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Spring Semester 2023

Date / Time Speaker Title Location
23 February 2023
15:15-16:15 		Iakovos Androulidakis
University of Athens
Event Details

Talks in Mathematical Physics

Title Hypoellipticity and the Helffer-Nourrigat conjecture
Speaker, Affiliation Iakovos Androulidakis, University of Athens
Date, Time 23 February 2023, 15:15-16:15
Location HG G 43
Abstract Hypoelliptic differential operators play a central role in various fields, from stochastic analysis to contact and sub-riemannian geometry. A computable criterion of hypoellipticity was proposed by Helffer and Nourrigat in 1979. In this lecture we will give an overview of hypoellipticity and present the proof of the Helffer-Nourrigat conjecture. This is joint work with Omar Mohsen and Robert Yuncken.
Hypoellipticity and the Helffer-Nourrigat conjectureread_more
HG G 43
23 March 2023
15:15-16:15 		Keyu Wang
Universitè Paris Cité
Event Details

Talks in Mathematical Physics

Title TQ and QQ~ systems for twisted quantum affine algebras
Speaker, Affiliation Keyu Wang, Universitè Paris Cité
Date, Time 23 March 2023, 15:15-16:15
Location HG G 43
Abstract As a part of Langlands duality, certain equations were found in two different areas of mathematics. They are known as Baxter’s TQ systems and the QQ type systems, as they trace back to Baxter’s study on integrable models in the 1970s. During the same decade, similar systems of equations were discovered in the area of ordinary differential equations (ODE) by Sibuya, Voros and others. Today, this remarkable correspondence is realized as a duality between representation theory of nontwisted quantum affine algebras (QAA) and the theory of opers for their Langlands dual Lie algebras. We are interested in this duality when the roles of the affine Lie algebra and its dual are exchanged. When the nontwisted QAA is of type BCFG, its dual will be a twisted QAA. To exchange their roles amounts to studying representations of twisted QAAs. In this talk, we will begin by reviewing this story. We will explain the representation theory of Borel subalgebras of twisted QAAs and the expected relationship between twisted and nontwisted types. We will establish TQ systems and QQ~ systems for twisted QAAs.
TQ and QQ~ systems for twisted quantum affine algebrasread_more
HG G 43
6 April 2023
15:15-16:15 		Tommaso Botta
ETH Zurich
Event Details

Talks in Mathematical Physics

Title Bow varieties, stable envelopes and 3d-mirror symmetry
Speaker, Affiliation Tommaso Botta, ETH Zurich
Date, Time 6 April 2023, 15:15-16:15
Location HG G 43
Abstract Mirror symmetry for 3d N=4 supersymmetric gauge theories has recently received much attention in geometry and representation theory. Theories within this class admit very interesting moduli spaces of vacua, whose most relevant components are called Higgs and Coulomb branches. The study of the mathematics of the Higgs branch was initiated by Nakajima in the 90s, and since then its geometry has proved to be intimately related to the representation theory of Kac-Moody Lie algebras and quantum groups. Only recently, mathematically accurate definitions of the Coulomb branch have been proposed, and their study has started. The main mathematical prediction of 3d mirror symmetry is that the Higgs and Coulomb branches of a pair of dual theories are interchanged. Hence, both pairs of homologous branches (Higgs-Higgs and Coulomb-Coulomb) are expected to share exceptional topological and geometric properties. One of the main predictions of mirror symmetry is that the elliptic stable envelopes, which are certain topological classes intimately related to elliptic quantum groups, are the same after appropriate identifications. In this talk, I will focus on Coulomb and Higgs branches of type A, which are collectively described by a class of varieties known as Cherkis bow varieties, and I will discuss the main ideas behind the proof of mirror symmetry of sable envelopes (joint work in preparation with Richard Rimanyi).
Bow varieties, stable envelopes and 3d-mirror symmetryread_more
HG G 43
20 April 2023
15:15-16:15 		Alexander Veselov
Loughborough University
Event Details

Talks in Mathematical Physics

Title Complex cobordisms, theta divisors and permutohedra
Speaker, Affiliation Alexander Veselov, Loughborough University
Date, Time 20 April 2023, 15:15-16:15
Location HG G 43
Abstract It was known since 1960s (Novikov, Mischenko) that the logarithm of the formal group in complex cobordisms can be written explicitly in terms of the complex projective spaces, but the algebro-geometric nature of the coefficients of the corresponding exponential was not clear until recently. In the talk I will explain that the answer can be given by the smooth theta divisors of principally polarised abelian varieties. It will be shown that the topological characteristics of the theta divisors and their intersections can be expressed in terms of the combinatorics of permutohedra. We reveal also interesting relations between the theta divisors, permutohedral varieties and Tomei manifolds from the Toda lattice theory. The talk is based on joint work with V.M. Buchstaber.
Complex cobordisms, theta divisors and permutohedraread_more
HG G 43
27 April 2023
15:15-16:15 		Miquel Cueca
University of Göttingen
Event Details

Talks in Mathematical Physics

Title Shifted Lagrangian structures in Poisson geometry
Speaker, Affiliation Miquel Cueca, University of Göttingen
Date, Time 27 April 2023, 15:15-16:15
Location HG G 43
Abstract It is well known that BG carries a 2-shifted symplectic structure. In this talk, I will study the shifted lagrangian groupoids of BG. I will show how many constructions on Poisson geometry unify using the language of shifted symplectic groupoids. This is work in progress with Daniel Alvarez and Henrique Bursztyn.
Shifted Lagrangian structures in Poisson geometryread_more
HG G 43
18 May 2023
15:15-16:15 		Camilo Arias Abad
Universitdad Nacional de Colombia en Medellin
Event Details

