear group. Theorems about the limit imply uniform behaviour of the members of the family.
This phenomenon is relevant in pure and applied mathematics, since many algebraic models come in infinite families with highly symmetric infinite-dimensional limits. Here the symmetry is typically captured by either the infinite symmetric group or the infinite general linear group. Theorems about the limit imply uniform behaviour of the members of the family.
I will present older and new results in this area, along with applications to algebraic statistics, tensor decomposition, and algebraic combinatorics.
I will present older and new results in this area, along with applications to algebraic statistics, tensor decomposition, and algebraic combinatorics.
Title: The critical curvature degree of an algebraic variety
Title: The critical curvature degree of an algebraic variety
Time: April 20, 2021, 4:15 pm (Rome Time)
Time: April 20, 2021, 4:15 pm (Rome Time)
Abstract: This topic is about the complexity involved in the computation of the reach in arbitrary dimensions and in particular the computation of the critical spherical curvature points of an arbitrary algebraic variety. We present properties of the critical spherical curvature points as well as an algorithm for computing them.
Abstract: This topic is about the complexity involved in the computation of the reach in arbitrary dimensions and in particular the computation of the critical spherical curvature points of an arbitrary algebraic variety. We present properties of the critical spherical curvature points as well as an algorithm for computing them.
Title: Signature tensors of paths
Title: Signature tensors of paths
Time: April 13, 2021, 4:15 pm (Rome Time)
Time: April 13, 2021, 4:15 pm (Rome Time)
Abstract: I'm interested in connections between algebraic geometry and other branches of math. In stochastic analysis, a standard method to study a path is to work with its signature. This is a sequence of tensors that encode information of the path in a compact form. When the path varies, such signatures parametrize an algebraic variety in the tensor space. My goal is to study the geometry of such varieties and to link it to properties of certain classes of paths.
Abstract: I'm interested in connections between algebraic geometry and other branches of math. In stochastic analysis, a standard method to study a path is to work with its signature. This is a sequence of tensors that encode information of the path in a compact form. When the path varies, such signatures parametrize an algebraic variety in the tensor space. My goal is to study the geometry of such varieties and to link it to properties of certain classes of paths.
Title: Expressive geometry
Title: Expressive geometry
Time: April 6, 2021, 4:15 pm (Rome Time)
Time: April 6, 2021, 4:15 pm (Rome Time)
Abstract: We review two so-called expressive models, morsifications of real plane curve singularities introduced in 70s by A'Campo and Gusein-Zade, and real affine expressive curves. These models are characterized by the property that their underlining polynomial has the smallest number of critical points allowed by the topology of the real point set. The classification of these objects is tightly related to the mutational equivalence of the corresponding quivers (which in turn naturally appear in the theory of cluster algebras). We discuss various problems in the geometry of morsifications and expressive curves, including related objects like planar divides, links of singularities and links of curves at infinity, combinatorics of quivers. Based on joint works with S. Fomin, P. Pylyavskyy, D. Thurston.
Abstract: We review two so-called expressive models, morsifications of real plane curve singularities introduced in 70s by A'Campo and Gusein-Zade, and real affine expressive curves. These models are characterized by the property that their underlining polynomial has the smallest number of critical points allowed by the topology of the real point set. The classification of these objects is tightly related to the mutational equivalence of the corresponding quivers (which in turn naturally appear in the theory of cluster algebras). We discuss various problems in the geometry of morsifications and expressive curves, including related objects like planar divides, links of singularities and links of curves at infinity, combinatorics of quivers. Based on joint works with S. Fomin, P. Pylyavskyy, D. Thurston.
Title: Poincaré-Reeb trees of real Milnor fibres
Title: Poincaré-Reeb trees of real Milnor fibres
Time: March 30, 2021, 4:15 pm (Rome Time)
Time: March 30, 2021, 4:15 pm (Rome Time)
Abstract: We study the real Milnor fibre of real bivariate polynomial functions vanishing at the origin, with an isolated local minimum at this point. We work in a neighbourhood of the origin in which its non-zero level sets are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries of convex disks. Otherwise, they may fail to be convex, as was shown by Coste.
Abstract: We study the real Milnor fibre of real bivariate polynomial functions vanishing at the origin, with an isolated local minimum at this point. We work in a neighbourhood of the origin in which its non-zero level sets are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries of convex disks. Otherwise, they may fail to be convex, as was shown by Coste.
