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