Edition 2025-2026

November 7th, 2025

Speaker: Valerio Tripodi (SISSA)

Title: Why does string theory require extra dimensions? 

Abstract: One of the many additional physical ingredients required by string theory in order to have a consistent framework is the presence of extra (spatial) dimensions. This concept is rather challenging for our human minds used to living in a 3+1 dimensional spacetime, but it opens up the possibility of exploring new interesting physics phenomena.

In this seminar, I will explain how the constraint on the number of spacetime dimensions arises in (bosonic) string theory. First, I will briefly review the general set up by analyzing the Polyakov action and sketching its BRST quantization procedure, which introduces the "ghost" fields in the theory. The corresponding dynamics of the degrees of freedom naturally yields to a 2d conformal field theory and so to a Virasoro algebra. Than I will argue why we require the total central charge of the theory to vanish and how this fixes the number of spacetime dimensions.

If time permits, I'll shortly illustrate what this phenomenon implies in superstring theory and how one can recover a 4-dimensional theory, perhaps sketching the easiest and earliest example: Kaluza-Klein theory. 

November 14th, 2025

Speaker: Luca Morstabilini (SISSA)

Title: Topology vs geometry vs algebra: the nonabelian Hodge correspondence

Abstract: The aim of this talk is to discuss a deep and elegant correspondence between objects of very different nature on a Riemann surface C: representations of the fundamental group (topological), flat connections (differential), and Higgs bundles (holomorphic/algebraic). We will also see that the correspondence can be made geometric, i.e. it upgrades to diffeomorphisms between the moduli spaces of the aforementioned objects. This provides a bridge between the topology of the Riemann surface and its complex structure. Time permitting, i will also mention some consequences of this dictionary.

During the talk I will introduce all the relevant notions and try to justify their importance. Despite the name, no knowledge of Hodge theory is required.

Edition 2024-2025

November 15th, 2024

Speaker: Alaa Boukholkhal (ENS Lyon)

Title: Isometric Immersions: A historical overview

Abstract: The problem of isometric embeddings is fundamental in differential geometry. In the 1950s, John Nash provided a complete solution to this problem, marking a significant turning point in the field. In this talk, we delve into the historical context of this problem, highlighting the distinction between the C^1 case and the smooth case. We then detail Nash's revolutionary results, providing insight into their impact and significance in the field. Finally, if time permits, we will explore the h-principle and other flexibility results.

November 22nd, 2024

Speaker: Lorenzo Cortelli (SISSA)

Title: Regular foliations on compact complex surfaces

Abstract: Arising naturally from a various mathematical problems, foliations are versatile objects, relevant to geometry and mathematical physics alike. Submersions, actions on manifolds and differential equations are just a few of the possible ways to define a foliation.

In this talk we'll focus on holomorphic foliations, introducing the most commonly used tools of complex foliation theory and ultimately giving a classification of all regular foliations on compact complex surfaces. Residue-like index formulae and Kodaira's work on complex surfaces will play a fundamental role.

November 27th, 2024

Speaker: Alessio Di Prisa (Scuola Normale Superiore di Pisa)

Title: Equivariant algebraic concordance of strongly invertible knots

Abstract: In 1969 Levine defined a surjective homomorphism from the knot concordance group to the so called algebraic concordance group, which is a Witt group of Seifert forms.

Studying symmetric knots and in particular strongly invertible knots, a natural question is whether it is possible to define an appropriate equivariant version of algebraic concordance.

In this talk we briefly recall Levine's construction and we highlight some of the difficulties occuring when trying to define its equivariant analogous.

Finally, we define a notion of equivariant algebraic concordance for strongly invertible knots, we show some of the differences with the classical case and (time permitting) we will give an application of this theory to the study of equivariant slice genus.

December 6th, 2024

Speaker: Simone Fabbri (SISSA)

Title: The Phi4 theory by bare hands

Abstract: The Phi4 moldel is a paradigmatic example of interacting quantum field theory, playing a remarkable role also in statistical mechanics and probability. The model itself is the theory of a quantum, real, scalar field whose Lagrangian is the sum of a standard (quadratic) kinetic term and a quartic self-interaction, which makes the overall system not exactly solvable.

Precisely, I will present a Euclidean version of the massive Phi4 model on lattice and I will focus on the regime of small coupling. I will initially show how standard perturbation theory fails at constructing the interacting model, giving rise to a power series in the coupling which is actually divergent. This motivates the search for an alternative small-coupling expansion for the observables, which should be convergent uniformly in the volume of the system. The goal of this seminar is to present a simple way to set up such a convergent expansion for the correlation functions of the theory. the beauty of such methods, based on standard cluster-expansion theory, relies on the nontriviality of their underlying ideas, which is at the same time realized with extreme simplicity and lightness.

December 13th, 2024

Speaker: Lorenzo Cecchi (SISSA)

Title: Optimal Transport in Algebraic Geometry

Abstract: In the attempt of solving Smale's 17th problem, Shub and Smale bumped into the following problem: is there a "natural" metric on the space of complex polynomials such that the subset of smooth polynomials is geodesically convex? Thirty years later, Antonini, Cavalletti and Lerario found an answer by trying to address the following question: what is the optimal way to deform a projective hypersurface into another one?

In this talk I will present their results, where optimal transport and symplectic geometry endow the (moduli) space of hypersurfaces in CP^n with a rich geometric structure. Time permitting, I shall also discuss the directions for further research arising from my work.

January 10th, 2025

Speaker: Andrea Rosana (SISSA)

Title: Typical ranks of random order-three tensors

Abstract: In the last 30 years tensors, which can be thought of as higher dimensional matrices, have become crucial for many applications, including statistics, signal processing, quantum computing and many more. One of the key concepts in applicative settings is that of tensor rank and rank decomposition. 