In order to measure the non-convexity of the level curves, we introduce a new combinatorial object, called the Poincaré-Reeb tree, and show that locally the shape stabilises and that no spiralling phenomena occur near the origin. Our main objective is to characterise all topological types of asymptotic Poincaré-Reeb trees. To this end, we construct a family of polynomials with non-Morse strict local minimum at the origin, realising a large class of such trees.
In order to measure the non-convexity of the level curves, we introduce a new combinatorial object, called the Poincaré-Reeb tree, and show that locally the shape stabilises and that no spiralling phenomena occur near the origin. Our main objective is to characterise all topological types of asymptotic Poincaré-Reeb trees. To this end, we construct a family of polynomials with non-Morse strict local minimum at the origin, realising a large class of such trees.
As a preliminary step, we reduce the problem to the univariate case, via the interplay between the polar curve and its discriminant. Here we give a new and constructive proof of the existence of Morse polynomials whose associated permutation (the so-called “Arnold snake”) is separable, using tools inspired from Ghys’s work.
As a preliminary step, we reduce the problem to the univariate case, via the interplay between the polar curve and its discriminant. Here we give a new and constructive proof of the existence of Morse polynomials whose associated permutation (the so-called “Arnold snake”) is separable, using tools inspired from Ghys’s work.
Title: Concentration of Measure in Integral Geometry
Title: Concentration of Measure in Integral Geometry
Time: March 23, 2021, 4:15 pm (Rome Time)
Time: March 23, 2021, 4:15 pm (Rome Time)
Abstract: Intrinsic volumes are fundamental geometric invariants that include the Euler characteristic and the volume. Important results in integral geometry relate the intrinsic volumes of random projections, intersections, and sums of convex bodies to those of the individual volumes. We present a new interpretations of classic results, based on the observation that intrinsic volumes (both in spherical and Euclidean settings) concentrate around certain indices. One consequence is, for example, that as the dimension of a subspace varies, the average intrinsic volume polynomial of a random projection of a convex body to this subspace is as large as possible or is negligible, and the exact location of the transition between these two cases can be expressed in terms of a summary parameter associated with the convex body. Similar phase transitions appear in related problems, including the rotation mean formula, the slicing (Crofton) formula, and the kinematic formula. This is joint work with Joel Tropp.
Abstract: Intrinsic volumes are fundamental geometric invariants that include the Euler characteristic and the volume. Important results in integral geometry relate the intrinsic volumes of random projections, intersections, and sums of convex bodies to those of the individual volumes. We present a new interpretations of classic results, based on the observation that intrinsic volumes (both in spherical and Euclidean settings) concentrate around certain indices. One consequence is, for example, that as the dimension of a subspace varies, the average intrinsic volume polynomial of a random projection of a convex body to this subspace is as large as possible or is negligible, and the exact location of the transition between these two cases can be expressed in terms of a summary parameter associated with the convex body. Similar phase transitions appear in related problems, including the rotation mean formula, the slicing (Crofton) formula, and the kinematic formula. This is joint work with Joel Tropp.
Title: The average condition number of different problems, from a geometric perspective
Title: The average condition number of different problems, from a geometric perspective
Time: March 16, 2021, 4:15 pm (Rome Time)
Time: March 16, 2021, 4:15 pm (Rome Time)
Abstract: I will present the condition number of problems from a general perspective as a measure of the stability of problems. Then I will discuss how does this condition number look and interact with other mathematical concepts in different problems which are very basic but are still full of mysteries: polynomial solving, eigenvalue/eigenvector problems or tensor decomposition problems. All the material will be presented for a general audience. Different parts of what will be presented has been done with different authors, for example the tensor decomposition part has been done with Paul Breiding and Nick Vannieuwenhoven. Check the following video for our result in 2 minutes:
Abstract: I will present the condition number of problems from a general perspective as a measure of the stability of problems. Then I will discuss how does this condition number look and interact with other mathematical concepts in different problems which are very basic but are still full of mysteries: polynomial solving, eigenvalue/eigenvector problems or tensor decomposition problems. All the material will be presented for a general audience. Different parts of what will be presented has been done with different authors, for example the tensor decomposition part has been done with Paul Breiding and Nick Vannieuwenhoven. Check the following video for our result in 2 minutes:
Title: On minimality of determinantal varieties
Title: On minimality of determinantal varieties
Time: March 9, 2021, 4:15 pm (Rome Time)
Time: March 9, 2021, 4:15 pm (Rome Time)
Abstract: Minimal submanifolds are mathematical abstractions of soap films: they minimize the Riemannian volume locally around every point. Finding minimal algebraic hypersurfaces in 𝑅𝑛 for each n is a long-standing open problem posed by Hsiang. In 2010 Tkachev gave a partial solution to this problem showing that the hypersurface of n x n real matrices of corank one is minimal. I will discuss the following generalization of this fact to all determinantal matrix varieties: for any m, n and r<m,n the (open) variety of m x n real matrices of rank r is minimal. More generally, I will show that real tensors of fixed multi-linear rank form a minimal submanifold in the Euclidean space of all tensors of a given format. The talk is partially based on a joint work with A. Heaton and L. Venturello.