Talks in Mathematical Physics

Title Singular Chains on Lie groups, the Cartan relations and Chern-Weil theory
Speaker, Affiliation Camilo Arias Abad, Universitdad Nacional de Colombia en Medellin
Date, Time 18 May 2023, 15:15-16:15
Location HG G 43
Abstract The Lie algebra of vector fields on a manifold acts on differential forms by Lie derivatives and contractions, and these operations are related by the Cartan relations. We will explain an interpretation of these relations from the point of view of Lie theory, and describe how this leads to a categorification of the Chern-Weil homomorphism. For a Lie group G, we consider the space of smooth singular chains C(G), which is a differential graded Hopf algebra. We show that the category of sufficiently local modules over C(G) can be described infinitesimally, as the category of representations of a dg-Lie algebra which is universal for the Cartan relations. If G is compact and simply connected, the equivalence of categories can be promoted to an A-infinity equivalence of dg-categories, which are also A-infinity equivalent to the category of infinity local systems on the classifying space BG. The equivalence can be realized explicitly to provide a categorification of the Chern-Weil homomorphism. The talk is based on joint works with A. Quintero and S. Pineda, and work in progress with M. Rivera and F. Bischoff.
Singular Chains on Lie groups, the Cartan relations and Chern-Weil theoryread_more
HG G 43
1 June 2023
14:00-15:00 		Ping Xu
Pennsylvania State University
Event Details

Talks in Mathematical Physics

Title Derived differentiable manifolds
Speaker, Affiliation Ping Xu, Pennsylvania State University
Date, Time 1 June 2023, 14:00-15:00
Location HG G 43
Abstract One of the main motivations behind derived differential geometry is to deal with singularities arising from zero loci or intersections of submanifolds. Both cases can be considered as fiber products of manifolds which may not be smooth in classical differential geometry. Thus we need to extend the category of differentiable manifolds to a larger category in which one can talk about "homotopy fiber products". In this talk, we will discuss a solution to this problem in terms of dg manifolds. The talk is mainly based on a joint work with Kai Behrend and Hsuan-Yi Liao.
Derived differentiable manifoldsread_more (CANCELLED)
HG G 43
1 June 2023
15:15-16:15 		Benoit Dherin
Google
Event Details

Talks in Mathematical Physics

Title Deep learning basics and the problem of implicit regularization
Speaker, Affiliation Benoit Dherin, Google
Date, Time 1 June 2023, 15:15-16:15
Location HG G 43
Abstract In the first part of this talk, we will recall the building blocks of deep learning, framing the learning problem as an optimization problem solved in practice by gradient descent. This first part will be very accessible and self-contained. Then we will attempt to convey how surprising it is that deep learning works so well given the extreme complexity of its solution space, pointing toward the existence of an implicit regularization mechanism self-selecting the simpler solutions that generalize best ahead of the more complex ones that do not perform well. At last, we will outline a recent approach attempting to uncover such an implicit regularization mechanism based on the backward error analysis of the gradient descent scheme.
Deep learning basics and the problem of implicit regularizationread_more
HG G 43
1 June 2023
17:15-18:15 		Benoit Dherin
Google
Event Details

Talks in Mathematical Physics

Title Why neural networks find (geometrically) simple solutions
Speaker, Affiliation Benoit Dherin, Google
Date, Time 1 June 2023, 17:15-18:15
Location Y27 H 12
Abstract We will start by defining a notion of geometric complexity for neural networks based on intuitive notions of volume and energy. This will be motivated by the visualization of training sequences in the case of simple 1d neural regressions. Then we will explain why for neural networks the optimization process creates a pressure to keep the network geometric complexity low. Additionally, we will see that many other common heuristics in the training of neural networks (from initialization schemes to explicit regularization strategies) have as a side effect to also keep the geometric complexity of the learned solutions low. We will conclude by explaining how this points toward a preference toward a form of harmonic maps built in the commonly used training and tuning heuristics in deep learning.
Why neural networks find (geometrically) simple solutionsread_more
Y27 H 12
8 June 2023
15:15-16:15 		Alexander Varchenko
University of North Carolina, Chapel Hill
Event Details

Talks in Mathematical Physics

Title Polynomial solutions of KZ equations modulo an integer
Speaker, Affiliation Alexander Varchenko, University of North Carolina, Chapel Hill
Date, Time 8 June 2023, 15:15-16:15
Location HG G 43
Abstract I will review the construction of polynomial solutions of KZ equations modulo an integer and the properties of the solutions, in particular, their p-adic limit.
Polynomial solutions of KZ equations modulo an integer read_more
HG G 43
22 June 2023
15:15-16:15 		Sergei Merkulov
Uni Luxembourg
Event Details

Talks in Mathematical Physics

Title On the interrelations between graph complexes
Speaker, Affiliation Sergei Merkulov, Uni Luxembourg
Date, Time 22 June 2023, 15:15-16:15
Location HG G 43
Abstract We study Maxim Kontsevich's graph complex as well as its oriented and targeted versions, and show a new proof of the theorems due to Thomas Willwacher and Marko Zivkovic stating isomorphisms of their cohomology groups. Both theorems follow from one and the same surprisingly short argument.
On the interrelations between graph complexesread_more
HG G 43

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