Date/Time : 15 Nov 2022 15:00
Venue : S17 #04-04 (SR3)
Speaker : Timo Takala, Aalto University

The John-Nirenberg space JNp is a space of functions with bounded oscillation depending on a parameter p. It is a generalization of the space of bounded mean oscillation BMO. The most profound result on JNp functions is probably the John-Nirenberg inequality, which shows that the space JNp is embedded in the weak Lp space. In this talk I will present the space JNp and some of its fundamental properties. I will also show my recent results on weak Lp type integrals of JNp functions.

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Date/Time : 11 Nov 2022 16:00
Venue : via Zoom
Speaker : David Hilditch, The University of Lisbon

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https://nus-sg.zoom.us/j/84455764091?pwd=WDF6dWJFVHVFNHZSWHY3VlNWM0xodz09

Meeting ID: 844 5576 4091
Passcode: 171086

In my talk I will consider the initial value problem in general relativity, with smooth initial data tuned to the threshold of black hole formation. Starting in the early 1990s with the work of Choptuik such setups have been studied numerically with various simplifying assumptions. For instance spherical symmetry has often been assumed and combined with a scalar field matter model to drive the dynamics. In that context a remarkably simple phenomenology manifests near the threshold of collapse, with a naked singularity appearing in the limit of perfect tuning. I will review recent progress in the more subtle setting in which symmetry assumptions are dropped, so as to include the fundamental case of pure gravitational waves collapsing under their own self-gravity.

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Date/Time : 11 Nov 2022 14:00
Venue : via Zoom
Speaker : Liping Li, Hunan Normal University

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https://nus-sg.zoom.us/j/85349169300?pwd=bE9Ed2ZLelNHdVpjSFV5YnEwdlJpUT09

Meeting ID: 853 4916 9300
Passcode: 176749

Abstract: In this talk we describe an approach to classify irreducible discrete representations of infinite linear groups, based on Artin’s theorem connecting discrete representations of topological groups and sheaves of modules over atomic sites as well as some recent development in representation stability theory.

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Date/Time : 10 Nov 2022 10:00
Venue : via Zoom
Speaker : Yoonsang Lee, Darthmouth College

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https://nus-sg.zoom.us/j/9775606531?pwd=cVZ6eTRYS3ZzVUlFNXRuSlhMVjJSUT09

Meeting ID: 977 560 6531
Passcode: 123321

We propose a neural network-based approach to the homogenization of multiscale problems. The proposed method uses a derivative-free formulation of a training loss, which incorporates Brownian walkers to find the macroscopic description of a multiscale PDE solution. Compared with other network-based approaches for multiscale problems, the proposed method is free from the design of hand-crafted neural network architecture and the cell problem to calculate the homogenization coefficient. The exploration neighborhood of the Brownian walkers affects the overall learning trajectory. We determine the bounds of micro- and macro-time steps that capture the local heterogeneous and global homogeneous solution behaviors, respectively, through a neural network. The bounds imply that the computational cost of the proposed method is independent of the microscale periodic structure for the standard periodic problems. We validate the efficiency and robustness of the proposed method through a suite of linear and nonlinear multiscale problems with periodic and random field coefficients.

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Date/Time : 10 Nov 2022 15:30
Venue : S17 #05-11 (SR5)
Speaker : Fabrizio Bianchi, University of Lille

Abstract:  We discuss the stability of holomorphic dynamical systems under perturbation. In dimension 1, the theory is now classical and is based on works by Lyubich, Mané-Sad-Sullivan and DeMarco. I will review this theory and present a recent generalisation valid in any dimension. Since classical 1-dimensional techniques no longer apply in higher dimensions, our approach is based on ergodic and pluripotential methods.

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Date/Time : 09 Nov 2022 14:00
Venue : S17 #04-04 (SR3)
Speaker : Debsoumya Chakraborti, Institute for Basic Science

Abstract:  There have been extensive studies on the following question: given k graphs G_1,… , G_k over a common vertex set of size n, what conditions on G_i ensure a rainbow copy of H, i.e., a copy of H with at most one edge from each G_i? In this talk, we will survey some results on this topic.

Joos and Kim showed that for k=n, the condition that each G_i has minimum degree at least n/2 confirms a rainbow Hamiltonian cycle. This generalizes the classical Dirac’s theorem. In this talk, we will discuss our recent rainbow generalizations of some results on embedding Hamiltonian cycles in tournaments. This part of the talk will be based on a joint work with Jaehoon Kim, Hyunwoo Lee, and Jaehyeon Seo.

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Date/Time : 09 Nov 2022 17:15
Venue : S17 #04-05 (SR2)
Speaker : Benjamin T Castle, Ben Gurion University of the Negev

Abstract: Motivated by recent progress toward Zilber’s Restricted Trichotomy Conjecture, we study reducts of the complex field up to interdefinability over parameters. Precisely, we will consider structures of the form (C, P_1,…,P_n), where C is set of complex numbers and the P_i are polynomial maps of potentially different arities.   Somewhat surprisingly, our main result states that almost all such structures (in a precise sense) are interdefinable. The proof uses a mix of tools from geometric stability theory, combinatorics, and algebraic geometry.

This is joint work with Chieu-Minh Tran.

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Date/Time : 04 Nov 2022 15:00
Venue : via Zoom
Speaker : Ting-Kam Leonard Wong, University of Toronto

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https://nus-sg.zoom.us/j/83007497441?pwd=cmJmSEpuTWRUVHhJc3JJZ2h3dFNZdz09

Meeting ID: 830 0749 7441
Passcode: 087943

Information geometry is the geometric study of statistical manifolds which are spaces of probability or nonnegative measures. It originated from statistical inference and has been applied in diverse fields including statistical physics and machine learning. This short course is an introduction to information geometry and some of its recent developments. Mathematical prerequisites such as differential geometry and convex analysis will be introduced as needed. In the first part, we will introduce statistical manifolds, divergence functionals and the induced dualistic geometry. In the second part, we will discuss some relations between information geometry and optimal transport.

Further update : Information and updates about this course (including current and future Zoom links) will also be available on the webpage https://ujangangopadhyay.github.io/information_geometry.html

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Date/Time : 04 Nov 2022 16:00
Venue : S17 #04-06 (SR1)
Speaker : Li Te, National University of Singapore

Hydrodynamic stability at high Reynolds number is a central topic in fluid mechanics, which is related to turbulence. The stability of general flows and their transition are challenging problems. We develop a systematic method to establish some sharp resolvent estimates for the linearized operator around some typical shear flows and rotating flows. One of the main difficulties is that the linearized operator is non-self-adjoint and non-local such that the spectral analysis becomes very complicated. Based on the resolvent estimates, we first show the sharp enhanced dissipation rate of the solution for the linearized system. We also derive the space-time estimates of the nonlinear part using the resolvent estimates. Combining this, we obtain the nonlinear transition threshold of some typical shear flows and rotating flows.

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