Abstract: Minimal submanifolds are mathematical abstractions of soap films: they minimize the Riemannian volume locally around every point. Finding minimal algebraic hypersurfaces in 𝑅𝑛 for each n is a long-standing open problem posed by Hsiang. In 2010 Tkachev gave a partial solution to this problem showing that the hypersurface of n x n real matrices of corank one is minimal. I will discuss the following generalization of this fact to all determinantal matrix varieties: for any m, n and r<m,n the (open) variety of m x n real matrices of rank r is minimal. More generally, I will show that real tensors of fixed multi-linear rank form a minimal submanifold in the Euclidean space of all tensors of a given format. The talk is partially based on a joint work with A. Heaton and L. Venturello.
Title: Cohomology rings of real flag manifolds
Title: Cohomology rings of real flag manifolds
Time: March 2, 2021, 4:15 pm (Rome Time)
Time: March 2, 2021, 4:15 pm (Rome Time)
Abstract: The cohomology ring of a complex (partial) flag manifold has two classical descriptions; a topological one (via characteristic classes) and a geometric one (via Schubert classes). Similar descriptions are well-known for real flag manifolds X with mod 2 coefficients. In this talk I will discuss some aspects of what can be said with rational, or integer coefficients. Namely, I will consider questions of the following type:
Abstract: The cohomology ring of a complex (partial) flag manifold has two classical descriptions; a topological one (via characteristic classes) and a geometric one (via Schubert classes). Similar descriptions are well-known for real flag manifolds X with mod 2 coefficients. In this talk I will discuss some aspects of what can be said with rational, or integer coefficients. Namely, I will consider questions of the following type:
1) Which Schubert varieties represent an integer cohomology class?
1) Which Schubert varieties represent an integer cohomology class?
2) What are their structure constants?
2) What are their structure constants?
3) What can be said about torsion in H^*(X;\Z)?
3) What can be said about torsion in H^*(X;\Z)?
I will also discuss some applications of the ring structure to real Schubert calculus.
I will also discuss some applications of the ring structure to real Schubert calculus.
Title: The (Symplectic) Geometry of Spaces of Frames
Title: The (Symplectic) Geometry of Spaces of Frames
Time: February 23, 2021, 4:15 pm (Rome Time)
Time: February 23, 2021, 4:15 pm (Rome Time)
Abstract: A frame for a finite-dimensional complex vector space is simply a spanning set. Frames include bases, but also spanning sets that are larger than a basis, and hence more suitable for applications where robustness to noise and erasures are important. In applications, frames are sometimes also called overcomplete dictionaries.
Abstract: A frame for a finite-dimensional complex vector space is simply a spanning set. Frames include bases, but also spanning sets that are larger than a basis, and hence more suitable for applications where robustness to noise and erasures are important. In applications, frames are sometimes also called overcomplete dictionaries.
In applications we typically want to impose some structure on our frame, for example by requiring each frame vector to be unit-length or by requiring that the frame satisfies the Parseval identity. While frames satisfying these and similar constraints form (real) algebraic varieties, it can be challenging to extract useful information from this perspective.
In applications we typically want to impose some structure on our frame, for example by requiring each frame vector to be unit-length or by requiring that the frame satisfies the Parseval identity. While frames satisfying these and similar constraints form (real) algebraic varieties, it can be challenging to extract useful information from this perspective.
