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 : 05 Mar 2025 17:00
Venue : S17 #04-04 (SR3)
Speaker : Subin Pulari, LaBRI

Measuring the informational content of real numbers has been a significant area of inquiry in algorithmic information theory. Finite-state compressibility (or finite-state dimension) of a real number is a value in [0, 1] which quantifies the amount of information/randomness in the real number as measured using finite-state automata. Finite-state dimension is the lower asymptotic ratio of compression achievable on an infinite string using information-lossless finite-state compressors. Interestingly, the finite-state dimension of a real number is also equal to the block Shannon entropy rate of the infinite sequence representing the expansion of the real number in a base b. A line of work, originating from Schnorr and Stimm (1972), has established that a number is Borel normal in base b if and only if its base b expansion has finite-state compressibility equal to 1, i.e., is incompressible. Hence, normal numbers are precisely the class of numbers that are incompressible using finite-state compressors.

Most of the prior research on finite-state dimension has relied on combinatorial methods. In this talk we explore how tools from Fourier analysis can be employed to gain new insights into the compressibility of real numbers by solving an open question posed by Lutz and Mayordomo regarding the existence of absolutely dimensioned numbers.

Absolutely normal numbers, being finite-state incompressible in every base of expansion, are precisely those numbers which have finite-state dimension equal to 1 in every base. At the other extreme, for example, every rational number has 0 finite-state dimension in every base. Generalizing this, Lutz and Mayordomo asked the following question: Does there exist any s strictly between 0 and 1 and a real number r such that r has finite state dimension equal to s in every base?. In this talk we use several tools involving exponential sums and techniques from Schmidt’s work in 1960 to construct, for any given s in (0,1], a real number r having finite-state compressibility equal to s in every base.

We also discuss further applications of exponential sums in analyzing and characterizing the finite-state compressibility of real numbers.

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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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Date/Time : 04 Mar 2025 14:00
Venue : S17 #05-11 (SR5)
Speaker : Xie Kaitao, The University of Hong Kong

The totally nonnegative part of flag varieties has been studied extensively for many years, and many remarkable properties have been established. Motivated by the study of higher Teichmuller theory, Guichard and Wienhard generalized Lusztig’s notion of total positivity and introduce theta positivity. In this talk, I will discuss the analogues for a family of theta nonnegative part of Lagrangian Grassmannian.

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Date/Time : 04 Mar 2025 15:00
Venue : S17 #04-05 (SR2)
Speaker : Anastasia Borovykh, Imperial College London

State-of-the-art foundation models are often seen as black boxes: we send a prompt in and we get out our – often useful – answer. But what happens inside the system as the prompt gets processed remains a bit of a mystery & our ability to control the processing into specific directions is limited.

In this walk we will discuss two concepts that can help us add additional controls into these models. First up, LogitLens, a method to gain more insight into the representations of hidden nodes. Second, steering vectors: by computing a vector that represents a particular feature or concept, we can use this to steer the model to include any property in the output we want. We end the talk by discussing whether the approach of ‘build a good model, then interpret it’ can work or whether we should shift our focus to ‘build an a priori interpretable model’ and discuss ways of potentially achieving this.

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Date/Time : 04 Mar 2025 15:00
Venue : S17 #05-11 (SR5)
Speaker : Max Gurevich, Technion - Israel Institute of Technology

For a split reductive p-adic group G, one can choose a hyperspecial maximal compact subgroup K and its Iwahori subgroup I. The resulting Hecke algebra H(I\K/I) is on one hand understood as a deformation of the Weyl group W of G; on the other, it acts on representations in the principal Bernstein block of G-representations.

With the p-adic story set aside, the geometric theory of Weyl group representations is expressed in the Springer correspondence, which parametrizes the spectrum of W in terms of local systems on unipotent conjugacy classes of the complex Langlands dual group to G.

When G is of classical type, the same local systems also arise as parameters in Arthur’s endoscopic approach to Langlands reciprocity. With Emile Okada, we explore the interplay between these two sets of parameters attached to irreducible representations, investigating their compatibility with natural restriction functors.

If time permits, I will explain one angle of broader motivation: an expectation relating local Arthur packets to the notion of ‘weak sphericity’ of representations. Interestingly, this property is most naturally analyzed through its specialization to the domain of Weyl group representations.

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Date/Time : 03 Mar 2025 16:00
Venue : S16 #03-06
Speaker : Li Zhuoyuan, Peking University

Data assimilation, as a key technique for merging physical models with observational to estimate state variables, has been widely explored in the fields of weather forecasting, oceanography, and ecology. However, conventional assimilation frameworks, such as variational methods (3D/4DVar) and sequential methods (EnKF), often struggle with the high dimensionality and pronounced nonlinearity inherent in both physical and observational models. In this talk, I will introduce our recent work on leveraging deep learning to address these challenges, focusing on data-driven reduced-order modeling and deep generative modeling.

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Date/Time : 26 Feb 2025 10:00
Venue : S17 #05-11 (SR5)
Speaker : Ma Jiajun, Xiamen University and Xiamen University Malaysia

This mini-course explores the fundamental aspects of local theta correspondence. The series consists of 5 lectures covering the following topics:

1. Models of the Heisenberg-Weil representation and the Metaplectic group
2. Dual pairs and general properties of theta correspondences
3. Theta correspondences over real fields
4. Theta correspondences over finite fields and p-adic fields
5. Geometric aspects of theta correspondence

This mini-course is designed for senior undergraduate students and graduate students.

This mini-course will live stream. Here is the zoom ID:
Meeting ID: 848 8485 0855
Passcode: 226224

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