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.

Add to calendar :

Date/Time : 11 Aug 2021 16:00
Venue : via Zoom
Speaker : Frank Stephan, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09 Meeting ID: 830 4925 8042 Password: 1729=x3+y3 Let a positive equivalence relation to be an r.e. equivalence relation on the set of natural numbers with infinitely many equivalence relations. Khoussainov initiated with coauthors a deep study of the following question: Given a positive equivalence relation eta, which structures from a given set of structures does this equivalence relation realise? Here realisation means that functions in the structure are recursive and relations are r.e. with the equality itself given by the equivalence relation eta. In other words, the given r.e. structure divided by eta is the structure realised by eta. Now questions studied by Khoussainov and his coworkers included questions like “What is the partial ordering on positive equivalence relations eta,rho where eta is below rho iff every structure of the given type realised by eta is also realised by rho? Besides algebraic structures and orders, it has also been studied how the learnability notions behave with respect to uniformly r.e. one-one families realised by positive equivalence relations.

Add to calendar :

Date/Time : 30 Jul 2021 09:00
Venue : via Zoom
Speaker : Hao Shen, University of Wisconsin – Madison

Join Zoom Meeting https://nus-sg.zoom.us/j/86176910318?pwd=NnQ5SFcxTk9BRFc1TklwSFRhYS9nZz09 Meeting ID: 861 7691 0318 Passcode: 016860 Systems with random fluctuations are ubiquitous in the real world. Stochastic PDEs are default models for these random systems, just as PDEs are default models for deterministic systems. However, a large class of such stochastic PDEs were poorly understood until very recently: the presence of very singular random forcing as well as nonlinearities render it challenging to interpret what one even means by a “solution”. The recent breakthroughs by M. Hairer, M. Gubinelli and other researchers including the speaker not only established solution theories for these singular SPDEs, but also led to an explosion of new questions. These include scaling limits of random microscopic models, development of numerical schemes, ergodicity of random dynamical systems and a new approach to quantum field theory. In this talk we will discuss the main ideas of the recent solution theories of singular SPDEs, and how these SPDEs arise as limits of various important physical models.

Add to calendar :

Date/Time : 23 Jul 2021 09:00
Venue : via Zoom
Speaker : Jingrui Cheng, State University of New York at Stony Brook

Join Zoom Meeting https://nus-sg.zoom.us/j/86843946871?pwd=eXk1ZmF6OElwY2s3MjdzM1piV2VQQT09 Meeting ID: 868 4394 6871 Passcode: 398037 I will explain the a priori estimates for the cscK equation on a compact manifold, and how to use these estimates to obtain existence when the associated energy functional is “coercive”. If time permits, I will also explain how we can hope to get existence from a more “algebraic” condition, which might be easier to check in practice.

Add to calendar :

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.

Add to calendar :

Date/Time : 18 Jun 2021 16:00
Venue : via Zoom
Speaker : Yash Lodha, Korea Institute for Advanced Study

Join Zoom Meeting https://nus-sg.zoom.us/j/84001105600?pwd=OUc1ZGtOb21KSGtaRHhjN3VWMHJsdz09 Meeting ID: 840 0110 5600 Passcode: 346800 The classification of finite simple groups is one of the most incredible achievements of 20th century mathematics. So what do we know about infinite, finitely generated (or even finitely presented), simple groups? Starting from a beautiful example of Higman from 1951: the first example of a finitely generated infinite simple group, to the striking examples of Thompson, we shall survey this territory full of beautiful phenomena. We shall also discuss some questions that remain mysterious.

Add to calendar :

Date/Time : 17 Jun 2021 10:00
Venue : via Zoom
Speaker : Rui Wang, University of California at San Diego

Join Zoom Meeting https://nus-sg.zoom.us/j/88213418159?pwd=K2lPSXEvemZGSXhiVGYxTVUzQ0d3QT09 Meeting ID: 882 1341 8159 Password: pcasvd Modelling complex physical dynamics is a fundamental task in science and engineering. Traditional physics-based models are interpretable but usually require significant computational resources and expertise. While deep learning (DL) provides alternatives for efficiently emulating nonlinear dynamics, it does not necessarily obey the physics laws or generalize well across different systems. Thus, there is a growing need for incorporating prior physics knowledge into DL models so that we can take the best of both types of approaches. We present three physics-guided DL approaches for spatiotemporal forecasting. We first present TF-Net, which marries the numerical techniques with custom-designed neural networks, that can predict accurate physical fields that obey desirable physical characteristics. We also provide two approaches to improve generalization across different systems. One is to design deep equivariant models that respect given symmetries of physical dynamics, and the other one is to infer shared time-invariant characteristics by leveraging meta-learning techniques.

Add to calendar :

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.

Add to calendar :

Date/Time : 03 Jun 2021 16:00
Venue : via Zoom
Speaker : Liu Yong, Institute of High Performance Computing

Join Zoom Meeting https://nus-sg.zoom.us/j/88213418159?pwd=K2lPSXEvemZGSXhiVGYxTVUzQ0d3QT09 Meeting ID: 882 1341 8159 Password: pcasvd Transformer has been developed as one of most impactful deep neural network architecture for Natural Language Processing (NLP) tasks such as machine translation, question & answer, and reading comprehension. Its effectiveness has been demonstrated by the impressive performance of large language model such as GPT-3 to produce convincing stream of human-like text. This talk will cover the mechanism of transformer and how it has been used for computer vision tasks such as image classification, followed by our recent work on transformer-based medical image segmentation for multiple diseases. Compared with representative existing methods, the proposed method consistently achieved the highest segmentation accuracy, and exhibited good cross-domain generalization capabilities.

Add to calendar :