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Organizer: Francesco Sala
Scientific Committee: Filippo Callegaro, Michele D'Adderio, Giovanni Gaiffi, Andrea Maffei, Francesco Sala, Mario Salvetti
Partially supported by: PRIN 2017YRA3LK "Moduli spaces and Lie Theory"
Organizer: Francesco Sala
Scientific Committee: Filippo Callegaro, Michele D'Adderio, Giovanni Gaiffi, Andrea Maffei, Francesco Sala, Mario Salvetti
Partially supported by: PRIN 2017YRA3LK "Moduli spaces and Lie Theory"
Winter Break
Old Talks
December 9, 2021 at 3.00 pm (GMT+1).
Giovanna Carnovale
Università di Padova, Italy 
Title: Approximations of a Nichols algebra from a geometric point of view
Abstract: The talk is based on an ongoing joint project with Francesco Esposito and Lleonard Rubio y Degrassi. Nichols (shuffle) algebras are a family of graded Hopf algebras (in a braided monoidal category) which includes symmetric algebras, exterior algebras and the positive part of quantized enveloping algebras. They are crucial in the classification of (pointed) Hopf algebras. However, it is very difficult to describe their relations or to estimate their dimensions in general and new tools are very welcome. We will exploit an equivalence due to Kapranov and Schechtman between a category of graded bialgebras in a braided monoidal category \(V\) and the category of factorizable systems of perverse sheaves on all symmetric products \(\mathsf{Sym}^n(\mathbb{C})\) with values in \(V\). I will describe the factorizable perverse sheaves counterpart of some algebraic constructions, including the \(n\)th approximation of a graded bialgebra. Since the image of Nichols algebras through Kapranov and Schechtman equivalence is very precise, we can translate into geometric statements when a Nichols algebra is finitely presented, or coincides with any of its approximations. 
December 2, 2021 at 3.00 pm (GMT+1).
Andrea Appel
Università di Parma, Italy 
Title: Generalized SchurWeyl duality for quantum affine symmetric pairs and orientifold KLR algebras
Abstract: In the work of Kang, Kashiwara, and Kim, the Schur–Weyl duality between quantum affine algebras and affine Hecke algebras is extended to certain KhovanovLaudaRouquier (KLR) algebras, whose defining com binatorial datum is given by the poles of the normalized R–matrix on a set of representations. In this talk, I will describe a boundary version of this construction, providing a Schur–Weyl duality between quantum symmetric pairs of affine type and KLR algebras arising from a framed quiver with an involution. With respect to the KangKashiwaraKim construction, the extra combinatorial datum we take into account is given by the poles of the \(k\)–matrix (that is, a solution of the reflection equation) of the quantum symmetric pair. This is based on joint work in progress with T. Przezdziecki. 
November 25, 2021 at 3.00 pm (GMT+1).
Kyoji Saito
RIMS and Kavli IPMU, Japan 
Title: Elliptic Artin Monoids and Higher Homotopy Groups
Abstract: A half century ago when simply elliptic singularity was introduced, it was a natural question whether its discriminant complement is a \(K(\pi, 1)\) space. At that time, Fulvio Lazzeri suggested a possibility of existence of a nontrivial \(\pi_2\) class by a heuristic argument on the real discriminant complement. In the present talk, I approach this problem from elliptic Artin monoids, where the monoid is defined by generalizing the classical Artin braid relations to the new relations, called elliptic braid relations, defined on elliptic diagrams. Contraly to the classical Artin monoids, the elliptic Artin monoids are not cancellative and their natural homomorphisms to elliptic Artin groups (=the fundamental groups of the elliptic discriminant complements) are not injective (except for rank 1 case). This fact leads to me to a construction of \(\pi_2\)classes in the complement of the discriminant. We conjecture that they are nonvanishing. Then, we reformulate the classical \(K(\pi, 1)\)conjecture for complements of discriminants, whether the discriminant complements are homotopic to classifying spaces associated to the elliptic Artin monoids. 
October 21, 2021 at 3.00 pm (GMT+2).