We will start by defining the tensor rank, highlighting the differences with the matrix case and the differences between complex and real tensors. This will lead to the definition of typical ranks of random real tensors, i.e. those ranks arising with positive probability. We will then focus on the case of three dimensional tensors and show how to characterize geometrically tensor rank through a result by Friedland. Using this result, we will link tensor rank with linear sections of the Segre variety of real rank-one matrices. Exploiting tools from integral geometry we will show some heuristics on typical ranks of a family of order-three tensors. If time allows, we will briefly show how the rank of a real random 3x3x5 tensor depends on the number of real lines on a random cubic surface. 

Part of this talk is based on joint work with P. Breiding and S. Eggleston (University of Osnabrück) https://arxiv.org/abs/2407.08371.

January 17th, 2025

Speaker: Matteo Montagnani (SISSA)

Title: Life after continuous K-theory

Abstract: After Efimov’s groundbreaking insight that K-theory can also be defined for dualizable categories, your life probably hasn’t changed all that much. However, by starting with an introduction to algebraic K-theory and revisiting some of the most important ideas in this context, I will attempt to convince you that Efimov’s definition enables algebraic K-theory to explore new paths that were previously inaccessible.

January 24th, 2025

Speaker: Thomas Nicosanti (SISSA)

Title: The simplest 3d Topological Field Theory I know

Abstract: After my last Junior GMP seminar, we resume the discussion on the axiomatisation of Topological Field Theories (TFTs). We will focus on just one example, that is abelian Chern-Simons theories, and explicitly construct the TFT as a functor from the category of 3-dimensional bordisms to the category of (finite-dimensional) vector spaces. Beware that the construction heavily relies on ideas from physics, like path integrals and quantisation.

Finally, we’ll go back to the classification problem of TFTs and show that the functor viewpoint does not capture all the information. In particular, a rich structure - a semi-simple braided monoidal category - emerges when we consider Wilson lines, i.e. operators supported on 1-dimensional submanifolds. 

This observation motivates the definition and the study of extended TFTs as a higher categorical version.

January 31st, 2025

Speaker: Tommaso Pedroni (SISSA)

Title: Studying Fuchsian Differential Equations through 2d CFT

Abstract: In this seminar, we will discuss the study of Fuchsian differential equations using methods derived from two-dimensional Conformal Field Theory (2d CFT). We will begin with a brief introduction to 2d CFT, including an overview of the representation theory of the Virasoro algebra, with a particular focus on the concept of degenerate states. We will then briefly examine correlation functions and the constraints they satisfy, along with the state-operator correspondence. Next, we will explore the Ward identities arising from conformal symmetry and introduce conformal blocks as fundamental solutions to these constraints.

Finally, we will present the Belavin–Polyakov–Zamolodchikov (BPZ) equations. In particular, we will focus on the simplest non-trivial BPZ equation and show how its classical limit yields the most general 2nd order Fuchsian ODE on the sphere. This provides a foundation for studying such equations through the framework of 2d CFT.

February 7th, 2025

Speaker: Leonardo Goller (SISSA)

Title: Monopoles, Laplacians and Lattice Gauge Theory

Abstract: The flux problem is a a famous problem, solved by Lieb, arised in the study of quantum frustrated antiferromagnetism.  He proved that the energy of a system of electrons on a lattice at half filling subjected to a static external magnetic field, which is mathematically described by the sum of the first half eigenvalues of the magnetic graph laplacian, is minimized by tuning a flux of π in each face of the lattice.

What happens when we couple the system to a dynamical Z_2 gauge field whose energy is minimized by 0-flux configurations (flat field configurations)?

In this talk, I will discuss the stability problem of the π-Flux Phase under gauging by showing, using Reflection Positivity techniques, that the energy of the fermions at half-filling in a background of N monopoles increases extensively in N. Furthermore, I will highlight some properties of this topological phase such as the ground state degeneracy depending on the number of spin structures on the torus and the non-trivial braiding properties of its excitations: monopoles and fermions.

Part of this talk is based on the joint work with M. Porta (SISSA) https://arxiv.org/pdf/2501.10065

February 20th, 2025

Speaker: Luca Fiorindo (Università di Genova)

Title: Coding theory and Vasconcelos invariant

Abstract: Coding theory studies methods and algorithms for detecting and correcting errors in message transmission. The Vasconcelos invariant is an algebraic invariant of graded modules over Noetherian graded rings. In this talk, we will follow the red thread of fate linking these two worlds, passing through algebraic geometry. Finally, we will present recent results on the asymptotic behaviour of the Vasconcelos invariant. This is joint work with Dipankar Ghosh.

February 28th, 2025

Speaker: Mohamed Aliouane (SISSA)

Title: Weil conjectures: a story of  math-fiction theorems

Abstract: In the 1940’s André Weil made arguably the most important conjectures in the 20th century, which relate Number Theory, Algebraic Geometry, and Topology.

The conjectures state that the Zeta function of a smooth projective variety (that we will define in the talk)   is a rational function; secondly, it satisfies a nice functional equation that encodes the Euler characteristic of the variety, and finally, this Zeta function satisfies an analogue of the Riemann Hypothesis.

Those conjectures were proved completely in 1974, today although they are not conjectures anymore, mathematicians still call them “Weil Conjectures” (maybe because they still don’t believe that such conjectures were proven!). 

In this talk, we are mostly going to see the history of Weil conjectures, with some applications.

