Date/Time : 10 Sep 2021 09:00
Venue : via Zoom
Speaker : YanYan Li, Rutgers University

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https://nus-sg.zoom.us/j/84328425455?pwd=MXpPQWYxZ0hvQ2UzOW1sbUREUG5XZz09

Meeting ID: 843 2842 5455
Passcode: 204458

Abstract:  We will first recall the Nirenberg problem and the Yamabe problem, together with some previous results on the existence and compactness of solutions to the problems. We will then discuss the σ_k-Nirenberg problem and the σ_k-Yamabe problem, the study of which requires solving fully nonlinear elliptic equations involving Schouten tensor on Riemannian manifolds. Some recent works and open problems will be discussed.

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Date/Time : 13 Aug 2021 09:00
Venue : via Zoom
Speaker : Toshiyuki Kobayashi, University of Tokyo

Join Zoom Meeting https://nus-sg.zoom.us/j/83343417958?pwd=S2lJU29jbldIUzFRaWVBYXhHWmhEdz09 Meeting ID: 833 4341 7958 Passcode: 149725 The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as pseudo-Riemannian geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry of general signature, surprisingly little is known about global properties of the geometry even if we impose a locally homogeneous structure. In this colloquium, I plan to discuss two topics. Global geometry: Existence problem of compact manifolds modelled locally on homogeneous spaces, and their deformation theory. Spectral analysis: Construction of periodic eigenfunctions for the (indefinite) Laplacian, and stability question of eigenvalues under deformation of geometric structure by using representation theory.

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Date/Time : 16 Jul 2021 16:00
Venue : via Zoom
Speaker : Anna Wienhard, Heidelberg University

Join Zoom Meeting https://nus-sg.zoom.us/j/81534494468?pwd=dHJqL1NOSjBVR09ZcG9iRk5EN3daZz09 Meeting ID: 815 3449 4468 Passcode: 717949 Semi-simple Lie groups play a role in many areas of mathematics. They are closely related to and naturally arise as isometry groups of symmetric spaces. A key tool in the investigation of discrete subgroups of semisimple Lie groups is the their action on this symmetric space, and on its compactifications. There is a basic dichotomy between rank one Lie groups, when the symmetric space is of strict negative curvature, and Lie groups of higher rank, when the symmetric space is only non-positively curved. For many years, starting with Margulis celebrated superrigidity theorem, rigidity has been the overarching paradigm in higher rank. In recent years we have seen a shift of this paradigm towards a combination of flexibility and rigidity which opened up new research directions. In this talk I will describe some of the key properties of symmetric spaces, describe some of the recent developments, and discuss how the intricate geometry of symmetric spaces can also be leveraged in the context of graph embeddings and in representation learning.

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Date/Time : 04 Jun 2021 16:00
Venue : via Zoom
Speaker : Yves Benoist, CNRS and Université Paris-Saclay

Join Zoom Meeting https://nus-sg.zoom.us/j/85852787585?pwd=MDZWUjV5SlNoaFJDM0trWnRJMWdIdz09 Meeting ID: 858 5278 7585 Passcode: 236000 In this expository talk, I will give three examples where p-adic Lie groups are crucial tools to understand dense subgroups in compact real Lie groups. The first one is the Tits alternative. The second one is the Prasad-Rapinchuk regularity theorem. The third one is the Sarnak spectral gap conjecture for random walk on subgroups with algebraic coefficients.

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Date/Time : 07 May 2021 09:00
Venue : via Zoom
Speaker : Weiqiang Wang, University of Virginia

Join Zoom Meeting https://nus-sg.zoom.us/j/82997442700?pwd=TzI5bGJEM2JMb1o0enFRMFlReEZLZz09 Meeting ID: 829 9744 2700 Passcode: 178346 The original quantum Schur duality (due to Jimbo) concerns about commuting actions on a tensor space of a quantum group and a Hecke algebra of type A. Several years ago, Huanchen Bao and I generalized this to a duality between a Hecke algebra of type B and an i-quantum group arising from quantum symmetric pairs. Both dualities are deeply connected to canonical basis, flag varieties, and Kazhdan-Lusztig theory of type ABC. In this talk, I will explain a new unification of both dualities, which leads to an unexpected generalization of Kazhdan-Lusztig construction. This is joint work with Yaolong Shen.

