Date/Time : 09 Feb 2022 16:30
Venue : via Zoom
Speaker : Marcin Lis, University of Vienna

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https://nus-sg.zoom.us/j/87181869950?pwd=L2xWTkV0SFJYUXFYbGJxQ1QyYVhSdz09

Meeting ID: 871 8186 9950
Passcode: 110158

Abstract: Using elementary methods we obtain a power-law lower bound on the two-point function of the planar XY spin model at low temperatures. This was famously first rigorously obtained by Fröhlich and Spencer and establishes a Berezinskii-Kosterlitz-Thouless phase transition in the model. Our argument relies on a new loop representation of spin correlations, a recent result of Lammers on delocalisation of integer-valued height functions, and classical correlation inequalities. This is joint work with Diederik van Engelenburg.

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Date/Time : 09 Feb 2022 16:00
Venue : via Zoom
Speaker : Xie Ruofei,

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https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09

Meeting ID: 830 4925 8042
Passcode: 1729=x3+y3

Abstract: Consider the sum of the series (-1)^{x_n}a_n over n, where x_n is a binary sequence and a_n is a sequence of real numbers.
We say the sequence x_n has convergence property if the sum above converges for every computable sequence of real numbers whose sum \sum_n a_n^2 converges computably.

Downey, Greenberg, and Tanggara showed in their unpublished paper that every Schnorr random series x_n has convergence property.
In this talk, we will focus on convergence property and show its similarities and differences with randomness.

This is joint work with Noam Greenberg.

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Date/Time : 08 Feb 2022 10:00
Venue : via Zoom
Speaker : Theodore Weisman, University of Texas, Austin

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https://nus-sg.zoom.us/j/87511476637?pwd=VURaUDVzaHpMYmduN2Fod0h0Y29sQT09

Meeting ID: 875 1147 6637
Passcode: 468399

Abstract: Anosov representations are a higher-rank generalization of convex cocompact subgroups of rank-one Lie groups. They are only defined for word-hyperbolic groups, but recently Kapovich-Leeb and Zhu have suggested possible definitions for an Anosov representation of a relatively hyperbolic group – aiming to give a higher-rank generalization of geometrical finiteness.

In this talk, we will introduce a more general version of relative Anosov representation which also interacts well with the theory of convex projective structures. In particular, the definition includes projectively convex cocompact representations of relatively hyperbolic groups, and allows for deformations of cusped convex projective manifolds (including hyperbolic manifolds) in which the cusp groups change in nontrivial ways.

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Date/Time : 08 Feb 2022 14:00
Venue : via Zoom
Speaker : Wang Zhicheng, Soochow University

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https://nus-sg.zoom.us/j/83614500983?pwd=U1psUGRnbVRiN0ZyZ2pzWU1aWjNDQT09

Meeting ID: 836 1450 0983
Passcode: 893893

Abstract: In this talk, we will discuss the Gan-Gross-Prasad problem for finite classical groups. Our results provide complete answers for unipotent representations, and we obtain the explicit branching laws for these representations. Moreover, for arbitrary representations, we give a formula to reduce the Gan-Gross-Prasad problem to the restriction problem of Deligne-Lusztig characters.

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Date/Time : 07 Feb 2022 10:00
Venue : via Zoom
Speaker : Xinliang An, National University of Singapore

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https://nus-sg.zoom.us/j/85439997818?pwd=d0xxKy9wZEVYU0dYNXlIRENXVE5GQT09

Meeting ID: 854 3999 7818
Passcode: 723884

Abstract: We construct counterexamples to the local existence of low-regularity solutions to elastic wave equations and to the ideal compressible magnetohydrodynamics (MHD) system in three spatial dimensions (3D). Inspired by the recent works of Christodoulou, we generalize Lindblad’s classic results on the scalar wave equation by showing that the Cauchy problems for 3D elastic waves and for 3D MHD system are ill-posed in $H^3(R^3)$ and $H^2(R^3)$, respectively. Both elastic waves and MHD are physical systems with multiple wave-speeds. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. In particular, when the magnetic field is absent in MHD, we also provide a desired low-regularity ill-posedness result for the 3D compressible Euler equations, and it is sharp with respect to the regularity of the fluid velocity. Our proofs for elastic waves and for MHD are based on a coalition of a carefully designed algebraic approach and a geometric approach. To trace the nonlinear interactions of various waves, we algebraically decompose the 3D elastic waves and the 3D ideal MHD equations into $6\times 6$ and $7\times 7$ non-strictly hyperbolic systems. Via detailed calculations, we reveal their hidden subtle structures. With them we give a complete description of solutions’ dynamics up to the earliest singular event, when a shock forms. This talk is based on joint works with Haoyang Chen and Silu Yin.