March 7th, 2025

Speaker: Armando Capasso

Title: CW-Complexes and Bigraded Poincaré-Gorenstein Algebras

Abstract: Poincaré-Gorenstein algebras appear as cohomology rings in several categories. For instance, real orientable manifolds, projective varieties, Kähler manifolds, convex polytopes, matroids, Coxeter groups and tropical varieties are examples of categories for which the ring of cohomology is a Poincaré-Gorenstein algebra over an algebraically closed field of characteristic zero. From another point of view, each of these algebras is the quotient of the differential operators ring of polynomial ring modulo the annihilator of a homogeneous polynomial. Focusing on so-called Nagata (bihomogeneous) polynomials, in the same way one has the bigraded Poincaré-Gorenstein algebras. A natural and classical problem consists in understanding their possible Hilbert function, sometimes also called Hilbert vector. A way to compute this vector is to associate a CW-complex to any (bi)homogeneous polynomial. Indeed, the s-vector of a CW-complex associated to a bihomogeneous polynomial coincides with the Hilbert vector of the bigraded Poincaré-Gorenstein algebra associated to the same bihomogeneous polynomial. In the homogeneous case, one has a bound on the entries of the Hilbert vector. This seminar is based on joint work with P. De Poi and G. Ilardi, and on another work in progress.

March 14th, 2025

Speaker: Christian Forero Pulido (SISSA)

Title: The field with One Element: An Oxymoron with Mathematical Depth

Abstract: The so-called "Field with One Element" F_1 is an oxymoron—there is no such thing as a field with just one element. However, various mathematical frameworks suggest the possibility of a generalized algebraic geometry where an object behaving like a field can meaningfully be considered as F_1. In this talk, I will first recall the Weil conjectures and highlight how Deligne’s proof—building on Weil’s ideas—offers the closest approach to the Riemann Hypothesis by interpreting the integers (or rather, a compactification of them) as a curve over F_1. The main focus will be on showing that different geometries can be constructed over F_1, following Tits’ original insight. By studying projective geometries over finite fields F_q and taking the limit as q→1, we arrive at meaningful combinatorial structures. If time permits, I will conclude with a formal discussion of one of the simplest models of geometry over F_1. In my view, these ideas should be part of the general mathematical folklore.

March 21st, 2025

Speaker: Filippo Fila Robattino (SISSA)

Title: Coisotropic Reduction via Cohomological Methods

Abstract: In this talk, I will explore the problem of symplectic reduction in the presence of coisotropic constraints and its resolution through cohomological techniques. Considering the well-known case in which a coisotropic submanifold C is obtained as the zero-locus of a moment map, I will proceed to explain how to obtain the BRST complex, showing that the algebra of functions on the symplectic reduction of C can be identified with the cohomology of an operator known as the BRST charge. Lastly, I will show how this procedure can be generalized to less restrictive cases by means of the BFV formalism. Time allowing, I will provide a paradigmatic application in field theory.

March 28th, 2025

Speaker: Anis Bousclet (SISSA)

Title: The Hitchin-Thorpe inequality: a topological obstruction for a 4-manifold to be Einstein

Abstract: We present an inequality due to Hitchin and Thorpe which relates the signature and Euler characteristic of an Einstein manifold. This gives examples of simply-connected 4 manifolds without Einstein metrics e.g blow up of CP^2 in 9 points. 

The inequality follows from the expression of the signature and Euler characteristic in terms of a suitable irreducible decomposition of the Riemann curvature tensor.

We study carefully the equality case in the Hitchin-Thorpe inequality which is attained if and only if the manifold is flat, or is covered by a K3 surface (simply connected Ricci flat Kahler surface). In this case the fundamental group is either trivial, Z/2Z or Klein four. 

April 4th, 2025

Speaker: Vanja Zuliani (Université Paris-Saclay)

Title: Antisymplectic involutions on moduli spaces of sheaves on a K3 surface

Abstract: This seminar will start with a gentle introduction to hyperkahler manifolds: many hyperkahler manifolds are moduli spaces of sheaves on a K3 surface. We will give examples of antisymplectic involutions on moduli spaces, study the fixed loci of  involutions and give some examples of birational modifications of moduli spaces.

If time permits, we will state the following theorem by Flapan, O'Grady, Macrì, Saccà: with some condition on the polarization of a hyperkahler manifold of K3-type, we have that the fixed locus of the antisymplectic involution has two connected components; one of them is an interesting Fano variety.

April 11th, 2025

Speaker: Pietro Ciusa (SISSA)

Title: An Introduction to Extremal Kahler Metrics

Abstract: A basic problem in differential geometry is to find canonical, or best, metrics on a given manifold. Extremal metrics were introduced by Calabi in the 1980s as an attempt to find canonical Kahler metrics on Kahler manifolds as critical points of a natural energy functional. The most important examples of extremal metrics are Kahler-Einstein metrics and constant scalar curvature Kahler (or cscK) metrics. In this seminar we will briefly introduce the basic objects of Kahler geometry. We will define the concept of extremal metric, with special emphasis on the constant scalar curvature equation.

May 9th, 2025

Speaker: Lorenzo Barbato (SISSA)

Title: A theorem by Duistermaat and Heckman, via equivariant cohomology

Abstract: In symplectic geometry one spontaneously searchs for a construction that makes the orbit space a symplectic manifold: symplectic reduction is a possible answer. This construction relies, above all, on the notion of momentum map for a Hamiltonian action, which generalizes familiar ideas such as linear and angular momentum. In particular, suitable quotients of the preimages of regular values of the momentum map are endowed with a reduded symplectic structure. How does the reduced symplectic form vary if one moves the regular value? Under suitable hypotheses, in the toric case, the Duistermaat-Heckman Theorem states that the difference between the cohomology classes of two reduced symplectic forms, which are compared via parallel translation, is a linear function of the difference of the corresponding two regular values. Equivariant cohomology provides an elegant proof of this theorem.

