Date/Time : 16 Apr 2025 16:00
Venue : S17 #04-04 (SR3)
Speaker : Weijun Xu, Peking University

We consider two types of stochastic perturbations to the one dimensional Allen-Cahn equation, one with spacetime white noise, and the other with more singular noise. In both models, the interface location, when suitably rescaled, converges to a diffusion process on the real line. Cancellations play important roles in both models, but they happen in very different manners.

In the first model, we construct “correctors” that are designed to cancel potential divergences. We thus recover classical results by Funaki (1995) and Brassesco-Butta-De Masi-Presutti (1998), but also shed some new thoughts. In the second model, the cancellation happens somewhat miraculously, and the result seems to be new. Based on joint works with Wenhao Zhao (EPFL) and Shuhan Zhou (PKU).

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Date/Time : 16 Apr 2025 17:00
Venue : S17 #04-04 (SR3)
Speaker : Isabella Scott,

Ziegler-reducibility (written ≤*) was introduced by Martin Ziegler in his investigations into finitely generated subgroups of existentially closed (e.c.) groups. It is a refinement of both enumeration- and Turing-reducibility and agrees with Q-reducibility on the c.e. sets. Remarkably, Ziegler showed that it can be characterised purely algebraically: for finitely generated G and H, W(H) ≤* W(G) if and only if H embeds in every e.c. group containing G.

In this talk, I shall discuss the connections between computability theory and existentially closed groups, and recent work on the structure of the Ziegler degrees. In particular, we show that there is a minimal Ziegler degree, and, generalising this construction, that the 3-quantifier theory of the Ziegler degrees is undecidable.

This is joint work with Steffen Lempp and Josiah Jacobsen-Grocott.

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Date/Time : 16 Apr 2025 14:00
Venue : S17 #04-05 (SR2)
Speaker : Paolo Cascini, Imperial College

I will review recent progress on extending the Minimal Model Program to algebraically integrable foliations. My focus will be on some recent results toward the boundedness of Fano foliations.

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Date/Time : 16 Apr 2025 15:00
Venue : S17 #05-12 (SR4)
Speaker : Mu-Tao Wang, Columbia University

Two black holes rotate about each other and eventually merge into a single black hole. How does one measure the angular momentum carried away by gravitational radiation during this merger? This has been a subtle issue since the 1960’s due to the existence of “supertranslation ambiguity”: the angular momentums recorded by two observers of the same system may not be the same.

In this talk, I shall describe how the mathematical theory of quasilocal mass and optimal isometric embedding identifies a new definition of angular momentum that is free of any supertranslation ambiguity. In addition, some recent development of the cross-section continuity of the angular momentum definition will also be discussed. This is based on joint work with Po-Ning Chen, Jordan Keller, Daniel Paraizo, Robert Wald, Ye-Kai Wang, and Shing-Tung Yau.

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Date/Time : 16 Apr 2025 9:30
Venue : S17 #05-11 (SR5)
Speaker : Jishnu Kundu, National University of Singapore

The Relative Trace formula is a generalization of the Arthur Selberg trace formula and has its role in better understanding the automorphic representations of a general reductive group G. The talk will concern about the terms contributing to the spectral and geometric sides of the formula and concluded by a number theoretic application of it.

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Date/Time : 15 Apr 2025 15:00
Venue : S17 #04-05 (SR2)
Speaker : Hemant Tyagi, Nanyang Technological University

We consider the problem of finite-time identification of linear dynamical systems from $T$ samples of a single trajectory. Recent results have predominantly focused on the setup where either (a) no structural assumption is made on the system matrix $A^*$, or (b) specific structural assumptions (e.g. sparsity, low-rankness etc) are made on $A^*$. We assume prior structural information on $A^*$ is available, which can be captured in the form of a convex set $\mathcal{K}$ . For the solution of the ensuing constrained least squares estimator, we derive non-asymptotic error bounds in the Frobenius norm that depend on the local size of $\mathcal{K}$ at $A^*$. The results are instantiated for four examples, namely when (i) $A^*$ is sparse and $\mathcal{K}$ is a suitably scaled $\ell_1$ ball; (ii) $\mathcal{K}$ is a subspace; (iii) $\mathcal{K}$ consists of matrices each of which is formed by sampling a bivariate convex function on a uniform $n \times n$ grid (convex regression); (iv) $\mathcal{K}$ consists of matrices each row of which is formed by uniform sampling (with step size $1/T$) of a univariate Lipschitz function. In all these situations, we show that $A^*$ can be reliably estimated for values of $T$ much smaller than what is needed for the unconstrained setting.