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Date/Time : 09 Apr 2021 09:00
Venue : via Zoom
Speaker : Irene M. Gamba, University of Texas at Austin

Join Zoom Meeting https://nus-sg.zoom.us/j/85868905071?pwd=SVJOYzdvY3NOSlZJOVVOS0lUelg4Zz09 Meeting ID: 858 6890 5071 Password: 422756 We will focus on recent advances on numerical schemes resolving recent approaches of hybrid computational solvers of two kinds: conservative spectral based and Finite Element Method (FEM) based for the Boltzmann equation binary interactions. We will show that recently developed functional analysis tools enable proofs of consistency schemes, construction of error estimates and convergence to statistical equilibrium results, both, in the conservative spectral method as well as on FEM approach. Finally, a new approach to grazing collision limits will be presented that enable us to calculate the approximation of the Boltzmann to the Fokker Plank Landau model for Coulomb interactions with a velocity dependent Coulomb logarithm as much envisioned by the original calculations of Rutherford and Landau in the first part of the 20th century. Numerical simulations show a clear difference between the classical constant angular cut-off versus the velocity dependent angular cut-off.

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Date/Time : 05 Mar 2021 16:00
Venue : via Zoom
Speaker : Eric Moulines, Ecole Polytechnique

Join Zoom Meeting Meeting ID: 822 6641 6347 Password: 603563 Abstract: Statistical learning is rich in inference problems involving probability distributions (e.g. requiring to sample complex distributions, compute normalizing constants, approximate high-dimensional integrals, etc.). Examples include (Bayesian) uncertainty quantification, inference and sampling of generative models, aggregation of predictors. The classical approach to address these challenging problems in computational statistics is to use Monte Carlo techniques. Markov Chain Monte Carlo algorithms (equipped with numerus bonuses) has long provided the gold standard in this area. The situation radically changes 10 years ago with the development of deep learning: classical MCMC algorithms (Gibbs sampler, simple Metropolis-Hastings methods) fare badly for models now routinely used in today AI. This has triggered a lot of efforts in the last decade, with some initial successes, but opening new research avenues. In this talk, I will describe two approaches to raise (partially !) these issues. I will present a novel generation of methods based on non-equilibrium distributions which might be seen as a way to combine either optimization and / or normalizing flows and can be used both to sample distributions and compute normalizing constants. I will also cover new research directions allowing to enrich variational approaches with MCMC technique, a way to bridge the gap between variational inference and “plain” Monte Carlo.

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Date/Time : 05 Feb 2021 09:00
Venue : via Zoom
Speaker : Chenyang Xu, Princeton University

Join Zoom Meeting Meeting ID: 884 1179 8539 Password: 528772 Abstract: From many view points, all varieties are built from three basic types, depending on the sign of their first Chern class. Those with positive first Chern class are historically named after Italian Mathematician G. Fano. For a long time, it is mysterious to people on how to parametrize Fano varieties, due to some fundamental pathological behaviors for families of such varieties. However, in the last decade, it was realized an extra notion called K-stability, which was first introduced by complex geometers to characterize the existence of KE metric, indeed gives a right framework to construct a ‘good’ parametrizing space, called the K-moduli space. In my talk, I will discuss this with an aim make some main ideas accessible to general audiences.

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Date/Time : 08 Jan 2021 09:00
Venue : via Zoom
Speaker : Wei Zhang , MIT

Join Zoom Meeting Meeting ID: 871 9433 1381 Password: 878103 Abstract: This will be an introductory talk to conjectures on algebraic cycles on varieties over number fields and their relation to L-functions, viewed as a version of the “local-to-global principle”. We will focus on “modular varieties” whose L-functions are known to be related to automorphic representations. They often admit a natural collection of algebraic cycles (e.g., arising from the theory of quadratic forms and hermitian forms over various integer rings). We will indicate a few applications to arithmetic and geometric questions on elliptic curves and K3 surfaces parameterized by modular functions on suitable classical hermitian symmetric domains.

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