Date/Time : 14 Mar 2025 11:00
Venue : S17 #04-04 (SR3)
Speaker : Wang Yutong, National University of Singapore

In the previous sections, we have introduced some recent LLM tools in Lean 4, especially the ones for auto-formalization and auto-proving. In this lecture, I will show you the basic concepts and techniques behind these tools, and how you can use them to improve your efficiency in both formal and informal research. I will also try to introduce to you some recent progress in the field of LLM reasoning in mathematics.

Zoom link: https://nus-sg.zoom.us/j/84932999912?pwd=tSka7Th1z2McUW3zhjquuUQ7hcLfab.1
Zoom ID: 849 3299 9912
Pwd: 475364

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Date/Time : 13 Mar 2025 14:00
Venue : S17 #06-11 (SR6)
Speaker : Didac Martinez Granado, University of Luxembourg

In geometric group theory, one typically studies whether a given group G admits particular geometric structures—well-behaved actions on real trees, measured wall structures, or admissible random walks—all of which give rise to (invariant) metrics on G. This mini course adopts a broader perspective: rather than focusing on each structure in isolation, we consider *all* possible metrics on G at once and investigate the global geometry of this “space of metrics.” Introduced by Furman in the early 2000s, it has recently undergone an intense study by Reyes, Cantrell-Tanaka, Cantrell-Reyes, and myself with collaborators, among others.

When G is a surface group, a motivating analogy comes from the Teichmüller space of a surface, which parametrizes all marked hyperbolic metrics and is a finite-dimensional ball. However, the space of metrics of a surface group is infinitely dimensional, incorporating diverse constructions such as hyperbolic metrics, word-metrics, geodesic currents and Green metrics. We will describe some of these metrics, focusing on geodesic currents as a motivating example. Another avenue to obtain metrics on G from measures is from certain “admissible” random walks with respect to probability measures on G. These induce “Green metrics” which naturally sit in this space. We will introduce and analyze a natural distance on the space of metrics, discussing in particular how both word-metrics and those arising from admissible random walks are dense (Cantrell-MG-Reyes).

We believe this course could be of interest to people interested in hyperbolic geometry, geometric group theory, dynamics, or probability.

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Date/Time : 12 Mar 2025 15:00
Venue : S17 #05-12 (SR4)
Speaker : Svetlana Katok, Pennsylvania State Unviersity

Given a closed, orientable surface of constant negative curvature and genus g ≥ 2, we study a family of generalized Bowen–Series boundary maps and their two dynamical invariants: the topological entropy and the measure-theoretic entropy with respect to their smooth invariant measure. Each such map is defined for a particular fundamental polygon and a particular multi-parameter. We prove two strikingly different results: rigidity of topological entropy and flexibility of measure-theoretic entropy. The topological entropy is constant in this family and depends only on the genus of the surface. We give an explicit formula for this entropy and show that it stays constant both within our parameter space and within the Teichmüller space of the surface. We obtain an explicit formula for the measure-theoretic entropy for the smooth invariant probability measure on the circle that only depends on the genus of the surface and the perimeter of the (8g−4)-sided fundamental polygon, and prove that it varies in the Teichmüller space and takes all values between 0 and maximum that is achieved on the surface which admits a regular fundamental (8g-4)-gon, and stays constant on a subset of the parameter space. The rigidity proof uses conjugation to maps of constant slope, while the flexibility proof – valid only for certain multi-parameters – uses the realization of the geodesic flow on the surface as a special flow over the natural extension of the boundary map, and the Reduction Theory for maps associated to Fuchsian groups.

In particular, we show that the smooth invariant probability measure is not the measure of maximal entropy (the Parry measure) for the considered boundary maps.

This is joint work with Adam Abrams and Ilie Ugarcovici.

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Date/Time : 12 Mar 2025 16:30
Venue : S17 #05-12 (SR4)
Speaker : Zhang Junyu, National University of Singapore

Mirror descent (MD) and its stochastic variants are well-understood first order methods in convex optimization under appropriate smoothness conditions and have been extended to general relative smooth and nonconvex problems in recent years. Despite being widely studied, there are still many issues in the existing research.

