Embedded Files
Gioacchino Antonelli (Courant Institute - NYU)
A new sharp and rigid spectral generalization of Bishop--Gromov volume estimate and applications
16 luglio 2024 alle ore 15:00: in aula 2AB45
If an n-dimensional smooth complete manifold M without boundary satisfies Ric>=n-1, then it is compact, its volume is bounded above by that of the round unit sphere, and the first fundamental group of M is finite.
In this talk I will discuss a new sharp and rigid generalization of the previous Bishop--Gromov volume estimate. Calling \Delta the Laplace operator on functions, I will prove that if on a compact n-manifold the first eigenvalue of the Schroedinger operator -(n-1)/(n-2)*\Delta+Ric is greater or equal than n-1, then the same comparison as stated above holds, verbatim.
I will discuss sharpness of the constants, rigidity, and applications of this result to geometric problems -- e.g., the stable Bernstein problem.
---------------
Nicola Paddeu (Université de Fribourg)
Metabelian distributions and sub-Riemannian geodesics
2 luglio 2024 alle ore 12:00: in aula 2AB40
This talk focuses on those particular sub-Riemannian structures whose distribution is metabelian. On the one hand, we present necessary conditions that infinite normal geodesics in Carnot groups must satisfy. On the other hand, we prove that the projection of abnormal curves to some lower dimensional manifold must stay inside an analytic variety. As a consequence, for rank-2 metabelian distributions, geodesics are continuously differentiable. This talk is based on joint works with Alejandro Bravo-Doddoli, Enrico Le Donne and Alessandro Socionovo.
---------------
Olivier Glass (Université Paris Dauphine)
2 x 2 hyperbolic systems of conservation laws in classes of functions of bounded p-variation
20 giugno 2024 alle ore 11:30: in aula 2BC30
In this talk, I will discuss the existence of entropy solutions for 2 x 2 hyperbolic systems of conservation laws in one space dimension. This is a classical topic, but the novelty here is that we consider solutions in Wiener's class of functions of bounded p-variation. This function space can be considered an intermediate between essentially bounded functions and functions with bounded variation (which correspond to p=1).
Precisely, I will describe a result showing that under an assumption on the characteristic fields that is slightly more general than Lax's genuine nonlinearity or linear degeneracy, one can obtain the existence of entropy solutions in such classes when p is between 1 and 3/2.
---------------
Daniele Cassani (Università dell'Insubria)
On the maximum principle for higher order operators in general domains and any dimension
19 giugno 2024 alle ore 11:00: in aula 1BC45
In the higher order case, it is well known how truncation methods in general fail. We prove a new Harnack type inequality demanding for augmented integrability on the function involved in place of being just a solution to a PDE, which usually yields Caccioppoli’s inequality and thus the solution belongs to the corresponding De Giorgi class. Indeed, in this context Di Benedetto-Trudinger ’84 prove a Harnack type inequality for functions with membership in suitable De Giorgi classes. We drop this assumption though we assume more regularity in terms of integrability which enables us to prove De Giorgi type pointwise level estimates. As a consequence, we prove the strong maximum principle for uniformly elliptic operators of any even order, which do contain lower order derivatives, in sufficiently smooth bounded domains which enjoy the interior sphere condition. This is done by a limiting procedure starting from compactly supported functions and then extending the results to the solutions of higher order PDEs subject to Dirichlet boundary conditions.
---------------
Felisia Angela Chiarello (Universita dell'Aquila)
Nonlocal conservation laws with p-norm, the singular limit problem and applications to traffic flow
6 giugno 2024 alle ore 15:00: in aula 2AB45
In this talk we present nonlocal conservation laws with the p-norm. Nonlocal refers to the fact that the velocity of the conservation law depends on an integral term in space. Typically, the nonlocal term consists of integrating the solution in an L^1 sense, while here we will study the case when the nonlocal term is realized via the L^p-norm. We show the existence and uniqueness of weak solution considering the initial datum bounded away from zero. Moreover, we consider the singular limit problem when the nonlocal operator converges to a Dirac distribution recovering the corresponding local entropy solution. An application to traffic flow modelling is reasonable, as a higher integrability reflects a more conservative driving behavior, braking sooner than when using the L^1-mass as integral.
---------------
Camillo Brena (Scuola Normale Superiore - Pisa)
Unique continuation for area minimizing currents
30 maggio 2024 alle ore 12:00: in aula 2AB40
Consider an m-dimensional area minimizing current and an m-dimensional minimal surface. If, in a integral sense, the current has infinite order of contact with the minimal surface at a point, then the current and the minimal surface coincide in a neighborhood of that point. This results is contained in a work with Stefano Decio.