May 16th, 2025

Speaker: Oliviero Malech (SISSA)

Title: An Elementary Proof of the Jordan Curve Theorem 

Abstract: The Jordan Curve Theorem (JCT) states that any simple closed curve separates the Euclidean plane into two path-connected components. If the curve is assumed to be merely continuous—without requiring smoothness or piecewise linearity—then the JCT becomes surprisingly challenging to prove, even though it may seem "obvious" intuitively. The goal of this talk is to present a direct, elementary, and elegant proof of the theorem. We will avoid any use of algebraic topology, relying instead on basic properties of compactness, connectedness, and continuity, along with a bit of graph theory. As an intermediate step, we will also recover Brouwer’s Fixed Point Theorem, which states that any continuous function from a disk to itself has a fixed point. 

May 23rd, 2025

Speaker: Leonardo Goller (SISSA)

Title: An Introduction to the Quantization Problem

Abstract: At the beginning of the 20th century, Planck, Einstein and Bohr realized that some observed phenomena (the spectrum of the Black Body Radiation, the photoelectric effect and the emission/absorption spectrum of the Hydrogen atom) could be explained only if one admitted that energy could exist as small fixed packages called quanta. Such observations posed a serious challenge to classical physics paradigms and led to the foundations of Quantum Mechanics by Schrodinger, Heisenberg, Dirac, Pauli, Born, Jordan and Von Neumann based on Hilbert spaces and Operators. At the root of Quantum Mechanics, there is the problem of how to obtain such objects. Quantization is a formal scheme proposed by Dirac to assign operators and Hilbert spaces to observables and states of a quantum system starting from known information about the Phase Space (Symplectic Manifold) describing the classical counterpart of such system.

In this talk, I will introduce the problem of quantization, starting from the simplest example of a particle on a line (Cotangent Bundle to R), highlighting some of the criticalities that can arise, and then following with the case of the Sphere. This is a simple example of the so-called geometrical quantization procedure of Kahler Manifolds.

Edition 2023-2024

October 27, 2023

Speaker: Armando Capasso (Università degli Studi Roma Tre)

Title: Positivity Conditions for (Higgs) vector bundles: an overview

Abstract: The positivity conditions for vector bundles in Complex Algebraic Geometry are useful tools which influence the geometry of the base complex variety. For example and just to give an idea: a manifold X is projective if and only if it carries a positive line bundle. 

So a natural question is: how can one extend these notions in presence of a non zero Higgs field on a vector bundle? Bruzzo, Graña Otero, Hernández Ruipérez and myself have attempted to furnish an answer to this problem, with success. We proved almost all the properties for Higgs vector bundles staisfying some of these notion; however the presence of a non zero Higgs field creates a break in the theory.

In this talk I will introduce the Higgs bundles, the positivity conditions for (Higgs) vector bundles, state the basic properties of these objects, and some applications to minimal surfaces of general type. This last part is a cojoint work with Ugo Bruzzo and Beatriz Graña Otero.

November 10, 2023

Speaker: Stefano Galanda (Università degli Studi di Genova)

Title: A disordered introduction to Entropy in QFT

Abstract: Entropy is a notion of interest both in applied mathematics (information theory and statistics) and in theoretical and mathematical physics (thermodynamics, statistical mechanics and its geometrical interpretation in Black-Hole spacetime). In this talk, Entanglement and Relative Entropy are discussed starting from quantum mechanics and generalized to relativistic quantum field theory. The generalization is done using the Algebraic approach to quantum theories and Tomita-Takesaki modular theory therefore, part of the talk is devoted to introducing this axiomatic framework. Finally, explicit expressions to compute relative entropy for free bosonic and fermionic QFT are given, allowing their use in concrete applications.

November 17, 2023

Speaker: Leonardo Goller (SISSA)

Title: Construction of Yang-Mills Theories on Surfaces

Abstract: 4d Yang-Mills theory is the Gauge theory that underlies the Standard Model of particle physics, describing both the weak and the strong forces. Mathematically, it has been used by Simon Donaldson to construct smooth invariants of 4-manifolds and proving the low energy spectrum of the theory is a 1 million dollar worth-problem. 

In this talk, following the work of Migdal and Witten, I will construct its baby version on Riemann surfaces computing explicitly the partition function. 

November 24, 2023

Speaker: Christian Forero Pulido (SISSA)

Title: The stack of triangles

Abstract: In this talk, I will present an introductory exploration and an invitation into the realm of Stacks, a mathematical concept with profound applications in geometry, topology, and theoretical physics. Following the book "Introduction to Algebraic Stacks" by K. Behrend, we will study a fundamental geometric example—the Stack of triangles—as our guide. Delving into the ingenious invention of stacks by the Grothendieck school of algebraic geometry in the 1960s, we will focus in understanding stacks as a powerful framework for studying moduli problems.

Throughout the seminar, we will dissect the stack of Euclidean triangles, and explore its various facets, including the absence of a fine moduli space due to symmetries within isosceles and equilateral triangles. The seminar will take a leisurely and elementary approach, making the content accessible to those with basic backgrounds in abstract algebra and topology. As we progress, we will encounter fine moduli spaces, coarse moduli spaces, and delve into the concept of a versal family, laying the groundwork for a deeper comprehension of stacks and their pivotal role in moduli problems.

December 1, 2023

Speaker: Jacopo Zanchettin (SISSA)

Title: Morita equivalence for the Erhesmann-Schauenburg Hopf algebroid

Abstract: In this talk, I will recall the notion of principal bibundle for commutative Hopf algebroids introduced by El Kaoutit and Kowalzig and adapt it to Schauenburg's Hopf algebroids. Eventually, I will show that any such Hopf algebroid admitting a principal bibundle with a Hopf algebra is isomorphic to the Ehresmann-Schauenburg Hopf algebroid associated with a Hopf-Galois extension. This reproduces the classical result that every Lie groupoid is Morita equivalent to a Lie group if and only if it is the gauge groupoid of a principal bundle. In the last part, I will discuss how to get the same result starting from a monoidal equivalence. This talk is part of a joint work with A. Chirvasitu and M. Tobolsky.