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Date/Time : 03 Feb 2022 16:00
Venue : via Zoom
Speaker : Andre Nies, The University of Auckland

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https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09

Meeting ID: 830 4925 8042
Passcode: 1729=x3+y3

Abstract: The K-trivial sets are antirandom in the sense that the initial segment complexity in terms of prefix-free Kolmogorov complexity K grows as slowly as possible. Chaitin introduced this notion in about 1975, and showed that each K-trivial is Turing below the halting set. Shortly after, Solovay proved that a K-trivial set can be noncomputable.

In the past two decades, many alternative characterisations of this class have been found: properties such as being low for K, low for Martin-Loef (ML) randomness, and a basis for ML randomness, which state in one way or the other that the set is close to computable.

Initially, the class of noncomputable K-trivials appeared to be amorphous. More recently, an internal structure has been found. Most of these results can be phrased in the language of a reducibility on the K-trivials which is weaker than Turing’s:

A is ML-below B if each ML-random computing B also computes A.

Bienvenu, Greenberg, Kucera, Nies and Turetsky (JEMS 2016) showed that there an ML complete K-trivial set. Greenberg, Miller and Nies (JML, 2019) established a dense hierarchy of subclasses of the K-trivials based on fragments of Omega computing the set, and each such subclass is an initial segment for ML. More recent results (see arxiv.org/abs/1707.00258, updated and submitted Dec 2020) generalise these approaches using cost functions. They also show that each K-trivial set is ML-equivalent to a c.e. K-trivial.

The extreme lowness of K-trivials, far from being an obstacle, allows for methods which don’t work in a wider setting. The talk provides an overview and discusses open questions. For instance, is ML-completeness an arithmetical property of K-trivials?

This is joint work with Noam Greenberg, Joseph Miller and Dan Turetsky.

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Date/Time : 28 Jan 2022 16:00
Venue : via Zoom
Speaker : Riddhipratim Basu, ICTS-TIFR

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https://nus-sg.zoom.us/j/81804554884?pwd=MEx6eVptVmxYZjNPMjZXZE1IaWxUQT09

Meeting ID: 818 0455 4884
Passcode: 743763

Abstract: First and last passage percolation on the two dimensional Euclidean lattice are canonical models of stochastic planar growth that are generically believed to exhibit universal behaviour in the Kardar-Parisi-Zhang (KPZ) class. This, however, is rigorously known only for a handful of integrable models, e.g., the exponential LPP on the plane. In this talk, I shall describe some results on understanding of the geometry of the geodesics in these exactly solvable models using estimates from integrable probability and illustrate how this geometric understanding, in turn, can be used to obtain more refined information on the passage time landscape.

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Date/Time : 27 Jan 2022 09:00
Venue : via Zoom
Speaker : Promit K. Ghosal, MIT

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https://nus-sg.zoom.us/j/83355567762?pwd=UTFGTTRKQlZxbGhYaU9ZSE1CZmp2Zz09

Meeting ID: 833 5556 7762
Passcode: 947225

Abstract: The Kardar-Parisi-Zhang (KPZ) equation is a fundamental stochastic PDE related to the KPZ universality class. In this talk, we focus on how the tall peaks and deep valleys of the KPZ height function grow as time increases. In particular, we will ask what is the appropriate scaling of the peaks and valleys of the (1+1)-d KPZ equation and whether they converge to any limit under those scaling. These questions will be answered via the law of iterated logarithms and fractal dimensions of the level sets. The talk will be based on joint works with Sayan Das and Jaeyun Yi. If time permits, I will also mention an ongoing work with Jaeyun Yi for the (2+1)-d case.

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Date/Time : 26 Jan 2022 16:00
Venue : via Zoom
Speaker : Zhang Jing, National University of Singapore

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https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09

Meeting ID: 830 4925 8042
Passcode: 1729=x3+y3

Abstract:  Structures like trees, linear orders, partial orders, graphs have been widely studied in different areas of mathematics. The Ramsey-type theorems we will discuss usually take the following form: for any given coloring of the structure, there exists a “large sub-structure” that intersects “very few” color classes. One example is a theorem of Laver that states for any finite coloring of Q x Q (ordered pairs of rationals), there exists X, Y order isomorphic to Q, such that X x Y intersects at most 2 color classes. We will discuss the uncountable analogues of these statements and their consistency. In particular, a diagonal version of the Halpern-Lauchli theorem plays a key role. The differences between countable combinatorics and uncountable combinatorics will be highlighted.

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