---------------
Jean Paul Gauthier (Université de Toulon)
Some results in Quantum control
23 maggio 2024 alle ore 14:30: in aula 2BC30
A finite dimensional control system is just a control affine skew-adjoint system on the n-dimensional complex space and therefore on the unit sphere of dimension 2n-1.
I shall talk about the following result, answer to a conjecture of Andrei Agrachev: there is a universal gap for the minimum-time problem on the sphere, that is (a unit of time being fixed, related with the "time of the drift"), there is a universal constant c>0 such that the supremum of the minimum time between two points of the unit sphere is always >=c.
This is joint work with F. Rossi.
If I have time I also talk about another problem of "partial controllability" for such systems, that is controllability for some controls fixed, with respect to the free others.
Both problems are related with the structure and the representations of compact connected Lie groups and their maximal subgroups.
---------------
Nikolay Pogodaev (Università di Padova)
Monotone vector fields in the Wasserstein space
28 maggio 2024 alle ore 11:15: in aula 1BC50
The concept of a maximal monotone operator is well-known in the Hilbert setting, where such operators are closely related to partial differential equations that generate nonexpansive semigroups. In many cases, maximal monotone operators exhibit a gradient structure, implying that they can be represented as subdifferentials of proper convex lower semicontinuous functions. The corresponding partial differential equations are commonly referred to as gradient flows. The theory of gradient flows was extended from the Hilbert to the Wasserstein setting in the early 2000s. This extension was initiated in the papers of R. Jordan, D. Kinderlehrer, and F. Otto, and it reached its classical form in the monograph by L. Ambrosio, N. Gigli and G. Savarè, opening up numerous applications since then.
In the present talk, we explore the possibility of defining a monotone vector field in the Wasserstein space without necessarily adhering to a gradient form. We discuss the corresponding variant of the JKO scheme, provide examples of monotone vector fields, and explore potential applications.
---------------
Francesco Solombrino (Università di Napoli Federico II)
Mean-field optimal control of multipopulation systems and related Pontryagin-type conditions
6 maggio 2024 alle ore 12:30: in aula 2AB40
This talk concerns the problem of parsimoniously controlling multi-agent systems where individuals, in togheter with a position in phase space, are labelled by their probability of belonging to a certain population. The dynamics features an activation function that allows a hypothetical policy maker to select a narrow pool of agents to act on. Such a choice may be based on, for example, their influence on the rest of the population, which may vary over time. A mean-field optimal control problem for the limit of such systems is identified via Gamma-convergence, ensuring the convergence of the optimal controls of particle systems to the optimum of the limit problem. The latter is governed by a diffusion-free continuity-type equation in a convex subset of a Banach space, possibly of infinite dimension. In the absence of a linear structure and compactness of the phase space, the usual tools of needle variations and linearized equations, used to derive Pontryagin-type conditions, must be generalized to the context at hand. This requires appropriate notions of differentials for functions defined on convex sets and on Wasserstein spaces, and a careful analysis of the involved functional spaces.
Based on joint works with with G. Albi (Verona), S. Almi (Naples "Federico II"), R. Durastanti (Naples "Federico II") and M. Morandotti (Polytechnic University of Turin).
---------------
Samuel Borza (SISSA - Trieste)
Synthetic curvature-dimension bounds in the sub-Finsler Heisenberg group
18 aprile 2024 alle ore 12:30: in aula 2BC60
The Heisenberg group is important to many fields in mathematics and physics, including quantum theory, metric geometry, and harmonic analysis. I will discuss the sub-Finsler geometry of the Heisenberg group and explain how it is related to the isoperimetric problem in the non-Euclidean (Finsler) plane. We will then explore various approaches to studying the curvature of the sub-Finsler Heisenberg group, focusing particularly on the Lott–Sturm–Villani curvature-dimension condition and the measure contraction property that appears in the analysis of metric measure spaces. We will see that the validity/failure of these synthetic curvature-dimension bounds depend on the regularity of the reference norm. This is a joint work with Kenshiro Tashiro, Mattia Magnabosco, and Tommaso Rossi.
