Schedule for 2024-2025

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 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.

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.

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.

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. 

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 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 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.

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.

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 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

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.

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 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 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.

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.

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.

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.

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 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.