Michele D'Adderio
Université Libre de Bruxelles, Belgium 
Title: Partial and global representations of finite groups
Abstract: The notions of partial actions and partial representations have been extensively studied in several algebraic contexts in the last 25 years. In this talk, we introduce these concepts and give a short overview of the results known for finite groups. We will briefly show how this theory extends naturally to the classical global theory, in particular in the important case of the symmetric group. This is joint work with William Hautekiet, Paolo Saracco, and Joost Vercruysse. 
October 7, 2021 at 3.00 pm (GMT+2).
Viola Siconolfi
Università di Pisa, Italy 
Title: Ricci curvature, graphs and Coxeter groups
Abstract: I will talk about a notion of curvature for graphs introduced by Schmuckenschläger which is defined as an analogue of Ricci curvature. This quantity can be computed explicitly for various graphs and allows to find bounds on the spectral gap of the graph and isoperimetrictype inequalities. I will present some general results on the computation of the discrete Ricci curvature of any locally finite graph. I will then focus on graphs associated with Coxeter groups: Bruhat graphs, weak order graphs and Hasse diagrams of the Bruhat order. 
December 9, 2020 at 3.00 pm (GMT+1).
Roberto Pagaria
Università di Bologna, Italy 
Title: Representation theory and configuration spaces
Abstract: We introduce the category of finite sets and we study their representations. Then we apply this tool to the study of cohomology of ordered configuration spaces of points on a manifold. In the case of configurations of points on an elliptic curve, we obtain some new results on the Betti numbers and their growth. 
November 11, 2020 at 3 pm (GMT+1).
Paolo Papi
Università di Roma Sapienza, Italy 
Title: On a quadratic polynomial attached to simple Lie algebras
Abstract: We will introduce a polynomial \(p_{\mathfrak{g}}(k)\) attached to a simple Lie algebra \(\mathfrak{g}\). It appears in several different contexts:
The results pertain to joint projects with Kac and Moseneder Frajria. 
May 26, 2020 at 4 pm (GMT+2).
Karim Adiprasito
University of Copenhagen, Denmark, and Einstein Institute for Mathematics, Jerusalem, Israel 
Title: Lefschetz theory beyond positivity
Abstract: I will survey and present techniques for Lefschetz theorems beyond the case of Kaehler manifolds, and in particular discuss the Lefschetz theorem for nonprojective toric varieties. Click here to see the video of the talk.

May 18, 2020 at 4 pm (GMT+2).
Joel Kamnitzer
University of Toronto, Canada 
Title: Levi restriction for Coulomb branch algebras and categorical \(\mathfrak{g}\)actions for truncated shifted Yangians
Abstract: Given a representation \(V\) of a reductive group \(G\), BravermanFinkelbergNakajima defined a Poisson variety called the Coulomb branch, using a convolution algebra construction. This variety comes with a natural deformation quantization, called a Coulomb branch algebra. Important cases of these Coulomb branches are (generalized) affine Grassmannian slices, and their quantizations are truncated shifted Yangians. Motivated by the geometric Satake correspondence, we define a categorical \(\mathfrak{g}\)action on modules for these truncated shifted Yangians. Our main tool is the study of how the Coulomb branch algebra changes when we pass from \(G\), \(V\) to \(L\), \(U\), where \(L\) is a Levi in \(G\) and \(U\) is the invariants for a coweight defining \(L\). 
April 28, 2020 at 5 pm (GMT+2).
Giovanni Paolini
Caltech, USA 
Title: The \(K(\pi, 1)\) conjecture for affine Artin groups
Abstract: Artin groups are a generalization of braid groups, and arise as the fundamental groups of configuration spaces associated with Coxeter groups. A longstanding open problem, called the \(K(\pi, 1)\) conjecture, states that the higher homotopy groups of these configuration spaces are trivial. For finite Coxeter groups, this was proved by Deligne in 1972. In the first part of this talk I will introduce Coxeter groups, Artin groups, and the \(K(\pi, 1)\) conjecture (so that only few topological and combinatorial prerequisites are needed). Then I will outline a recent proof of the \(K(\pi, 1)\) conjecture in the affine case, which is a joint work with Mario Salvetti. 