Date/Time : 27 Aug 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Guolei Zhong, Institute for Basic Science, Korea

Let X be a smooth projective variety and G an abelian subgroup of automorphisms of X over the field of algebraic numbers. Suppose that G is of positive entropy. It was proved by Dinh and Sibony that the group is then free abelian of rank no more than dim(X)-1. In the past decade, from algebraic dynamics, the geometry of the extremal case when the rank achieves the maximal has been well-understood by Zhang. In this talk, we study the maximal rank case in the aspect of arithmetic dynamics. More precisely, we construct a canonical height function associated with the group, corresponding to a nef and big divisor, satisfying the Northcott property. By characterizing its null locus, we prove the Kawaguchi–Silverman conjecture for each group element. Some further discussion towards its application and the existence of Zariski dense orbits will also be presented. This is based on a joint work with Fei Hu.

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Date/Time : 26 Aug 2025 10:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Marcel Nutz, Columbia University

Entropic optimal transport — the optimal transport problem regularized by KL divergence — is highly successful in statistical applications. Thanks to the smoothness of the entropic coupling, its sample complexity avoids the curse of dimensionality suffered by unregularized optimal transport. The flip side of smoothness is overspreading: the entropic coupling always has full support, whereas the unregularized coupling that it approximates is usually sparse, even given by a map. Regularizing optimal transport by less-smooth divergences such as L^p regularization is known to allow for sparse approximations, but is often thought to suffer from the curse of dimensionality as the couplings have limited differentiability and the dual is not strongly concave. We refute this conventional wisdom and show, for a broad family of divergences, that the key empirical quantities converge at the parametric rate, independently of the dimension. More precisely, we provide central limit theorems for the optimal cost, the optimal coupling, and the dual potentials induced by i.i.d. samples from the marginals. (Joint work with Alberto Gonzalez-Sanz and Stephan Eckstein.)

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Date/Time : 25 Aug 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Li Qianxiao, National University of Singapore

In this talk we discuss connections between deep learning and the study of dynamical processes. We begin by examining how dynamics and control provide a natural framework for understanding how complexity emerges through composition—a hallmark of modern deep learning architectures, including both feed forward and recurrent models. In the reverse direction, we highlight how deep learning techniques can advance the modelling and simulation of dynamical processes in science and engineering.

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Date/Time : 25 Aug 2025 14:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Adeel Khan, Academia Sinica

I will discuss a new type of pullback operation on perverse sheaves, which is defined for morphisms of complex algebraic stacks equipped with a certain extra structure, namely that of a so-called relative exact (-1)-shifted symplectic fibration. These perverse pullbacks are closely related to classical operations such as vanishing cycles functors and the Fourier-Sato transform. They also vastly generalize the theory of Donaldson-Thomas invariants of Calabi-Yau threefolds, and we will sketch how they lead to a proof of a conjecture of Joyce about their functoriality. Time permitting, I will attempt to briefly survey some applications of these constructions to topics like cohomological Hall algebras, curve counting on Landau-Ginzburg models, and the period sheaves for Hamiltonian G-spaces conjectured by Ben-Zvi, Sakellaridis, and Venkatesh. This is a report on joint work in progress with Tasuki Kinjo, Hyeonjun Park, and Pavel Safronov.

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Date/Time : 22 Aug 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Marcel Nutz, Columbia University

Passive liquidity providers (LPs) in automated market makers (AMMs) face losses due to adverse selection, which static trading fees often fail to offset in practice. We study the key determinants of LP profitability in a dynamic reduced-form model where an AMM operates in parallel with a centralized exchange (CEX), traders route their orders optimally to the venue offering the better price, and arbitrageurs exploit price discrepancies. Using large-scale simulations and real market data, we analyze how LP profits vary with market conditions such as volatility and trading volume, and characterize the optimal AMM fee as a function of these conditions. The mechanisms driving these relationships are highlighted through extensive comparative statics, and the model’s relevance is confirmed through calibration to market data. We find that under normal market conditions, the optimal AMM fee is competitive with the trading cost on the CEX and remarkably stable, whereas in periods of very high volatility, a high fee protects passive LPs from severe losses. These findings suggest a threshold-type dynamic fee schedule to improve LP outcomes. (Joint work with Steven Campbell, Philippe Bergault, Jason Milionis.)