My goal in this talk is to introduce a symplectic perspective on frames, and in particular to describe how many of the constraints on frames which are used in applications arise quite naturally in symplectic geometry, giving new insight into some important problems in frame theory, including the frame homotopy conjecture and the genericity of full spark frames. This is joint work with Tom Needham.
My goal in this talk is to introduce a symplectic perspective on frames, and in particular to describe how many of the constraints on frames which are used in applications arise quite naturally in symplectic geometry, giving new insight into some important problems in frame theory, including the frame homotopy conjecture and the genericity of full spark frames. This is joint work with Tom Needham.
Title: Tiling billiards in periodic tilings by equal triangles (and quadrilaterals)
Title: Tiling billiards in periodic tilings by equal triangles (and quadrilaterals)
Time: February 16, 2021, 4:15 pm (Rome Time)
Time: February 16, 2021, 4:15 pm (Rome Time)
Abstract: I will make an elementary introduction to tiling billiards — model of a light moving through a tiling under refraction laws.
Abstract: I will make an elementary introduction to tiling billiards — model of a light moving through a tiling under refraction laws.
This class of dynamical systems is new to mathematicians, simple to define as well as connected to already existing areas of research such as ergodic theory of interval exchange transformations and Novikov's problem on plane sections of 3-periodic surfaces.
This class of dynamical systems is new to mathematicians, simple to define as well as connected to already existing areas of research such as ergodic theory of interval exchange transformations and Novikov's problem on plane sections of 3-periodic surfaces.
I hope you will love it as much as I love it !
I hope you will love it as much as I love it !
(Not a very hard) Homework before the talk :
(Not a very hard) Homework before the talk :
Watch a following 5-MIN movie (here is a link to Youtube) by a wonderful mathematician and animator Ofir David.
Watch a following 5-MIN movie (here is a link to Youtube) by a wonderful mathematician and animator Ofir David.
Title: On the equivalence problem in sub-Riemannian geometry
Title: On the equivalence problem in sub-Riemannian geometry
Time: February 9, 2021, 4:15 pm (Rome Time)
Time: February 9, 2021, 4:15 pm (Rome Time)
Abstract: In mathematics, we are always interested in understanding when two objects are essentially the same. For the context of geometric structures, such as Riemannian and sub-Riemannian manifolds, "essentially the same" means being connected by an isometry.
Abstract: In mathematics, we are always interested in understanding when two objects are essentially the same. For the context of geometric structures, such as Riemannian and sub-Riemannian manifolds, "essentially the same" means being connected by an isometry.
For a Riemannian geometry, the central object measuring the local obstruction to the existence of an isometry is the curvature tensor of the Levi-Civita connection. We want to show that similar objects can be found on sub-Riemannian manifolds with constant nilpotentization, based on the work of T. Morimoto.
For a Riemannian geometry, the central object measuring the local obstruction to the existence of an isometry is the curvature tensor of the Levi-Civita connection. We want to show that similar objects can be found on sub-Riemannian manifolds with constant nilpotentization, based on the work of T. Morimoto.
We will show explicit formulas for a canonical choice of grading and connection for sub-Riemannian manifolds in some explicit cases.
We will show explicit formulas for a canonical choice of grading and connection for sub-Riemannian manifolds in some explicit cases.
Title: Complement of the discriminant variety, Gauss–skizze operads and hidden symmetries
Title: Complement of the discriminant variety, Gauss–skizze operads and hidden symmetries
Time: February 2, 2021, 4:00 pm (Rome Time)
Time: February 2, 2021, 4:00 pm (Rome Time)
Abstract: In this talk, the configuration space of marked points on the complex plane is considered. We investigate a decomposition of this space by so-called Gauss-skizze i.e. a class of graphs being forests. These Gauss-skizze, reminiscent of Grothendieck's dessins d'enfant, provide a totally different real geometric insight on this complex configuration space, which under the light of classical complex geometry tools, remains invisible. Topologically speaking, this stratification is shown to be a Goresky–MacPherson stratification.
Abstract: In this talk, the configuration space of marked points on the complex plane is considered. We investigate a decomposition of this space by so-called Gauss-skizze i.e. a class of graphs being forests. These Gauss-skizze, reminiscent of Grothendieck's dessins d'enfant, provide a totally different real geometric insight on this complex configuration space, which under the light of classical complex geometry tools, remains invisible. Topologically speaking, this stratification is shown to be a Goresky–MacPherson stratification.
