Date/Time : 21 Apr 2025 10:00
Venue : S17 #04-04 (SR3)
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 : 17 Apr 2025 13:00
Venue : S16 #03-06
Speaker : Abdul Basit, Monash University

The independent set sequence of a graph is the sequence whose k^th term is the number of independent sets of size k in the graph. Alavi, Malde, Schwenk and Erdős studied this sequence in 1987, and showed that it can exhibit any pattern of rises and falls. They raised the question, however, of whether the independent set sequence of a tree, or more generally a forest, is always unimodal — that is, has a single peak. This intriguing question is still very much open. I’ll survey some recent results about this question. (Some of this is joint work with David Galvin and Misha Isarev.)

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Date/Time : 17 Apr 2025 14:00
Venue : S16 #03-06
Speaker : Cheng Yui To, National University of Singapore

Freiman’s theorem is a cornerstone of additive combinatorics, providing a structural description of finite sets of integers with small doubling. In this talk, I will present an extension of Freiman’s theorem to infinite subsets of the integers by replacing cardinality with density. Our main tool is the abelian Lie model theorem, which harnesses the powerful machinery of ultraproducts to pass from density—a less well-behaved notion—to Haar measures, where the classical proof strategy of Freiman’s theorem can be adapted. (Joint work with Peng Fei and Tran Chieu-Minh, part of undergrad honor project requirement).

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Date/Time : 17 Apr 2025 15:00
Venue : S16 #03-06
Speaker : Kong Zuxiang, National University of Singapore

Convex sets in Euclidean space enjoy the elegant property that convexity is preserved under quotient projections. From the perspective of additive combinatorics, small doubling—a condition where a set grows slowly under addition—is a natural generalization of convexity, as illustrated by equality condition in the famous Brunn–Minkowski inequality. In this talk, I will discuss recent progress on how small doubling behaves under quotients in the setting of nonabelian locally compact groups. As part of my joint paper with Fei and Tran as well as my honors thesis, we improve on a work by An, Jing, Zhang, and Tran, by proving new bounds relating the doubling constant of a set to that of its image under a quotient map. In particular, we establish a sharp quadratic bound when the set is symmetric, a cubic bound in general, and a structural result showing that a large subset with linear doubling survives the quotient. These results offer a refined understanding of how additive structure interacts with quotient projections, and bridges geometry, analysis, and combinatorics in nonabelian settings. (Joint work with Peng Fei and Tran Chieu-Minh, part of undergrad honor project requirement).

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Date/Time : 17 Apr 2025 16:30
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 : 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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