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Date/Time : 21 Aug 2025 16:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Liu Sheng, Stanford University

While modern machine learning models and systems, including most advanced large language models, have achieved impressive results, their success is still predicated on massive collections of human-curated, carefully filtered data. This reliance exposes a fundamental vulnerability: when deployed in the real world, these models encounter data that is noisy, incomplete, or ambiguous, leading to unpredictable and unreliable behavior. This challenge is more urgent in high-stakes domains such as medicine, where robustness and trustworthiness are paramount.
In this talk, I will present how we can move beyond the limitations of curated data by developing machine learning algorithms and theory that enable models to learn and adapt directly from the complexity of real-world environments. I will share new insights into the underlying mechanisms of failure for current models under imperfect data, and introduce principled methods that address these vulnerabilities. Finally, I will illustrate how these advances empower reliable AI systems for critical medical applications, including early Alzheimer’s disease detection and automated radiotherapy treatment planning.

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Date/Time : 21 Aug 2025 14:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Wooyeon Kim, Korea Institute for Advanced Study

We study an effective version of Ratner’s equidistribution theorem in the space of affine lattices. For d>1, let H be a minimal horospherical subgroup of SL(d,R), embedded in ASL(d,R), with respect to a one-dimensional diagonal flow a(t). We show that the a(t)-push-forwards of a segment of H-orbit become equidistributed at a polynomial rate in the space of affine lattices, under a suitable Diophantine condition on the initial point. This extends the previous result of Strömbergsson for d=2 to higher dimensions.

Our proof exploits the SL(d,Z)-monodromy action on each fiber torus T^d induced by the H-orbits, and uses techniques from Bourgain-Furman-Lindenstrauss-Mozes on random walks in the torus setting, adapted here to a fiberwise context.

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Date/Time : 20 Aug 2025 17:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Tran Chieu Minh, National University of Singapore

In the classic paper “ Henselianity implies large’’ , Pop showed that a field is large if it is elementarily equivalent to the fraction field of a proper Henselian local domain. We show that it can, in fact, be improved to an “if and only if’’ statement.

We will explain how this result arises from the effort to introduce  canonical topologies over a field, namely, a result comparing the etale-open topology with a newly introduced finite-closed topology. (Joint with Will Johnson, Erik Walsberg, and Jinhe Ye)

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Date/Time : 20 Aug 2025 13:30
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Nguyen Tat Thang, Institute of Mathematics, Vietnam Academy of Science and Technology

Let $f:\R^n\to \R^m$ be a real polynomial mapping, $\Sing(f)$ be the set of singular points of $f$ in $\R^n$, and $K_0(f)=f(\Sing(f))$. A {\it bifurcation set} of $f$ is the smallest set of values in $\R^m$ outside which $f$ is a locally trivial fibration. A regular value $\lambda\in f(\R^n)\setminus K_0(f)$ is called a {\it typical value at $\infty$} of $f$ if there is an open neighborhood over which $f$ is a trivial fibration. Otherwise, $\lambda$ is called an {\it atypical value at $\infty$} of $f$. In this talk, we characterize atypical values at infinity of a real polynomial function of three variables by a certain sum of indices of the gradient vector field of the function restricted to a sphere with a sufficiently large radius. This is an analogy of a result of Coste and de la Puente for real polynomial functions with two variables. We also give a characterization of atypical values at infinity of a real polynomial map whose regular fibers are 2-dimensional surfaces. This is a joint work with Masaharu Ishikawa.

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