Date/Time : 10 Jul 2025 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Si-Jong Kwak, Korea Advanced Institute of Science & Technology

The gonality conjecture of a smooth projective curve raised by Green and Lazarsfeld. has been one of the leading problems in geometry and syzygy. The conjecture says that the gonality of a curve can be encoded in the Betti table if a given curve is embedded by the line bundle of sufficiently large degree. It was proved by Ein-Lazarsfeld in 2016.

In this talk, we generalize this conjecture to the secant varieties of a curve. The gonality sequence d_1, d_2, . . . , d_g are encoded in syzygies of the secant varieties of a curve. This is a joint work with J. Choe and J. Park.

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Date/Time : 08 Jul 2025 13:30
Venue : S17 Level 3 (IMS Executive Seminar Room)
Speaker : Lou van den Dries, The University of Illinois Urbana-Champaign

This will be a survey of our work on these topics. No familiarity with transseries or Hardy fields is required, but I mention here that both are natural settings for solving ordinary differential equations and studying their asymptotics (and for much more). Transseries are formal objects, and Hardy fields consist of actual functions (or rather germs of such). Around 2015 we showed that the differential field of transseries has a good model theory. We went on to prove that sufficiently rich Hardy fields have the same (complete) model theory, with interesting consequences even on the level of linear differential equations. More recently we extended this even to analytic Hardy fields. Some of the ideas involved go back to Sturm and Liouville (1830’s).

This is joint work with Aschenbrenner and Van der Hoeven.

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Date/Time : 04 Jul 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Ye Jieping, Alibaba

The rapid advancement of large language models (LLMs) showcases unprecedented capabilities, yet their opaque internal decision-making remains a critical challenge. Mechanistic interpretability has emerged as a critical frontier for enhancing the safety, reliability, and controllability of LLMs. In this talk, we synthesize cutting-edge research that reverse-engineers LLM internals to establish foundational principles and deployable applications. Through mechanistic analysis of diverse capabilities—including arithmetic computation, multilingual processing, and factual recall—we reveal how LLMs represent and process information as discrete circuits. These insights enable novel applications: a) Precise Control: Token-, activation-, neuron- and module-level intervention to adjust LLM’s behaviour; b) Precise Tuning: Head- and neuron-level tuning to enhance LLM’s performance c) Precise Quantization: Interpretability-guided quantization strategies achieving 3.4-bit parity with 4-bit baselines.

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Date/Time : 02 Jul 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : An Huang, Brandies University

What is the p-adic counterpart of the derivative operator, or the Laplacian? I shall describe a number of Non-Archimedean pseudo-differential operators arising in part from considerations in p-adic string theory. I shall explain the connections to Tate’s thesis in genus zero, and then in genus one, explain the computation of the Green’s function on the Tate curve. If time permits, questions regarding the spectrum of p-adic strings shall be mentioned.

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Date/Time : 25 Jun 2025 10:00
Venue : IMS Executive Seminar Room (S17 Level 3)
Speaker : François Loeser, Sorbonne University

We will provide a general overview of our work with E. Hrushovski on tame geometry over valued fields, using the stable completion of algebraic varieties over valued fields. A key result is the existence of retractions to piecewise-linear objects called skeleta, which are definably isomorphic to definable subsets of the value group. We will conclude by presenting a recent finiteness result obtained with A. Ducros, E. Hrushovski and J. Ye for tropical functions on such skeleta.

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Date/Time : 23 Jun 2025 14:00
Venue : S17 #04-04 (SR3)
Speaker : Chris Leininger, Rice University

I will discuss joint work with Autumn Kent in which we establish the existence of purely pseudo-Anosov surface subgroups of mapping class groups. We do this by constructing a type-preserving representation of the figure eight knot group into the mapping class group of the thrice-punctured sphere. As a corollary we obtain the first examples of closed atoroidal surface bundles over surfaces.

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Date/Time : 18 Jun 2025 14:00
Venue : S17 #04-04 (SR3)
Speaker : Khanh Le, Rice University

For a hyperbolic surface X with non-empty boundary, the ortho-length spectrum of X is the multi-set of lengths of (ortho)-geodesic arcs meeting the boundary of X perpendicularly at both endpoints. The spectrum was introduced by Basmajian for hyperbolic n-manifolds in the context of studying co-dimensional one embedded geodesic submanifolds. The ortho-length spectrum has been studied extensively in relation to geometric identities, hyperbolic geometry of surfaces and inverse geometry. In this work, we initiate a new investigation into the arithmetic properties of the ortho-length spectrum. In particular, we consider hyperbolic surfaces whose cosh-of-length of ortho-geodesics are all integers, ortho-integral surfaces. We prove that there are only finitely many ortho-integral surfaces for a fixed topological type. We also constructed infinitely many pairwise non-commensurable ortho-integral surfaces. In the talk, I will motivate the consideration of such surfaces and discuss some ideas and tools going into the proof. If time permits, I will also discuss some further questions that we would like to investigate. This is joint work with Minh Doan.

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Date/Time : 18 Jun 2025 16:00
Venue : S17 #06-11 (SR6)
Speaker : Jaydeep Singh, Princeton University

In this talk we will discuss self-similar solutions to the Einstein-scalar field system, first studied by Christodoulou (’94). These are a family of C^{1,\alpha} solutions modeling spacetime singularities without an event horizon. In BV spaces their instability is well understood via a geometric blue-shift effect, and leads to the formation of black holes. We will share progress on extending these results to Hölder spaces, and discuss some of the analytical and conceptual obstacles to doing so.

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Date/Time : 06 Jun 2025 14:00
Venue : S17 #04-05 (SR2)
Speaker : Xie Xingyu, National University of Singapore

Large language models excel at pattern completion yet struggle with multi-step mathematics, logical deduction, and tool-assisted problem solving. Recent work shows that reinforcement learning can provide the “slow-thinking” mechanisms needed for such tasks. This talk formulates reasoning as a sequential decision process and explains how policy-gradient objectives are adapted to text generation. The main case study, DeepSeek-R1, is examined in detail, covering its Group Relative Policy Optimisation objective, reward design, and the resulting gains on contest-level mathematics. The talk also surveys subsequent refinements that build on R1’s principles, illustrating how practical adjustments to reward formulation and output control further enhance reasoning performance. Together these methods illustrate how RL is currently advancing the reasoning abilities of state-of-the-art language models.

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