---------------
Andrea Kubin (Technische Universität München)
A notion of s-fractional mass for 1-currents in higher codimension
28 marzo 2024 alle ore 11:30: in aula 2AB40
In this talk I will discuss a notion of s-fractional mass for 1-currents in high codimension. Such a notion generalizes the notion of s-fractional perimeter for sets in the plane to higher-codimension, one-dimensional singularities. In particular, I will talk about some properties of 1-currents with finite s-fractional mass. These results are in collaboration with M. Cicalese, T. Heilmann, F. Onoue, M. Ponsiglione.
---------------
Giulia Bevilacqua (Università di Pisa)
A capillarity approach for the regularity of soap films
14 marzo 2024 alle ore 14:00: in aula 2BC30
Joint work with Salvatore Stuvard (UNIMI) and Bozhidar Velichkov (UNIPI).
We characterize boundary regularity for a variational model of soap film spanning a tubular neighborhood of a curve. Inspired by [1], soap films are chosen to be sets of finite perimeter containing a fixed volume and satisfying a topological spanning condition. In this talk, for a planar curve as midline of a tubular neighborhood, we show that minimizers are normal smooth graphs with constant mean curvature constructed over the plane and forming a contact angle equal to π/2 [2].
References
[1] D. King, F. Maggi, S. Stuvard, Plateau’s problem as a singular limit of capillarity problems, Communications on Pure and Applied Mathematics, 75:5 2022, pp. 895–969.
[2] G. Bevilacqua, S. Stuvard, B. Velichkov; Regularity of a free-boundary Plateau problem, in preparation.
---------------
Xavier Lamy (Université de Toulouse)
On the stability of Möbius maps of the n-sphere
11 marzo 2024 alle ore 13:30: in aula 2AB40
A classical theorem of Liouville asserts that if a map from the sphere to itself is conformal, then it must be a Möbius transform: a composition of dilations, rotations, inversions and translations (identifying sphere and euclidean space via stereographic projection). There is a long history of studying stability of this rigidity statement: if a map is nearly conformal, must it be close to a Möbius transform? One can also ask what happens if the image of the map is only nearly spherical. I will present optimal stability estimates obtained with André Guerra and Kostantinos Zemas, which generalize to higher dimensions recent results for the 2-sphere (where, unlike higher dimensions, the problem can be directly linearized).
---------------
Carlotta Donadello (Université de Franche-Comté)
Conservation laws on a star-shaped network
31 gennaio 2024 alle ore 12:30: in aula 2AB40
Hyperbolic conservation laws defined on oriented graphs are widely used in the modeling of a variety of phenomena such as vehicular and pedestrian traffic, irrigation channels, blood circulation, gas pipelines, structured population dynamics.
From the point of view of the mathematical analysis each of these situations demands for a different definition of admissible solution, encoding in particular the node coupling between incoming and outgoing edges which is the most coherent with physical observations.
A comprehensive study of the necessary and sufficient properties of the coupling conditions which lead to well-posedness of the corresponding admissible solutions is available in the framework of conservation laws with discontinuous flux, which can be seen as a simple 1-1 network, see (2).
A similar theory for conservation laws on star-shaped graph is at its beginning.
In particular, the characterization of family of solutions obtained as limits of regularizing approximations, such as vanishing viscosity limits, is still a partially open problem.
In this talk we’ll provide a general introduction to the topic, an overview of the most recent results and some explicit examples.
---------------
Giuseppe Maria Coclite (Università di Bari)
Nonlinear Peridynamic Models
9 gennaio 2024 alle ore 12:30: in aula 2AB40
Some materials may naturally exhibit singularity phenomena such as cracks or other manifestations as a result of scale effects and long range interactions. Peridynamic theory models such behavior introducing a new nonlocal framework for the basic equations of continuum mechanics. In this lecture we consider a nonlinear peridynamic model and discuss its qualitative properties and well-posedness in suitable fractional Sobolev spaces.
Those results were obtained in collaboration with A. Coclite (Bari), N. Dimola (Milano), S. Dipierro (Perth), G. Fanizza (Lisbona), L. Lopez (Bari), F. Maddalena (Bari), M. Romano (Bari), S. Pellegrino (Bari), T. Politi (Bari), and E. Valdinoci (Perth).
---------------
Luigi Ambrosio (Scuola Normale Superiore - Pisa)
On some variational problems involving functions with bounded Hessian
15 dicembre 2023 alle ore 14:30: in aula 2AB40
In this talk I will illustrate new questions, at the interface between Calculus of Variations and Geometric Measure Theory, motivated in a broad sense by the more applied machine learning community. More specifically, I will deal with second-order variational problems in the setting of functions with bounded variation and study the extremality properties of continuous and piecewise affine functions in the unit ball of a suitable energy, used as a regularization term in the applied community. In connection with this, a fine approximation result by piecewise affine function has been investigated. Papers in collaboration with C.Brena, S.Conti, S.Aziznejad and M.Unser.