We prove that for Gauss-skizze, classical tools from deformation theory, ruled by a Maurer--Cartan equation can be used only locally.
We prove that for Gauss-skizze, classical tools from deformation theory, ruled by a Maurer--Cartan equation can be used only locally.
We show as well, that the deformation of the Gauss-skizze is governed by a Hamilton--Jacobi differential equation.
We show as well, that the deformation of the Gauss-skizze is governed by a Hamilton--Jacobi differential equation.
Finally, a Gauss-skizze operad is introduced which can be seen as an enriched Fulton--MacPherson operad, topologically equivalent to the little 2-disc operad.
Finally, a Gauss-skizze operad is introduced which can be seen as an enriched Fulton--MacPherson operad, topologically equivalent to the little 2-disc operad.
The combinatorial flavour of this tool allows not only a new interpretation of the moduli space of genus 0 curves with n marked points, but gives a very geometric understanding of the Grothendieck--Teichmuller group.
The combinatorial flavour of this tool allows not only a new interpretation of the moduli space of genus 0 curves with n marked points, but gives a very geometric understanding of the Grothendieck--Teichmuller group.
Title: The adjoint of a polytope
Title: The adjoint of a polytope
Time: January 19, 2021, 4:00 pm (Rome Time)
Time: January 19, 2021, 4:00 pm (Rome Time)
Abstract: This talk brings many areas together: discrete geometry, statistics, intersection theory, classical algebraic geometry, geometric modeling, and physics. First, we recall the definition of the adjoint of a polytope given by Warren in 1996 in the context of geometric modeling. He defined this polynomial to generalize barycentric coordinates from simplices to arbitrary polytopes. Secondly, we show how this polynomial appears in statistics. It is the numerator of a generating function over all moments of the uniform probability distribution on a polytope. Thirdly, we prove the conjecture that the adjoint is the unique polynomial of minimal degree which vanishes on the non-faces of a simple polytope. In addition, we see that the adjoint appears as the central piece in Segre classes of monomial schemes, and in the study of scattering amplitudes in particle physics. Finally, we observe that adjoints of polytopes are special cases of the classical notion of adjoints of divisors. Since the adjoint of a simple polytope is unique, the corresponding divisors have unique canonical curves. In the case of three-dimensional polytopes, we show that these divisors are either K3 - or elliptic surfaces.
Abstract: This talk brings many areas together: discrete geometry, statistics, intersection theory, classical algebraic geometry, geometric modeling, and physics. First, we recall the definition of the adjoint of a polytope given by Warren in 1996 in the context of geometric modeling. He defined this polynomial to generalize barycentric coordinates from simplices to arbitrary polytopes. Secondly, we show how this polynomial appears in statistics. It is the numerator of a generating function over all moments of the uniform probability distribution on a polytope. Thirdly, we prove the conjecture that the adjoint is the unique polynomial of minimal degree which vanishes on the non-faces of a simple polytope. In addition, we see that the adjoint appears as the central piece in Segre classes of monomial schemes, and in the study of scattering amplitudes in particle physics. Finally, we observe that adjoints of polytopes are special cases of the classical notion of adjoints of divisors. Since the adjoint of a simple polytope is unique, the corresponding divisors have unique canonical curves. In the case of three-dimensional polytopes, we show that these divisors are either K3 - or elliptic surfaces.
This talk is based on joint works with Kristian Ranestad, Boris Shapiro and Bernd Sturmfels.
This talk is based on joint works with Kristian Ranestad, Boris Shapiro and Bernd Sturmfels.
Title: Singular solutions spaces of rolling balls problem
Title: Singular solutions spaces of rolling balls problem
Time: January 12, 2021, 4:00 pm (Rome Time)
Time: January 12, 2021, 4:00 pm (Rome Time)
Abstract: The "Rolling Balls Model", the model describing a pair of spheres of different ray rolling one on another without slipping or twisting, is a classical example of sub-Riemannian problem. The symmetries of the distribution associated with the system depend on the ratio of the rays and radically change when the ratio equals 3. Indeed, for this value of the ratio (and only for this value) it extends to the exceptional simple Lie group G2 which acts, still for this value of the ratio, also on the singular solutions related to the problem.
