Date/Time : 08 Oct 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Peter Markowich, King Abdullah University of Science and Technology

We propose a partial differential-integral equation (PDE) framework for deep neural networks (DNNs) and their associated learning problems by taking the continuum limits of both network width and depth. The proposed model captures the complex interactions among hidden nodes, overcoming limitations of traditional discrete and ordinary differential equation (ODE)-based models. We explore the well-posedness of the forward propagation problem, analyze the existence and properties of minimizers for the learning task, and provide a detailed examination of necessary and sufficient conditions for the existence of critical points. Controllability and optimality conditions for the learning task with its associated PDE forward problem are established using variational calculus, the Pontryagin Maximum Principle, and the Hamilton-Jacobi-Bellman equation, framing the deep learning process as a PDE-constrained optimization problem. In this context, we prove the existence of viscosity solutions for the latter and we establish optimal feedback controls based on the value functional. This approach facilitates the development of new network architectures and numerical methods that improve
upon traditional layer-by-layer gradient descent techniques. The paper provides a mathematical foundation for connecting neural networks, PDE theory, variational analysis, and optimal control. By integrating these fields, we offer a robust framework that enhances deep learning models’ stability, efficiency, and interpretability.

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Date/Time : 07 Oct 2025 13:00
Venue : S16 #05-18
Speaker : Jiakun Liu, The University of Sydney

In this talk, we will introduce the optimal transportation with some interesting applications in various fields including a reconstruction problem in cosmology; a brief proof of isoperimetric and Brunn-Minkowski inequalities in geometry; and an application in image recognition relating to a transport between hypercubes. We will also discuss our recent research on the regularity of free boundaries in optimal transportation.

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Date/Time : 07 Oct 2025 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Lo Chi-Heng, National University of Singapore

Last week, Jing Quan introduced three operators ui, dual-ui-dual, and partial duals (together with their inverses), which preserve the representations associated to extended multi-segments. Among these six types of operators, half of them raise the “temperedness” of the corresponding local Arthur parameters. We refer to these as the raising operators.

In this talk, I will present three orderings on A-parameters that measure temperedness. I will then show that for any representation π of Arthur type, the set

Ψ(π):={ ψ | π is in Π_ψ }

contains unique elements ψ^{max}(π) and ψ^{min}(π), which are respectively the maximal and minimal members under all of these 3 orderings. Finally, I will discuss several representation theoretic properties of these distinguished members. If time permits, I will introduce the fourth ordering: the closure ordering.

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Date/Time : 03 Oct 2025 14:00
Venue : S16 #04-35
Speaker : Song Linliang, Tongji University

In this talk, we introduce two new diagrammatic monoidal categories—the affine q-web category and the affine q-Schur category—along with their respective cyclotomic quotient categories. We construct diagrammatic integral bases for the Hom-spaces in all these categories. Furthermore, we establish the following isomorphisms, which provide the first diagrammatic presentations of these important algebras: (i) the path algebras of the affine q-web category are isomorphic to R. Green’s affine q-Schur algebras; (ii) the path algebras of the affine q-Schur category are isomorphic to Maksimau-Stroppel’s higher level affine q-Schur algebras; and most significantly, (iii) the path algebras of the cyclotomic q-Schur categories are isomorphic to Dipper–James–Mathas’ cyclotomic q-Schur algebras. This is based on joint projects with Yaolong Shen and Weiqiang Wang.

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Date/Time : 03 Oct 2025 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Mihir Mittal, Indian Institute of Technology Kanpur

Zarankiewicz’s problem asks: Given a bipartite graph on n vertices without a K_{k,k} subgraph, how many edges can it have? The classical Kővári–Sós–Turán Theorem gives us the bound O(n^{2-1/k}); this remains the best possible bound for general situations. In geometric settings, however, stronger bounds can be achieved. Recent work of Basit, Chernikov, Starchenko, Tao, and Tran showed that semi-linear incidence graphs in real vector spaces admit almost-linear Zarankiewicz bounds.

In this talk I will describe a counterpart of their result for the additive group of the p-adic numbers and more generally in valued vector spaces. This is obtained through introducing a new purely combinatorial notion of laminarity. As a corollary, we get almost-linear bounds for K_{k,k}-free point–box and point–polytope incidence graphs in these settings.

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Date/Time : 01 Oct 2025 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Homin Lee, Korea Institute for Advanced Study

In this talk, we will discuss about smooth higher rank lattice actions on manifolds with positive entropy. For instance, when lattices in SL(n,R) act on an n-dimensional manifold with positive entropy with n at least 3, we will see that the lattice is commensurable with SL(n,Z) and a topological information on manifolds. This is a joint work with Aaron Brown.

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Date/Time : 01 Oct 2025 14:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Akshay Hegde , National University of Singapore

Analysing the critical point structure of smooth Gaussian fields on $\mathbb{R}^d$ has been paramount in various probability problems. For example, the excursion probability P( sup f >u) can be precisely estimated by counting the number of critical points above the level u. Other examples include optimization of spherical spin glass models, percolation of level sets of smooth Gaussian fields. In this talk, we’ll explore the Poisson structure of high critical points of stationary smooth Gaussian fields. This is joint work with Dmitry Belyaev. 

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Date/Time : 30 Sep 2025 15:00
Venue : S16 #05-17
Speaker : Zhifei Zhu, Tsinghua University

In this talk, we show that the length of a shortest closed geodesic on a Riemannian manifold of dimension 4 with diameter D, volume v, and |Ric|<3 can be bounded by a function of v and D. In particular, this function can be explicitly computed if the manifold is Einstein. The proof of this result depends on a structural theorem proven by J. Cheeger and A. Naber. This is joint work with N. Wu.

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Date/Time : 30 Sep 2025 16:00
Venue : S16 #05-17
Speaker : Wenze Su, Tsinghua University

In this talk, I will present a recent result on the finite-time shock formation for the compressible Euler equations on the two-dimensional sphere $\mathbb{S}^2$. Unlike the flat Euclidean case $\mathbb{R}^2$, the geometry imposes new difficulties and the fluid dynamics is affected by the curved background.

To overcome these challenges, we modified the existing modulation method and employ a set of carefully constructed, time-dependent coordinates that precisely track the shock formation on $\mathbb{S}^2$. In particular, we apply time-dependent rotations on the sphere, stereographic projections for the rotated sphere, and straighten the steepening shock front to construct the desired coordinate system. Under these coordinates, the Euler’s equations on $\mathbb{S}^2$ can be recast into a new form that is suitable for self-similar analysis.

Within this framework, we then implement detailed bootstrap arguments and establish global well-posedness for the self-similar system. After translating these results back to the original physical system, we thereby demonstrate the finite-time shock formation on $\mathbb{S}^2$.

This talk is based on joint work with Xinliang An, Haoyang Chen, and Fulin Qi.

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