December 15, 2023

Speaker: Thomas Nicosanti (SISSA)

Title: Topological Quantum Field Theory for Everybody

Abstract: Topological Quantum Field Theories (TQFTs) are a rich subclass of Quantum Field Theories (QFTs), originally arisen as the zero-energy of QFTs. Due to their simplicity, which lies in the finite-dimensionality of their Hilbert space of states, these theories are exactly solvable. Nonetheless, they exhibit non-trivial phenomena, making them the perfect toy models for studying properties of QFTs.

Remarkably, TQFTs are of great interest even in mathematics, in particular in the study of topological invariants of manifolds.

While Quantum Field Theories (QFTs) still lack a rigorous mathematical treatment, I will show how one can define a TQFT. We will discuss the desired properties of a TQFT, basing our intuition on the path integral, and then compare them with the rigorous definition, trying to avoid any unnecessary technicality. 

After a quick discussion about the classification of TQFTs, we will deal with the problem of constructing theories in practice. Finally, if time permits, we will apply this procedure by extending a topological quantum invariant of 3-manifolds to a 3-dimensional TQFT.

January 19, 2024

Speaker: Adriano Chialastri (SISSA)

Title: Operator monotone functions and the magic of Loewner's theorem

Abstract: Operator monotone functions are the extension to self-adjoint matrices of the usual concept of monotonically increasing functions over the reals. While such a generalization may seem trivial at first glance, it presents some unexpected difficulties. First, the class of operator monotone functions over some domain is much smaller than the one of monotonically increasing ones! Many of the usual examples fail even for 2x2 matrices. Second, it is in general exceptionally hard to prove that some function is operator monotone. Luckily, we have Loewner's theorem to help us: in this talk, starting from the basics and going through some examples, we will see how operator monotonicity is equivalent to a powerful analytic condition, which makes it much easier to handle these issues.

January 26, 2024

Speaker: Elisa Vitale (SISSA)

Title: A brief Grothendieck ring of varieties

Abstract: The Grothendieck ring is generated by isomorphism classes of varieties up to "cut-and-paste" relations, with multiplication given by product of varieties. These defining properties make it a "universal" Euler characteristic, in the sense that it refines other additive invariants. Despite its elementary definition, the Grothendieck ring turns out to be a surprisingly interesting object because of its interplay with nontrivial geometric results. Moreover, the structure of this ring can be enriched to become a powerful and practical tool with applications to Geometry and Number Theory.

This talk is meant to be a guide to the Grothendieck ring of varieties, starting from its definition and basic properties, accompanied by plenty of examples. In the first part of the seminar we will explore some natural questions arising from these properties. The second part of the seminar will be reserved for more technical remarks including: a proof, some applications and generalizations.

February 2, 2024

Speaker: Matteo Testa (SISSA)

Title: Homotopy reconstruction from point clouds

Abstract: When studying point cloud data, it is sometimes important to look at more geometrical aspect. The most famous tool to describe the topology of point clouds is Persistence Homology. In this talk I will present an alternative approach based on the work of Smale Niyogi and Weinberger. Their method requires more precise information on the underlying manifold but gives more quantitative probability estimates for the required sample size. In the end I will talk of how this result can be generalized to obtain topological information also on maps between manifolds, we will also see, as a more theretical application of this approach, a bound for the number of homotopy types of Lipschitz maps between two fixed manifolds.

February 9, 2024

Speaker: Younes El Maamoun Benyahia (SISSA)

Title: Tailoring a 4-manifold for the everyday group

Abstract: Given a class of manifolds it is sometimes natural to ask what their possible homotopy types are. In particular, what groups possibly arise as their fundamental groups?

In this talk, we will introduce and discuss some cut and paste constructions which we will use to see how any finitely presented group G arises as a fundamental group for: a closed 4-manifold, a closed symplectic 4-manifold and then a closed exotic 4-manifold (a manifold admitting inequivalent smooth structures).

February 16, 2024

Speaker: Ian Selvaggi (SISSA) 

Title: Quick-and-dirty introduction to Chow groups

Abstract: For a projective variety, the interactions between Hodge theoretic invariants and its Chow ring is a surprisingly rich and still mysterious topic within complex algebraic geometry. After a short introduction to the language, I will show some know examples on algebraic curves and motivate various issues happening in higher dimension. In particular, I will focus on Mumford's result on Chow group of zero-cycles for surfaces of positive geometric genus and their non representability, and time permitting ideas on Bloch-Beilinson conjectures. 

February 23, 2024

Speaker: Sarah Eggleston (Osnabrück University)

Title: The amoeba dimension of a linear space

Abstract: Given a complex vector subspace V of C^n, the dimension of the amoeba of V∩(C^∗)^n depends only on the matroid that V defines on the ground set {1, . . . , n}. Here we prove that this dimension is given by the minimum of a certain function over all partitions of the ground set, as previously conjectured by Rau. We also prove that this formula can be evaluated in polynomial time.

March 1, 2024

Speaker: Giulio Grammatica (Sorbonne University)

Title: De Rham cohomology of algebraic varieties

Abstract: let X be a smooth complex algebraic variety. Grothendieck established the stunning fact that the de Rham cohomology of X - defined in a purely analytic way - can be recovered using only Kähler differentials. Only two years later, he found another algebraic description of de Rham cohomology which avoids differential forms altogether and also works for a singular X. I wish to explain these two results in my talk, and develop the formalism of Grothendieck topologies which is essential to the latter. Time permitting, I will mention a third, stacky description of de Rham cohomology that lays ground for very recent advances in arithmetic geometry.

March 8, 2024

Speaker: Mohamed Aliouane (SISSA) 

Title: The amazing story of cubic surfaces

Abstract: The story started  in 1849 when Caylay and Salmon proved that every cubic surface (projective and smooth) over C contains 27 lines . Nine years later Schläfli studied cubic surfaces over R and  gave an answer about the possible number of lines in such surfaces. Then in 1949 somewhere in Italy, Segre gave a more general theorem on the number of lines in cubic surfaces over any field.

