Date/Time : 29 Jul 2025 14:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Vilas Sreenivasan Winstein, University of California, Berkeley

Independent sets have long been studied as a model for gases in statistical mechanics, and Sapozhenko’s famous graph container method was developed to understand their quantity in the hypercube, where they correspond to error-detecting codes. When removing edges at random (i.e. percolating the hypercube), the number of independent sets in the resulting graph becomes the partition function of a disordered system, akin to popular models like spin glasses. I will discuss a new probabilistic framework which establishes sharp distributional characterizations of this partition function and the underlying disordered system, settling several open problems and explaining various phase transitions observed in this model. This is joint work with Mriganka Basu Roy Chowdhury and Shirshendu Ganguly.

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Date/Time : 28 Jul 2025 16:00
Venue : S17 #04-04 (SR3)
Speaker : Vilas Winstein , University of California, Berkeley

This is part 2 of a two-talk series. We will discuss some recent results on the limiting Gaussian nature of fluctuations of various observables under exponential random graph models; in particular, we obtain quantitative error bounds via Stein’s method. A key novelty is that our results apply in the low-temperature parameter regime which has previously been unexplored and requires new methods. These include the application of novel concentration inequalities derived from carefully analyzing the mixing behavior of relevant Markov chains.

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Date/Time : 25 Jul 2025 16:00
Venue : S17 #05-12 (Seminar Room 4)
Speaker : Vilas Winstein , University of California, Berkeley

This is part 1 of a two-talk series. We will introduce exponential random graph models, which are exponential tilts of Erdős–Rényi models by functions which increase or decrease the number of certain small subgraphs (e.g. triangles). We will also discuss relevant tools for studying these modelThis is part 1 of a two-talk series. We will introduce exponential random graph models, which are exponential tilts of Erdős–Rényi models by functions which increase or decrease the number of certain small subgraphs (e.g. triangles). We will also discuss relevant tools for studying these models, such as large deviations principles, mixing results, and concentration inequalities.s, such as large deviations principles, mixing results, and concentration inequalities.

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Date/Time : 16 Jul 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : Naihuan Jing, North Carolina State University

I will discuss some polynomials parameterized by Young diagrams, whose coefficients are central elements of the quantized enveloping algebra of the general linear algebra. These polynomials are q-immanants and are shown to provide the higher quantum Capelli identities and generalize Drinfeld-Reshetikhin’s construction of central elements in Z(U_q(g)) as well as q-analogues of Okounkov’s quantum immanants for U(gl(n)). The Harish-Chandra image of the these central elements are certain factorial Schur polynomials. The main results are joint works with Ming Liu and Alex Molev.

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Date/Time : 10 Jul 2025 10:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Nimish Shah, The Ohio State University

For the space of unimodular lattices in a Euclidean space, we give necessary and sufficient conditions for equidistribution of expanding translates of any real-analytic submanifold under a diagonal flow. This extends the earlier result of Shah in the case of non-degenerate submanifolds.

We apply the above dynamical result to show that if the affine span of a real-analytic submanifold in a Euclidean space satisfies certain Diophantine and arithmetic conditions, then almost every point on the manifold is not Dirichlet-improvable.

This is a joint work with Pengyu Yang

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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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