Abstract: The "Rolling Balls Model", the model describing a pair of spheres of different ray rolling one on another without slipping or twisting, is a classical example of sub-Riemannian problem. The symmetries of the distribution associated with the system depend on the ratio of the rays and radically change when the ratio equals 3. Indeed, for this value of the ratio (and only for this value) it extends to the exceptional simple Lie group G2 which acts, still for this value of the ratio, also on the singular solutions related to the problem.
In this talk we show how it is possible to describe the spaces of such singular solutions in a geometric way, as a family of 5-dimensional manifolds depending on the ratio. For rational values of the ratio such manifolds have a structure of SO(2)-principal bundles which are not topologically distinguished by their homology, homotopy and de Rham cohomology groups. In addition, we show that for integer values of the ratio the configuration manifold of the problem is a branched covering of each of such manifolds and how the covering maps associated allow to relate them with another known family of topological spaces, the lens spaces.
In this talk we show how it is possible to describe the spaces of such singular solutions in a geometric way, as a family of 5-dimensional manifolds depending on the ratio. For rational values of the ratio such manifolds have a structure of SO(2)-principal bundles which are not topologically distinguished by their homology, homotopy and de Rham cohomology groups. In addition, we show that for integer values of the ratio the configuration manifold of the problem is a branched covering of each of such manifolds and how the covering maps associated allow to relate them with another known family of topological spaces, the lens spaces.
This talk is based on the research works developed in my master thesis at University of Trieste, under the supervision of the professor A.Agrachev.
This talk is based on the research works developed in my master thesis at University of Trieste, under the supervision of the professor A.Agrachev.
Title: Critical points of eigenfunctions
Title: Critical points of eigenfunctions
Time: December 15, 2020, 4:00 pm (Rome Time)
Time: December 15, 2020, 4:00 pm (Rome Time)
Abstract: On a closed Riemannian manifold, the Courant nodal domain theorem gives an upper bound on the number of nodal domains of n-th eigenfunction of the Laplacian. In contrast to that, there does not exist such bound on the number of isolated critical points of an eigenfunction. I will try to sketch a proof of the existence of a Riemannian metric on the 2-dimensional torus, whose Laplacian has infinitely many eigenfunctions, each of which has infinitely many isolated critical points. Based on a joint work with A. Logunov and M. Sodin.
Abstract: On a closed Riemannian manifold, the Courant nodal domain theorem gives an upper bound on the number of nodal domains of n-th eigenfunction of the Laplacian. In contrast to that, there does not exist such bound on the number of isolated critical points of an eigenfunction. I will try to sketch a proof of the existence of a Riemannian metric on the 2-dimensional torus, whose Laplacian has infinitely many eigenfunctions, each of which has infinitely many isolated critical points. Based on a joint work with A. Logunov and M. Sodin.
Title: Geometrical properties of random eigenfunctions
Title: Geometrical properties of random eigenfunctions
Time: December 7, 2020, 4:00 pm (Rome Time) ----->Please note the exceptional change of date.<-----
Time: December 7, 2020, 4:00 pm (Rome Time) ----->Please note the exceptional change of date.<-----
Abstract: In this talk we deal with the geometry of random eigenfunctions on manifolds (the round sphere, the standard flat torus, the Euclidean plane...) motivated by both Yau's conjecture and Berry's ansatz. In particular, we investigate the asymptotic behavior (in the high-energy limit) of the so-called nodal length for random spherical harmonics and (un)correlation phenomena between the latter and other Lipschitz-Killing curvatures of their excursion sets at any level.
Abstract: In this talk we deal with the geometry of random eigenfunctions on manifolds (the round sphere, the standard flat torus, the Euclidean plane...) motivated by both Yau's conjecture and Berry's ansatz. In particular, we investigate the asymptotic behavior (in the high-energy limit) of the so-called nodal length for random spherical harmonics and (un)correlation phenomena between the latter and other Lipschitz-Killing curvatures of their excursion sets at any level.
This talk is mainly based on joint works with V. Cammarota, D. Marinucci, I. Nourdin, G. Peccati and I. Wigman.
This talk is mainly based on joint works with V. Cammarota, D. Marinucci, I. Nourdin, G. Peccati and I. Wigman.