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Date/Time : 15 Apr 2025 14:00
Venue : S17 #05-11 (SR5)
Speaker : Yotsanan Meemark, Chulalongkorn University

For a finite ring R with identity the unitary Cayley graph of R, C_R, is the graph with vertex set R and for each x, y in R, x and y are adjacent if and only if x-y is a unit of R. In this talk we will discuss many results on the structure parameters (e.g., clique number, chromatic number) and eigenvalues of this graph. We analyze the energy (the sum of eigenvalues) and characterize the ring R such that C_R is Ramanujan. Our combinatorial number-theoretic approaches mainly come from properties of finite rings and fields and their characters.

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Date/Time : 14 Apr 2025 10:00
Venue : S17 #05-11 (SR5)
Speaker : Weijun XU, Peking University

Many stochastic PDEs arising from natural probabilistic models are classically “wrong” in the sense that they involve products between distributions. This type of ill-posedness has been existing even in ODEs. Ito’s theory of SDEs is probabilistic in nature. There is a lack of analytic stability, as can be seen from the existence of Ito-Stratonovich corrections.

We start from an old idea by Terry Lyons, now known as rough paths, that aimed to understand the nature of the analytic ill-posedness in SDEs. We will then see how these early attempts are gradually developed into a quite complete picture today for singular PDEs (regularity structures and para-controlled distributions). We shall focus on one particular example of singular SPDE, the 2D generalised parabolic Anderson model, to illustrate the coherence of this series of ideas.

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Date/Time : 09 Apr 2025 17:00
Venue : S17 #04-04 (SR3)
Speaker : Frank Stephan, National University of Singapore

For an unfriendly partition, one colours the nodes of a graph using k colours (mostly, k in {2,3}) such that for each two distinct colours a,b and each node of colour a, this node has more neighbours of colour b than of colour a. If one requires “strictly more” then the colouring is called “strictly
unfriendly” and if one requires “more or at least as many” then the colouring is just called “unfriendly”. The sets of the the nodes of each colour form then a partition of the graph. For two colours, one knows that every finite graph has an unfriendly partition while for countable graphs this is an open problem and for uncountable graphs, this is false, as shown by Shelah and Milner [1990]. Joint work with Belanger, Cipriani, Goh, Richter
and Tang shows that for certain recursive graphs, constructing an unfriendly colouring with two colours cannot be solved by any hyperarithmetic oracle, though a more complicated colouring still exists. Thus one cannot expect to find an easy algorithm solving the task and this might be one of the reasons that the problem is still open. Haynes, Hedetniemi and Vasylieva [2015] showed that for two colours, even finite graphs might fail to have a strictly unfriendly colouring. Indeed, it is shown that constructing strictly unfriendly colourings for two or three colours on finite graphs requires, under the Exponential Time Hypothesis, exponential time. Furthermore, one can reduce the problem 3occur3SAT with n variables to the problem of finding an unfriendly colouring of a finite graph with n’ nodes such that the lower bound c^n for 3occur3SAT translates into the lower bound (c^2/11)^n’. These constant c is not explicitly known, but under the assumption ETH it is strictly between 1 and 2. If one only considers graphs of bounded degree, also nontrivial bounds on the runtime can be obtained. Similar results are obtained for three colours. This is joint work with Sanjay Jain and Haoyun Tang. Furthermore, the three authors joint with Ong Weng Qi investigated the complexity of finding unfriendly colourings for two or three colours when for some nodes some colours are preassigned. This was known to be NP-hard, but the finegrained complexity was not studied before.

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