Date/Time : 23 Apr 2025 15:00
Venue : S16 #05-17 (ACTIVE LEARNING ROOM 1)
Speaker : Jared P. Whitehead, Brigham Young University

In classical statistics, predictive error is understood as a trade-off between the bias and the variance that balances model simplicity with its ability to fit complex functions. However, over-parameterized models exhibit counterintuitive behaviors, such as “double descent” in which models of increasing complexity exhibit decreasing generalization error. Others may exhibit more complicated patterns of predictive error with multiple peaks and valleys. Neither double descent nor multiple descent phenomena are well explained by the bias-variance decomposition.

We introduce a novel decomposition that we call the generalized aliasing decomposition (GAD) to explain the relationship between predictive performance and model complexity. The GAD decomposes the predictive error into three parts: 1) model insufficiency, which dominates when the number of parameters is much smaller than the number of data points, 2) data insufficiency, which dominates when the number of parameters is much greater than the number of data points, and 3) generalized aliasing, which dominates between these two extremes.

We demonstrate the applicability of the GAD to diverse applications, showing that it accurately captures the non-monotonic nature of generalization curves in several different settings. GAD can be explicitly calculated from the relationship between model class and samples without seeing any data labels, so that this decomposition can answer questions related to experimental design and model selection before collecting data or performing experiments.

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Date/Time : 23 Apr 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : Luca Spolaor, University of California San Diego

I will discuss the boundary regularity for minimizers of the so-called Alt-Caffarelli problem and how it is related to a boundary unique continuation problem. This is joint work with L. Ferreri (SNS) and Bozhidar Velichkov (Pisa).

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Date/Time : 23 Apr 2025 15:00
Venue : S17 #04-05 (SR2)
Speaker : Jialun Li, Ecole Polytechnique de Paris

I will begin with the classical setting of the modular surface: the unit tangent bundle of the modular curve T^1(SL(2,Z)\H)=SL(2,Z)\SL(2,R), where periodic geodesics correspond to periodic orbits of the diagonal subgroup. Margulis and Bowen established dynamical frameworks for studying these orbits, while Selberg and Sarnak developed spectral and number-theoretic approaches.

We then turn to the higher-rank generalization, periodic diagonal orbits on SL(3,Z)\SL(3,R). In this case, each periodic diagonal orbit is a two-dimensional flat torus embedded in the space. Different counting problems emerge, depending on how one orders these tori–whether by dynamical ordering, geometric ordering or arithmetic ordering. The dynamical and geometric ordering are closely connected through the shapes of the flat tori. Finally, I will discuss my recent joint work with Thi Dang and Nihar Gargava on the density of shapes of periodic tori.

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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 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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