Date/Time : 15 Oct 2020 10:00
Venue : via Zoom
Speaker : Zhilin Luo, University of Minnesota--Twin Cities

Join Zoom Meeting https://nus-sg.zoom.us/j/94022303433?pwd=b3hSZWpNTVN2VW9sU21XdThEdlJ0QT09 Meeting ID: 940 2230 3433 Password: 100202 Abstract: The local Gan-Gross-Prasad conjecture studies the restrictioin and branching problems for representations of classical and metaplectic groups . In this talk, I will talk about my proof for the tempered part of the local Gan-Gross-Prasad conejecture (multiplicity one in Vogan packets) for special orthogonal groups over any local fields of characteristic zero, which combines the work of Waldspurger (for the tempered part of the conjecture for special orthogonal groups over p-adic fields) and Beuzart-Plessis (for the tempered part of the conjecture for unitary groups over real fields) in a non-trivial way. In the proof, an indispensable result which is also of independent interest is a formula expressing the regular nilpotent germs of quasi-split reductive Lie algebras over any local fields of characteristic zero via endoscopic invariants, which was previously proved by Shelstad over $p$-adic fields. We also relate the formula with the Kostant’s sections.

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Date/Time : 14 Oct 2020 10:00
Venue : via Zoom
Speaker : Junliang Shen, MIT

Join Zoom Meeting https://nus-sg.zoom.us/j/82961016557?pwd=SC85byszaFlHNDMzMmRBR1p6clNvdz09 Meeting ID: 829 6101 6557 Password: 706382 Abstract:  The moduli space of Higgs bundles and Hitchin‘s integrable system lie at the crossroads of mathematical physics, representaton theory, and geometry. In this talk, we focus on the structure of cohomology for the moduli of Higgs bundle from the viewpoints of the non-abelian Hodge theory, hyper-Kähler geometry, and mirror symmetry. In particular, we will discuss recent progress on the P=W conjecture and Hausel-Thaddeus’ topological mirror symmetry conjecture. Based on joint work with Mark de Cataldo and Davesh Maulik.

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Date/Time : 14 Oct 2020 17:00
Venue : via Zoom
Speaker : Tran Chieu-Minh,

Join Zoom Meeting https://nus-sg.zoom.us/j/96860201432?pwd=cVdwZmd2clVFaEhaTmJjaGdXMFdmdz09 Meeting ID: 968 6020 1432 Password: Is P=NP? Abstract:  Zarankiewicz’s problem in graph theory asks to determine the largest possible number of edges |E| in a bipartite graph G = (V_1, V_2; E) with the parts V_1 and V_2 containing n_1 and n_2 vertices, respectively, and such that G contains no complete bipartite subgraphs on k vertices. Graphs definable in o-minimal (or more generally distal structures) enjoy stronger bounds than general graphs, providing an abstract setting for the Szemeredi-Trotter theorem and related incidence bounds. We obtain almost optimal upper and lower bounds for hypergraphs definable in locally modular o-minimal structures, along with some applications to incidence counting (e.g. the number of incidences between points and boxes with axis parallel sides on the plane whose incidence graph is K_{k,k}-free is almost linear). We explain how the exponent appearing in these bounds is tightly connected to the trichotomy principle in o-minimal structures. Joint work with Abdul Basit, Artem Chernikov, Sergei Starchenko and Terence Tao.

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Date/Time : 14 Oct 2020 16:00
Venue : via Zoom
Speaker : Tiangang Cui, Monash University

Join Zoom Meeting https://nus-sg.zoom.us/j/83653279396?pwd=L2hNSzI5dENjYWpRSkZpOEFMT2NzZz09 Meeting ID: 836 5327 9396 Password: 150910 Characterising intractable high-dimensional random variables is one of the fundamental challenges in stochastic computation. It has broad applications in statistical physics, machine learning, uncertainty quantification, econometrics, and beyond. The recent surge of transport maps offers a mathematical foundation and new insights for tackling this challenge. In this talk, we present a functional tensor-train (FTT) based monotonicity-preserving construction of inverse Rosenblatt transport in high dimensions. It characterises intractable random variables via couplings with tractable reference random variables. By integrating our FTT-based approach into a nested approximation framework inspired by deep neural networks, we are able to significantly expand its capability to random variables with complicated nonlinear interactions and concentrated density functions. We demonstrate the efficacy of the FTT-based inverse Rosenblatt transport on a range of applications in statistical learning and uncertainty quantification, including parameter estimation for dynamical systems, PDE-constrained inverse problems, and Bayesian filtering. This is joint work with Dr. Sergey Dolgov (Bath) and Mr. Yiran Zhao (Monash).

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Date/Time : 09 Oct 2020 09:00
Venue : via Zoom
Speaker : Yilin Wang, MIT

Join Zoom Meeting Meeting ID: 947 9035 5755 Password: 167685  We introduce the Loewner energy for Jordan curves and relate this quantity to ideas and concepts coming from random conformal geometry, geometric function theory, and Teichmuller theory. Many of those concepts arise from mathematical physics, ranging from statistical mechanics, quantum field theory, and string theory. One motivation for the definition of the Loewner energy is that it describes the large deviations of a family of simple random curves called Schramm-Loewner evolutions (SLE).  This provides a probabilistic interpretation of the Loewner energy that allows us to establish several interesting invariance properties. We further derive an equivalent characterization of the Loewner energy using zeta-regularized determinants of Laplacians. This then identifies the family of finite energy curves to the family of so-called Weil-Petersson quasicircles. Moreover, the Loewner energy is shown to equal the Kahler potential, introduced by Takhtajan and Teo, of the unique Kahler metric on the corresponding Weil-Petersson Teichmuller space.