Title: Cylindrical decompositions in real and complex geometry
Title: Cylindrical decompositions in real and complex geometry
Time: December 1, 2020, 3:00 pm (Rome Time) ----->Please note the exceptional change of hour.<-----
Time: December 1, 2020, 3:00 pm (Rome Time) ----->Please note the exceptional change of hour.<-----
Abstract: The decomposition of a set into "cylinders" in one of the fundamental tools of semi-algebraic geometry (as well as subanalytic geometry and o-minimal geometry). Defined by means of intervals, these cylinders are an essentially real-geometric construct.
Abstract: The decomposition of a set into "cylinders" in one of the fundamental tools of semi-algebraic geometry (as well as subanalytic geometry and o-minimal geometry). Defined by means of intervals, these cylinders are an essentially real-geometric construct.
In a recent paper wit Novikov we introduce a notion of "complex cells", that form a complexification of real cylinders. It turns out that such complex cells admit a rich hyperbolic geometry, which is not directly visible in their real counterparts. I will sketch some of this theory, and how it can be used to prove some new results in real geometry (for instance a sharpening of the Yomdin-Gromov lemma).
In a recent paper wit Novikov we introduce a notion of "complex cells", that form a complexification of real cylinders. It turns out that such complex cells admit a rich hyperbolic geometry, which is not directly visible in their real counterparts. I will sketch some of this theory, and how it can be used to prove some new results in real geometry (for instance a sharpening of the Yomdin-Gromov lemma).
Title: Normally inscribable polytopes, routed trajectories, and reflection arrangements
Title: Normally inscribable polytopes, routed trajectories, and reflection arrangements
Time: November 24, 2020, 4:00 pm (Rome Time)
Time: November 24, 2020, 4:00 pm (Rome Time)
Abstract: Steiner posed the question if any 3-dimensional polytope had a realization with vertices on a sphere. Steinitz constructed the first counter examples and Rivin gave a complete complete answer to Steiner's question. In dimensions 4 and up, the Universality Theorem renders the question for inscribable combinatorial types hopeless. In this talk, I will address the following refined question: Given a polytope P, is there a continuous deformation of P into an inscribed polytope that keeps corresponding faces parallel?
Abstract: Steiner posed the question if any 3-dimensional polytope had a realization with vertices on a sphere. Steinitz constructed the first counter examples and Rivin gave a complete complete answer to Steiner's question. In dimensions 4 and up, the Universality Theorem renders the question for inscribable combinatorial types hopeless. In this talk, I will address the following refined question: Given a polytope P, is there a continuous deformation of P into an inscribed polytope that keeps corresponding faces parallel?
This question has strong ties to deformations of Delaunay subdivisions and ideal hyperbolic polyhedra and its study reveals a rich interplay of algebra, geometry, and combinatorics. In the first part of the talk, I will discuss relations to routed trajectories of particles in a ball and reflection groupoids and show that that the question is polynomial time decidable.
This question has strong ties to deformations of Delaunay subdivisions and ideal hyperbolic polyhedra and its study reveals a rich interplay of algebra, geometry, and combinatorics. In the first part of the talk, I will discuss relations to routed trajectories of particles in a ball and reflection groupoids and show that that the question is polynomial time decidable.
In the second part of the talk, we will focus on class of zonotopes, that is, polytopes representing hyperplane arrangements. It turns out that inscribable zonotopes are rare and intimately related to reflection groups and Gr\"unbaum's quest for simplicial arrangements. This is based on joint work with Sebastian Manecke.
In the second part of the talk, we will focus on class of zonotopes, that is, polytopes representing hyperplane arrangements. It turns out that inscribable zonotopes are rare and intimately related to reflection groups and Gr\"unbaum's quest for simplicial arrangements. This is based on joint work with Sebastian Manecke.
Title: Disguised toric dynamical systems
Title: Disguised toric dynamical systems
Time: November 17, 2020, 4:00 pm (Rome Time)
Time: November 17, 2020, 4:00 pm (Rome Time)
Abstract: We study families of polynomial dynamical systems inspired by biochemical reaction networks. We focus on complex balanced mass-action systems, which have also been called toric dynamical systems, by Craciun, Dickenstein, Shiu and Sturmfels. These systems are known or conjectured to enjoy very strong dynamical properties, such as existence and uniqueness of positive steady states, local and global stability, persistence, and permanence. We consider the class of disguised toric dynamical systems, which contains toric dynamical systems, and to which all dynamical properties mentioned above extend naturally. We show that, for some families of reaction networks, this new class is much larger than the class of toric systems. For example, for some networks we may even go from an empty locus of toric systems in parameter space to a positive-measure locus of disguised toric systems. We focus on the characterization of the disguised toric locus by means of real algebraic geometry. Joint work with Gheorghe Craciun and Laura Brustenga i Moncusí.