In this talk, we will first reveal the inappropriateness of the currently widely adopted optimality measures, which casts doubts about the validity of current complexity results, for both deterministic and stochastic MD. Second, due to the absence of Lipschitz continuity, most existing results on nonconvex stochastic mirror descent (SMD) do not allow popular acceleration techniques such as random shuffling and variance reduction, etc. To fix these issues, we propose a novel regularity condition that is widely satisfied by most of the popular kernels. Combined with the dual space perspective of mirror descent, a new optimality measure has been proposed and a Lipschitz-like gradient bound has been established. We elaborate how these concepts can help the analysis SMD by establishing new and improved complexity results for SMD with variance reduction.

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Date/Time : 12 Mar 2025 14:00
Venue : S17 #04-04 (SR3)
Speaker : Luo Muhan, National University of Singapore

We prove that under the natural assumption over the dynamical degrees, the saddle periodic points of a Henon-like map in any dimension equidistribute with respect to the equilibrium measure. Our work generalizes results of Bedford-Lyubich-Smillie, Dujardin and Dinh-Sibony along with improvements of their techniques. We also investigate some fine properties of Green currents associated with the map. This is a joint work with Qi Zhou.

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Date/Time : 07 Mar 2025 16:00
Venue : S17 #05-11 (SR5)
Speaker : Sheng MENG, East China Normal University

Let f be a surjective endomorphism of a projective variety X defined over a number field. The Kawaguchi-Silverman conjecture asserts that the arithmetic degree of a point with a Zariski dense forward orbit under f equals the first dynamical degree of f. In this talk, I will first survey previous results (primarily those based on Junyi Xie’s recursive approach) and then focus on our recent progress concerning the case where X is a smooth projective variety with Picard number 2.

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Date/Time : 07 Mar 2025 11:00
Venue : S17 #04-04 (SR3)
Speaker : Wang Yutong, National University of Singapore

In this lecture, we will continue our unfinished discussion on structures and typical proof logic. We will resume our journey to introduce more advanced techniques in recursion and induction. In addition to theorem proving, I will also talk about how can you design syntax and notations to make your formalization more readable and maintainable.

Zoom link: https://nus-sg.zoom.us/j/84932999912?pwd=tSka7Th1z2McUW3zhjquuUQ7hcLfab.1
Zoom ID: 849 3299 9912
Pwd: 475364

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Date/Time : 05 Mar 2025 16:30
Venue : S17 #05-12 (SR4)
Speaker : Anthony Chen, University of Michigan

Fast Summation refers to a family of techniques for the fast approximation of N-body sums. While traditionally fast summation has been applied to problems coming from astrophysics or electrodynamics, many problems in geophysical fluid dynamics can be rewritten as the computation of a spherical convolution, and when these integrals are discretized, the resulting problem is an N-body problem. In this talk, I discuss a novel spherical tree code/fast multipole method based on barycentric Lagrange interpolation, as well as applications to problems coming from geophysical fluid dynamics, including tidal modeling and vortex dynamics.

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Date/Time : 05 Mar 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : James Farre, Max Planck Institute for Mathematics in the Sciences

The geodesic flow on the unit tangent bundle of a hyperbolic surface is a prototypical example of an Anosov flow, exhibiting rich, structurally stable dynamics. While invariant measures and closed sets for the geodesic flow can be quite complicated, the closely related horocycle flow exhibits striking rigidity. For example, on a closed surface, every horocycle is dense and equidistributed in its unit tangent bundle; this is very far from being true for geodesics.

Despite this rigidity, the behavior of horocycles remains mysterious in the setting of hyperbolic surfaces with infinite topology. In this talk, I will explore the dynamics of horocyclic trajectories for a particular class of infinite-area regular covers of closed hyperbolic surfaces. Surprisingly, these trajectories are intimately connected to an exceptional set for the geodesic flow: the distance-minimizing geodesic lamination. Both objects arise naturally in a geometric optimization problem: what does an optimal Lipschitz map to the circle in a given homotopy class look like?

This is joint work with Or Landesberg and Yair Minsky.

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