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