Schedule for 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.