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Date/Time : 08 Oct 2020 10:00
Venue : via Zoom
Speaker : Zhilin Luo, University of Minnesota--Twin Cities

Join Zoom Meeting https://nus-sg.zoom.us/j/91356784967?pwd=Q0ZxYlc5VU9UT3NCczJJNUNYTjlOZz09 Meeting ID: 913 5678 4967 Password: 256173 Abstract: I am going to talk about my joint work with D. Jiang and L. Zhang. We develop a new type of harmonic analysis on an extended symplectic groups G=G_m x Sp_2n over p-adic fields. It is associated to the Langlands gamma-functions attached to irreducible admissbile representations of G(F) and the standard representation rho of the dual group. Our work can be viewed as an extension of the work of Godement-Jacquet (which is a generalization of Tate’s thesis). We confirm a series of conjectures in the local theory of the Braverman-Kazhdan proposal in this setting.

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Date/Time : 07 Oct 2020 09:00
Venue : via Zoom
Speaker : Sara Maloni, University of Virginia

Join Zoom Meeting https://nus-sg.zoom.us/j/86085470675?pwd=VFVkOVJoMGcrWmZpenZBenRScVozdz09 Meeting ID: 860 8547 0675 Password: 058406 Abstract:  Thurston conjectured that quasi-Fuchsian manifolds are determined by the induced hyperbolic metrics on the boundary of their convex core and Mess generalized those conjectures to the context of globally hyperbolic AdS spacetimes. In this talk I will discuss a universal version of these conjectures (and prove the existence part) by considering convex sets spanning quasicircles in the boundary at infinity of hyperbolic and anti-de Sitter space. This work generalizes Alexandrov and Pogorelov’s results about the characterization metrics induced on the boundary of a compact convex subset of hyperbolic space. In addition, in anti-de Sitter space quasicircles can be characterize by the width of their convex hulls, as Bonsante and Schlenker showed. Time permitting, we will discuss why this is not true in hyperbolic space, except when the hulls have small width. (This is joint work with Bonsante, Danciger and Schlenker.)

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Date/Time : 01 Oct 2020 10:00
Venue : via Zoom
Speaker : Hang Xue, University of Arizona

Join Zoom Meeting https://nus-sg.zoom.us/j/97939482481?pwd=OW10bDJ2T2l0M2VFc0oyek1pZXNmUT09 Meeting ID: 979 3948 2481 Password: 721722 Abstract: A classical branching theorem of Weyl determines the decomposition of an irreducible representation of (compact) U(n+1) when restricted to U(n). The local Gan–Gross–Prasad conjecture puts this in the framework of local Langlands correspondence and extends it to the setting of noncompact unitary groups (and also p-adic unitary groups). We present a very simple proof of this conjecture for tempered packets of real unitary groups based on theta correspondences.  

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Date/Time : 30 Sep 2020 17:00
Venue : via Zoom
Speaker : Vasco Brattka, Universtiaet der Bundeswehr, Muenchen

Join Zoom Meeting https://nus-sg.zoom.us/j/96860201432?pwd=cVdwZmd2clVFaEhaTmJjaGdXMFdmdz09 Meeting ID: 968 6020 1432 Password: Is P=NP? Abstract:   We discuss the question whether there is a weakest unsolvable mathematical problem. In recent years the Weihrauch lattice has been established as a suitable computability theoretic framework to analyze the uniform computational content of problems from many different fields of mathematics. Here we answer a question by Schroeder, whether there is a weakest discontinuous problem with respect to the continuous version of Weihrauch reducibility. We introduce the discontinuity problem, and we show that it is reducible exactly to the effectively  discontinuous problems, defined in a suitable way. However, in which sense this answers Schroeder’s question sensitively depends on the axiomatic framework that is chosen and it is a positive answer if we work in Zermelo- Fraenkel set theory with dependent choice and the axiom of determinacy. On the other hand, using the axiom of choice, one can construct problems which are discontinuous, but not effectively so.   Hence, the exact structure of the “bottom” of the Weihrauch lattice sensitively depends on the axiomatic setting that we choose.   We prove our result using Wadge games for mathematical problems and while the existence of a winning strategy for player II characterizes continuity of the problem (as already shown by Nobrega and Pauly), the existence of a winning strategy for player I characterizes effective discontinuity of the problem. We also provide further insights into the algebraic nature of the discontinuity problem. For one we show that the parallelization of the discontinuity problem is exactly the non-computability problem that was studied before. One the other hand, we introduce a new algebraic operation in the Weihrauch lattice that we call summation and which is the dual operation to parallelization. While parallelization can be seen as an analogue of the bang operator in linear logic, summation can be seen as an analogue of the question mark operator. It turns out that the discontinuity problem can be obtained as summation of a number of well-known problems in the Weihrauch lattice, such as the (lesser) limited problem of omniscience and variants thereof. More generally, we study the action of the monoid formed by parallelization and summation on the Weihrauch lattice, and we prove that this action can lead to at most five different Weihrauch degrees, which (in the maximal case) are always organized in a pentagon. We show that the discontinuity problem appears as the bottom of several natural such pentagons. This leads to further interesting characterizations of the discontinuity problem.

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