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 : 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 : 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 : 20 Aug 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Andreas Wieser, Institute for Advanced Study

A classical question in number theory asks: which integers are represented by a given quadratic form? More generally, one can ask when an integral quadratic form q in m variables can be represented by another integral quadratic form Q in n > m variables—that is, whether there exists an integer matrix T such that Q(Tx) = q(x).

Naturally, a necessary condition is that such a representation exists locally, meaning over the real numbers and modulo N for every positive integer N. In the absence of local obstructions, does a (global) representation of q by Q necessarily exist?

A breakthrough result of Ellenberg and Venkatesh in 2008 used deep methods from homogeneous dynamics to prove such a local-global principle when the codimension n – m is at least 5. In this talk, we present joint work with Manfred Einsiedler, Elon Lindenstrauss, and Amir Mohammadi in which we establish a local-global principle in codimension n – m at least 3. The key new ingredient is an effective equidistribution result for periodic orbits of semisimple groups on compact adelic homogeneous spaces.

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Date/Time : 20 Aug 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Jia Jia, Tsinghua University

We study the moduli space of higher rank marginally stable pairs (E,s), where E is a torsion free coherent sheaf of rank r on a smooth projective surface and s = (s_1, …, s_r) is a collection of r sections of E. Fixing the Chern character of E, the moduli space is realised as a subscheme of an appropriate Quot–scheme that parametrises quotient sheaves with the corresponding Hilbert polynomial.

We establish a precise link between these moduli spaces and the stable minimal models determined by E and its sections, together with the (analytic) minimal log resolutions of the base surface. Using the birational geometry of such minimal models, we analyse in detail the components of the Hilbert–Chow morphism from the moduli space to the Hilbert scheme of Cartier divisors on the surface. This is a work in progress.

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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 : 19 Aug 2025 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : York Ye, National University of Singapore

Configuration spaces of flags can be used to describe many algebraic varieties appearing in representation theory, such as double Bruhat cells and surface character varieties. In this talk, I will discuss the relationship between configuration spaces and cluster algebras. Then I will introduce my joint work with Huanchen Bao on the upper cluster structure of Kac-Moody Richardson varieties and explain some of the new techniques.

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Date/Time : 13 Aug 2025 17:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Kenny Gill, La Salle University

A countable structure S is indivisible if it satisfies a pigeonhole principle: for any bounded coloring of any presentation of S, there is a monochromatic induced substructure isomorphic to S. This naturally gives rise to an instance-solution problem Ind(S) which outputs such a substructure given a presentation and coloring. We seek to understand both (1) the computational difficulty of solving Ind(S) for various specific S; and (2) how properties of S can translate to properties of Ind(S) and vice versa. This talk will survey what is known about both (1) and (2), highlighting some of the many open problems in the area. In particular, we propose several notions of uniform embeddability of one structure in another which allow one to obtain desirable properties of Ind(S) for large classes of S. This is in part ongoing joint work with Damir Dzhafarov and Reed Solomon.

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Date/Time : 13 Aug 2025 16:30
Venue : S17 #04-04 (SR3)
Speaker : Liu Wenjie, Harbin Institute of Technology

In this talk, we introduce a new theoretical framework built upon fractional spaces for optimal error estimates of Chebyshev/Legendre polynomial approximations to functions with limited regularity. It naturally arises from exact representations of orthogonal polynomial expansion coefficients. Here, the essential pieces of the puzzle for the error analysis include (i) fractional integration by parts (under the weakest possible conditions); (ii) generalised Gegenbauer functions of fractional degree (GGF-Fs): a new family of special functions with notable fractional calculus properties, and (iii) asymptotic formulas or uniform upper bounds of GGF-Fs. Finally we introduce the hypothesis of Babuska and Hakula. The talk is based on joint work with Li-Lian Wang, Huiyuan Li, Boying Wu and Ruiyi Xie.

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