Since that time cubic surfaces have been studied intensively, under several aspects. In this talk, we will discuss some interesting geometric and arithmetic facts about cubic surfaces and, hopefully, by the end, cubic surfaces will be some of your favorite mathematical objects.

March 15, 2024

Speaker: Filippo Fila Robattino (SISSA) 

Title: Reduced phase space for Lagrangian field theories on manifolds with boundary

Abstract: In the context of Lagrangian field theory, the reduced phase space (RPS) corresponds, roughly speaking, to the space of Cauchy data of the theory. It can be endowed with the structure of symplectic manifold (the phase space), on which the corresponding Hamiltonian description of the theory can be studied, providing the starting point towards quantization.

In this talk, I will present a systematical method to obtain the RPS from a Lagrangian field theory on a manifold with boundary, using a construction due to Kijowski and Tulcjiew. In the remainder of the talk, I will consider a particular case (General Relativity in d=4), for which the RPS turns out to be non-smooth. If time allows it, I will show how in this case it is possible to recover the algebra of functions on the RPS as the degree zero cohomology of a differential operator defined on a (super)extension of the original RPS, employing the BFV (Batalin-Fradkin-Vilkovisky) formalism.

March 22, 2024

Speaker: Filippo Bianchi (Università degli Studi di Pisa) 

Title: Spin 4-manifolds from below

Abstract: 4-manifolds are complicated; surfaces, not so much. Therefore, it is great when you have a tool to encode some 4-dimensional information into 2-dimensional data. Lefschetz fibrations are 4-manifolds equipped with precisely such a tool, and they will be the main topic of this talk. In particular, I will explain how this works for spin structures.

April 5, 2024

Speaker: Emanuele Pavia (SISSA) 

Title: Derived methods in deformation theory or: how I learned to stop worrying and love derived geometry

Abstract: For more than half a century, deformation theory has represented one of the most relevant areas of research in algebraic geometry and mathematical physics. The quest for extending the domain of definition of some kinds of structures over arbitrary geometric objects via small or infinitesimal perturbations has provided key insights into both the study of local properties of moduli spaces and to the problem of quantization of classical mechanics. What is more, deformation theory has been one of the main reasons why algebraic geometers started heavily employing homotopical and derived methods in their work. This talk has to be interpreted as a brief fairy tale, without any pretension of completeness and technical rigor whatsoever, on how ideas from deformation theory, homotopical algebra, and derived algebraic geometry inexorably converged paving the way for the birth of derived deformation theory, starting from the groundbreaking work of Kodaira and Spencer in the Fifties and (hopefully) getting to some of the new frontiers on the subject.

April 19, 2024

Speaker: Ayush Singh (SISSA) 

Title: A Quantum Theory of... Knots?

Abstract: The standard story of quantum field theory and knot invariants is usually an account of Witten's, rather unexpected, discovery that vacuum expectation value of a Wilson loop in a Chern-Simons theory computes the Jones polynomial of a knot at a root of unity. However, in this talk---with the benefit of hindsight---I would like to argue that this relationship is natural. And by describing a TQFT of knots, I will demonstrate that the desired features of knot polynomials and (topological) QFT path integrals are the same.

May 10, 2024

Speaker: Bruno Renzi (Universita' degli studi di Milano Statale) 

Title: Universality in interacting dimers at the liquid-frozen transition.

Abstract: A central issue in equilibrium statistical mechanics is the universality of critical phenomena.  In the context of two-dimensional lattice models, we will examine the dimer model. A simple model for (among other things) discrete random surfaces, it was first solved in 1961 for planar graphs by Kasteleyn and Temperley and Fisher. In more recent years (2006), Kenyon, Okounkov and Sheffield [KOS] provided a complete characterization, for doubly periodic planar graphs, of its phase diagram in terms of correlations and fluctuations of the associated surface. They unveiled a deep geometric structure and, in the so-called rough phase, a universal Gaussian limit for surface fluctuations. In the presence of perturbations that break integrability, there are far fewer rigorous results. In very recent years it has been shown that a weaker form of universality is to be expected: in the rough phase, the critical exponents may vary with the interaction strength, but they are remarkably related by simple algebraic relations, so that a Gaussian limit for the surface can still be found [GMT-GRT]. In this talk, after an accessible overview of statistical mechanics and the topic of universality, we discuss the rough-to-frozen transition of the dimer model and a connected (strong-) universality result for the free energy.

[KOS] R. Kenyon, A. Okounkov, S. Sheffield, Dimers and Amoebae, AM 163 (2006).

[GMT] A. Giuliani, V. Mastropietro, F. L. Toninelli Non-integrable Dimers: Universal Fluctuations of Tilted Height Profiles, CMP 377 (2020).

[GRT] A. Giuliani, B. R., F. L. Toninelli, Weakly non-planar dimers, PMP 4 (2023).

May 13, 2024

Speaker: Feliz Thimm (University of Oslo)

Title: Wall-Crossing in Enumerative Geometry

Abstract: In enumerative geometry, we count various types of objects, for example curves, vector bundles, or quiver representations. To do so, we consider so-called moduli spaces, which parametrize such objects and consider invariants counting the "size" of the moduli space. To obtain well-defined invariants, we restrict to sub-moduli spaces, which parametrize geometrically meaningful objects. Abstractly, this is done by restricting to so-called semi-stable objects for a certain stability condition. We will introduce stability conditions, primarily focused on simple examples. When varying parameters of the stability condition, we change the moduli space, i.e. the objects we count. This leads to the natural question of how invariants change under a change of stability condition. The answer to this is given by so-called wall-crossing formulas. These can be used to show relations between different invariants and even help compute certain invariants. We will introduce the basic ideas of wall-crossing and present some applications. 