---------------
David Jesus (Università di Bologna)
Fully nonlinear equations degenerating as a Muckenhoupt weight
5 dicembre 2023 alle ore 12:30: in aula 2AB40
We will discuss the regularity of viscosity solutions for fully nonlinear Hessian equations with coefficients in some Muckenhoupt class. We prove Holder and higher regularity under mild assumptions on the coefficients. In particular, surprisingly, we achieve Holder differentiability across the submanifold where the ellipticity vanishes. This is a consequence of the uniqueness of suitably defined viscosity solutions. We use approximation techniques in the spirit of Caffarelli's seminal approach. This is a joint work with Yannick Sire from JHU.
---------------
Marco Pozzetta (Università di Napoli Federico II)
Uniqueness of blowups for the motion by curvature of networks
28 novembre 2023 alle ore 12:30: in aula 2AB40
A network is a finite union of embedded open curves in the plane such that their endpoints are joined at triple junctions. A motion by curvature is a one-parameter family of networks evolving so that the normal driving velocity at any point is given by the curvature vector. Hence, a motion by curvature corresponds to the L^2-gradient flow of the length functional on networks.
Like higher dimensional mean curvature flows, the motion by curvature of networks may develop singularities, whose behavior can be understood by studying blowups at points where the singularity occurs. A blowup is a flow obtained as the limit of a sequence of space-time dilations centered at the singularity point.
In this talk we discuss a result that proves uniqueness of blowups for the motion by curvature of networks, that is the fact that the blowup at a singularity does not depend on the chosen sequence of rescaling factors.
The proof is based on the application of a Lojasiewicz--Simon gradient inequality.
The talk is based on a work in collaboration with Carlo Mantegazza and Alessandra Pluda.
---------------
Raffaele Grande (UTIA Prague)
Stochastic control problems and a convergence result for horizontal mean curvature flow
14 novembre 2023 alle ore 12:30: in aula 1BC45
The evolution by horizontal mean curvature flow was broadly studied for its applications in neurogeometry and in image processing (e.g. Citti-Sarti model). It represents thecontracting evolution of a hypersurface embedded in a particular geometrical setting, called sub-Riemannian geometry, in which only some curves (called horizontal curves) are admissible by definition. This may lead the existence of some points of the hypersurface, called characteristic, in which is not possible to define the so called horizontal normal.
In order to avoid this problem, it is possible to use the notion of Riemannian approximation of a sub-Riemannian geometry applied to the horizontal mean curvature flow.
I will show the connection between the evolution of a generic hypersurface in this setting and the associated stochastic optimal control problem.
Then, I will show some results asymptotic optimal controls in the Heisenberg group and use them to later show a convergence result between the solutions of the approximated mean curvature flow and the horizontal ones.
This is from some joint works with N. Dirr and F. Dragoni.
---------------
Maxim Staritsyn (Matrosov Institute)
Exact increment formulas in optimal control problems: from classical to mean field control
7 novembre 2023 alle ore 12:30: in aula 2AB40
In the classical Calculus of Variations, there is a formula that allows one to express the difference between two values of a functional (i.e., its increment) as an integral of a certain function. This formula is known as the Weierstrass formula, and the integral is called the invariant Hilbert integral.
It turns out that the same can be done (in different ways) for optimal control problems, yielding exact representations of the cost increment, i.e., increment formulas devoid of residual terms.
These formulas allow, in turn, to develop new necessary optimality conditions, not equivalent to the paradigmatic Pontryagin maximum principle, and descent algorithms without free parameters.
It will be shown that the same approach can also be applied to optimal control problems for the nonlocal continuity equation and the Fokker-Planck-Kolmogorov equation in the space of probability measures.
---------------
Matteo Novaga (Università di Pisa)
Periodic partitions with minimal perimeter
19 ottobre 2023 alle ore 12:30: in aula 2AB45
I will discuss existence and regularity of fundamental domains which minimize a general perimeter functional in a homogeneous space. In the planar case I will give a more detailed description of minimal domains. This is a work in collaboration with Annalisa Cesaroni and Ilaria Fragalà.
Page updated
Google Sites
Report abuse