Date/Time : 25 Aug 2021 14:00
Venue : via Zoom
Speaker : Korkeat Korkeathikhun, University of North Carolina, Chapel Hill

Join Zoom Meeting https://nus-sg.zoom.us/j/89816394951?pwd=OXRvc1pXY0kxSGhYRkloMnFkL0dSUT09 Meeting ID: 898 1639 4951 Passcode: 903999 Let $G$ be a simply-connected simple algebraic group, $\mathfrak{g}$ its Lie algebra. Let $\mathrm{Gr}_G$ be the affine Grassmannian of $G$, and $\mathrm{Gr}_0^-$ the opposite open Schubert cell in $\mathrm{Gr}_G$. Achar and Henderson defined a map $\pi:\mathrm{Gr}_0^- \to \mathfrak{g}$ and used it to exhibit the relation between Schubert varieties in $\mathrm{Gr}_G$ and nilpotent cone in $\mathfrak{g}$. In our work, we consider a twisted analouge. We relate the geometry of Schubert varieties in twisted affine Grassmannian and the nilpotent varieties in symmetric spaces. This extends some results of Achar-Henderson in the twisted setting. We also get some applications for the geometry and topology of the order 2 nilpotent varieties in certain classical symmetric spaces.

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Date/Time : 25 Aug 2021 16:00
Venue : via Zoom
Speaker : Ng Keng Meng,

Join Zoom Meeting https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09 Meeting ID: 830 4925 8042 Passcode: 1729=x3+y3 There has been a recent revival in the interest in sub-computable mathematics. One of these approaches is to consider “primitive recursive” or punctual structures. This has led to a greater understanding in the effective content of well-known objects and proofs in classical computability theory. When considering the punctual anaologies of classical computabilitiy we often obtain strange and surprising results. I will discuss some recent work in progress in this area, focussing particularly on structural results. URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html  

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Date/Time : 18 Aug 2021 16:00
Venue : via Zoom
Speaker : Yu Liang, Nanjing University

Join Zoom Meeting https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09 Meeting ID: 830 4925 8042 Passcode: 1729=x3+y3 Besicovitch-Davis theorem says that the Hausdorff dimension of every analytic set can be approximated by its closed subset. But the Besicovitch-Davis theorem fails for co-analytic sets under the assumption V=L as observed by Slaman. We prove that the theorem holds for arbitrary sets under ZF+sTD. We also prove that the theorem holds for Sigma-1-2-sets under Martin’s axiom. This is joint work with Peng Yinhe and Wu Liuzhen.

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Date/Time : 18 Aug 2021 14:00
Venue : via Zoom
Speaker : Jinfeng Song, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/82727387952?pwd=WEo0M0lsd0tETHdhZWZIRGJUaEhEUT09 Meeting ID: 827 2738 7952 Passcode: 557651 Any variety over a field of characteristic p>0 possesses a Frobenius endomorphism F, which fixes all the points and raises the function to p-th power. A variety X is called Frobenius split if the p-th power map splits as a morphism of sheaves of modules. The most important class of examples of varieties which are Frobenius split are flag varieties. This is firstly proved by Mehta and Ramanathan using geometric methods. Later, Kumar and Littelmann construct an alternative proof using the methods of quantum groups. In this talk, I plan to introduce Kumar and Littelmann’s method, and explain its potential generalisation to quantum symmetric pairs.

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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 : 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.

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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.

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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.

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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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