May 31, 2024

Speaker: Chiara Meroni (ETH-ITS Zurich) 

Title: The best ways to slice a polytope 

Abstract: What is the largest slice of a cube in any dimension? What if we substitute the cube with any other polytope? The goal of this presentation is to answer these and similar questions. In a joint work with Marie-Charlotte Brandenburg and Jesús A. De Loera, we obtain a parametric, semialgebraic description of properties of the hyperplane sections of a polytope. Using this structure, we provide algorithms for the optimization of several combinatorial and metric properties over all hyperplane slices of a polytope. This relies on four fundamental hyperplane arrangements, and connects to current hot topics in convex analysis. 

Edition 2022-2023

January 20, 2023

Speaker: Alessandro Lehmann (UAntwerp, SISSA)

Title: Hochschild cohomology, algebraic deformation theory and the curvature problem (Part 1)

Abstract: I will introduce the basics of the deformation theory of algebras, and its relation to the Hochschild complex. I will then show how this generalizes to differential graded objects, and explain what is the curvature problem in this setting.

Format: first of two 1h talks, introductory.

January 27, 2023

Speaker: Alessandro Lehmann (UAntwerp, SISSA)

Title: Hochschild cohomology, algebraic deformation theory and the curvature problem (Part 2)

Abstract: In this second talk I will explain some further properties and characterizations of the Hochschild complex, and its geometric interpretation via the HKR theorem. Finally, I will define the Hochschild cohomology of a differential graded algebra and its relation to their deformation theory.

Format: second of two 1h talks, specialized.

February 6, 2023

Speaker: Valentina Bais (SISSA)

Title: Stiefel's parallelizability theorem for closed orientable 3 manifolds

Abstract: A theorem by Stiefel asserts that any closed orientable 3 manifold Is parallelizable. One can find many proofs of such a result in the literature. After giving some preliminary notions about Heegaard splittings of closed orientable 3 manifolds, I will sketch a Heegaard splitting-based proof of this result. This work was developed under the supervision of prof. Daniele Zuddas (UNITS).

Format: 1h

February 13, 2023

Speaker: Hamza Ounesli (ICTP, SISSA)

Title: Geometric dynamical systems and ergodic theory

Abstract: In this seminar, during the first part I will mainly introduce the field of dynamical systems, how it immerged from natural phenomenon in celestial mechanics, weather ..etc into pure mathematics! The second part will be more focused on my work: I will present a class of maps on closed manifolds (which can be continuous, diffeomorphisms, homeomorphisms, flows...) and explain what dynamical information one would like to know about them. I will present some classical results and some of the questions I am particularly interested in. This seminar is informal, meaning any interactions within the seminar are most welcome! My hope is to convince you that dynamical systems is a dense subset of mathematics, since it is sometimes perceived as a closed subset 😄 .

Format: 45 min (introductory) + 10 min break + 45 min (specialized).

February 20, 2023

Speaker: Riccardo Ontani (SISSA)

Title: Equivariant cohomology and Atiyah-Bott localization

Abstract: Equivariant cohomology is, informally, a cohomology theory for spaces with a group action. In this talk I’ll introduce what equivariant cohomology is and I’ll prove an important related result, the Atiyah-Bott localization theorem, which gives a useful formula to integrate cohomology classes on smooth manifolds with a torus action. Finally I’ll discuss simple examples and, depending on time, some applications to current research.

Format: 45 min + 10 min break + 45 min

February 22, 2023

Speaker: Sara Perletti (UniMiB)

Title: The hierarchy of topological type and the equivalence conjecture

Abstract: Cohomological field theories (CohFTs) were introduced by Kontsevich and Manin in 1994. In 2001 Dubrovin and Zhang defined an integrable hierarchy starting from a semisimple CohFT. In 2015 a new construction of an integrable hierarchy by Buryak appeared. It is conjectured that these two constructions are related by a so-called Miura transformation. In the first part of this seminar, we will define the hierarchy introduced by Dubrovin and Zhang and we discuss the above equivalence conjecture. In the second part, we will explicitly show the equivalence in some specific examples.

Format: 1 hour.

Note: the second part of the talk will be given by Lorenzo Cecchi as a part of the course "CohFT and Integrable hierarchies" by Danilo Lewanski.

February 27, 2023

Speaker: Lorenzo Cecchi (SISSA)

Title: An invitation to infinite dimensional manifolds

Abstract: An infinite dimensional manifold is a smooth manifold locally modelled on topological vector spaces. The idea of infinite dimensional manifold was explicitly mentioned by Riemann in 1854, but it took several years to have a suitable treatment. In this talk I will discuss smooth calculus in TVS, (non-)properties of Fréchet manifolds and Riemannian metrics on them which conjecturally lead to a relationship between unbounded curvature and vanishing geodesic distance.

The talk will be very introductory, so everyone is welcome!

Format: 1h

March 6, 2023

Speaker: Paolo Tomasini (SISSA)

Title: Lie algebras, Lie groups and...?

Abstract: given a Lie group, we can extract it's Lie algebra. In the talk, we'll see how these two structures (Lie algebra and Lie group) are related to equivariant cohomology and equivariant K-theory respectively (and explain what they are in the process), and argue that there is a third player in this game, related to a misterious third example of equivariant cohomology theory. In the second part of the talk, we will see how this topological picture can be translated into algebro-geometric language.

Format: 45 min + 10 min break + 45 min

March 13, 2023

Speaker: Harman Preet Singh (SISSA)

Title: A geometric picture of topological phases of matter

Abstract: Since the discovery of the integer Quantum Hall Effect at the beginning of the ‘80s, a lot of developments occurred in the classification of insulating phases of matter characterised by physical quantities with values independent of the microscopic nature of the material. In this introductory talk, we will see how, given a (non-interacting, spectrally gapped) crystal, to each such phase we may associate, via the Bloch-Floquet transform, a vector bundle over its Brillouin zone, called the Bloch bundle. Then, the physical response functions parametrising the phase may be obtained as topological invariants of this bundle, thus showing manifestly their robustness to smooth deformations. In particular, we will apply these notions to analyse a simple case study in one dimension, the Su-Schrieffer-Heeger (SSH) model.