Abstract: We study families of polynomial dynamical systems inspired by biochemical reaction networks. We focus on complex balanced mass-action systems, which have also been called toric dynamical systems, by Craciun, Dickenstein, Shiu and Sturmfels. These systems are known or conjectured to enjoy very strong dynamical properties, such as existence and uniqueness of positive steady states, local and global stability, persistence, and permanence. We consider the class of disguised toric dynamical systems, which contains toric dynamical systems, and to which all dynamical properties mentioned above extend naturally. We show that, for some families of reaction networks, this new class is much larger than the class of toric systems. For example, for some networks we may even go from an empty locus of toric systems in parameter space to a positive-measure locus of disguised toric systems. We focus on the characterization of the disguised toric locus by means of real algebraic geometry. Joint work with Gheorghe Craciun and Laura Brustenga i Moncusí.
Title: Carnot groups and abnormal dynamics
Title: Carnot groups and abnormal dynamics
Time: November 3, 2020, 4:00 pm (Rome Time)
Time: November 3, 2020, 4:00 pm (Rome Time)
Abstract: The existence of so called abnormal curves is one of the features distinguishing sub-Riemannian geometry from Riemannian geometry. The need to understand (or avoid) abnormal curves appears in many sub-Riemannian problems, such as the regularity of length-minimizing curves and the Sard problem. Some recent progress in both of these problems has been obtained by studying abnormal curves as trajectories of dynamical systems. In this talk, I will present some of the story of abnormal dynamics in the setting of Carnot groups. In particular, I will cover how to lift an arbitrary trajectory of an arbitrary polynomial ODE to an abnormal curve in some Carnot group.
Abstract: The existence of so called abnormal curves is one of the features distinguishing sub-Riemannian geometry from Riemannian geometry. The need to understand (or avoid) abnormal curves appears in many sub-Riemannian problems, such as the regularity of length-minimizing curves and the Sard problem. Some recent progress in both of these problems has been obtained by studying abnormal curves as trajectories of dynamical systems. In this talk, I will present some of the story of abnormal dynamics in the setting of Carnot groups. In particular, I will cover how to lift an arbitrary trajectory of an arbitrary polynomial ODE to an abnormal curve in some Carnot group.
Title: The topology of the nodal sets of eigenfunctions and a problem of Michael Berry
Title: The topology of the nodal sets of eigenfunctions and a problem of Michael Berry
Time: October 27, 2020, 4:00 pm (Rome Time)
Time: October 27, 2020, 4:00 pm (Rome Time)
Abstract: In 2001, Sir Michael Berry conjectured that given any knot there should exist a (complex-valued) eigenfunction of the harmonic oscillator (or the hydrogen atom) whose nodal set contains a component of such a knot type. This is a particular instance of the following problem: how is the topology of the nodal sets of eigenfunctions of Schrodinger operators? In this talk I will focus on the flexibility aspects of the problem: either you construct a suitable Riemannian metric adapted to the submanifold you want to realize, or you consider operators with a large group of symmetries (e.g., the Laplacian on the round sphere, or the harmonic quantum oscillator), and exploit the large multiplicity of the high eigenvalues. In particular, I will show how to prove Berry's conjecture using an inverse localization property. This talk is based on different joint works with A. Enciso, D. Hartley and F. Torres de Lizaur.
Abstract: In 2001, Sir Michael Berry conjectured that given any knot there should exist a (complex-valued) eigenfunction of the harmonic oscillator (or the hydrogen atom) whose nodal set contains a component of such a knot type. This is a particular instance of the following problem: how is the topology of the nodal sets of eigenfunctions of Schrodinger operators? In this talk I will focus on the flexibility aspects of the problem: either you construct a suitable Riemannian metric adapted to the submanifold you want to realize, or you consider operators with a large group of symmetries (e.g., the Laplacian on the round sphere, or the harmonic quantum oscillator), and exploit the large multiplicity of the high eigenvalues. In particular, I will show how to prove Berry's conjecture using an inverse localization property. This talk is based on different joint works with A. Enciso, D. Hartley and F. Torres de Lizaur.