Format: 1hh

March 20, 2023

Speaker: Vadym Kurylenko (ICTP, SISSA)

Title: Hodge-Deligne Numbers of Hypersurfaces and Newton Polytopes

Abstract: According to Deligne, cohomology of any complex algebraic variety is equipped with a mixed Hodge structure. During the talk we will consider the case of hypersurfaces in algebraic tori. We will follow a paper by Danilov and Khovanskii and compute the Hodge-Deligne numbers using geometry and arithmetic of the corresponding Newton polytopes.

Format: 45 min + 10 min break + 45 min

March 27, 2023

Speaker: Giuliamaria Menara (UniTs)

Title: Magnitude homology for subgraph analysis

Abstract: Hepworth, Willerton, Leinster and Shulman introduced the magnitude homology groups for enriched categories, and in particular for graphs. Although the construction of the boundary map suggests magnitude homology groups are strongly influenced by graph substructures it is not straightforward to detect such subgraphs. In this talk we will introduce magnitude homology, define the eulerian magnitude complex and show how it enables a more accurate analysis of graph substructures.

Format: 1h

April 3, 2023

Speaker: Oliviero Malech (SISSA)

Title: The good, the bad and the geometric: an orbifold story

Abstract: An orbifold is a nice topological space that locally looks like a quotient of the euclidian space by some finite group action, but a good way to think about it is as a manifolds with singularities.  In this thalk I will introduce the notions of good, bad and geometric orbifold. I will present some examples of orbifolds that are quotient of the Euclidian plane and the 2-sphere by some discrete group of isometry and as application I will sketch a classification of those actions via orbifold topology.

Format: 1 hour

April 17, 2023

Speaker: Giuseppe Orsatti (SISSA)

Title: One, one thousand and infinity solitons: the concept of soliton gas and new prospective

Abstract: Solitons are particular traveling waves that appears as special solutions of Integrable PDEs (such as Korteweg-de Vires, Sine-Gordon, Nonlinear Schrödinger equation,...). In 1971, V. Zakharov introduced the idea of studying how one big soliton interacts with infinitely many other smaller solitons for the Korteweg-de Vires equation, starting the idea of soliton gas. This concept was resumed 30 years later by G. El and, in recent years, studied for other Integrable systems, such as the Nonlinear Schrödinger equation (NLS). In this talk, we will discuss a new way to study soliton gas for the NLS equation by using techniques from Inverse Scattering Theory.

Format: 1 hour

April 28, 2023

Speaker: Dmitrii Rachenkov (SISSA)

Title: Cluster Algebras and beyond 

Abstract: Cluster algebras are commutative rings with a set of distinguished generators having a remarkable combinatorial structure. They were introduced by S. Fomin and A. Zelevinsky in 2000 in the context of Lie theory, but have since appeared in many other contexts, from Poisson geometry to triangulations of surfaces and Teichmüller theory. In my talk I will give an introduction to the subject, with definitions and examples, and prove some significant properties, such as the Lorantness phenomenon on cluster coordinates. If time permits, we will discuss the beyond part.

References: A. Zelevinsky, What is ... a Cluster Algebra, L. K. Williams, Cluster algebras: an introduction

Format: 1 hour

May 8, 2023

Speaker: Lorenzo Sillari (SISSA)

Title: Bott-Chern and Aeppli cohomologies of (almost) complex manifolds

Abstract: Bott-Chern and Aeppli cohomologies encode geometrical and analytical properties of complex manifolds. For example, they naturally appear in the study of Hermitian vector bundles, they characterize ∂∂ ̄-manifolds, and they build a bridge between Dolbeault and de Rham cohomologies. In the first part of the talk we will introduce these cohomologies and give several contexts in which they become useful, focusing on their relations with Dolbeault and de Rham cohomologies. In the second part of the talk, we will discuss the problems occurring in defining a similar object for almost complex manifolds and propose a definition of Bott-Chern and Aeppli cohomologies of almost complex manifolds.

Format: 45 min + 10 min break + 45 min

May 29, 2023

Speaker: Adriano Chialastri (SISSA)

Title: Quantum Mechanics 101: from the basics to the algebras

Abstract: Quantum mechanics is a mathematically well-founded theory, whose formalism mainly relies on functional analysis and the properties of linear operators on infinite-dimensional Hilbert spaces. Where's the catch, then? Well, this is true in the non-relativistic case, where we work with a finite number of degrees of freedom. Stepping into quantum field theory, the mathematical ground gets way muddier, asking for a more careful characterization of the algebraic structures under scrutiny. We will see, by talking about von Neumann entropy, why the situation in QFT is especially tricky and why it is not in other more familiar cases. Join me in this brief but intense journey from the very basics of quantum theory to its mathematical limits.

Format: 1 hour

June 5, 2023

Speaker: Dmitrii Rachenkov (SISSA)

Title: The Dimers, or There and Back Again 

Abstract: A perfect matching of a graph is a subset of edges which covers every vertex exactly once, that is, for every vertex there is exactly one edge in the set with that vertex as endpoint. The dimer model is the study of the set of perfect matchings of a (possibly infinite) graph.

Being a statistical model, in periodic cases it can be also associated with the integrable system. In my talk I will show this construction and its full integrability. Precisely, I will provide enough Hamiltonian functions describing them in terms of Newton polygons. If time permits, we will discuss how to come back from a polygon to a dimer. 

References: arXiv:1107.5588

Format: 1 hour