Date/Time : 13 Nov 2024 14:00
Venue : S17 #04-06 (SR1)
Speaker : Guillaume Bal, The University of Chicago

Surprising asymmetric transport phenomena along interfaces separating insulating bulks have been observed in many areas of applied sciences, e.g., electronics, photonics, and geophysics. Such transport displays quite strong robustness to perturbations as an obstruction to Anderson localization. In fact, the phenomenon affords a topological origin: systems in the same topological class display the same robust, quantized, edge/interface transport. Perturbations may then be interpreted as continuous deformations that preserve the topological class.

This talk considers such systems modeled by general elliptic partial differential operators on the Euclidean plane. We review a recent simple topological classification, which provides an explicit computation of a topological invariant, technically the index of a Fredholm operator obtained by means of confining domain walls. We next introduce a physical observable that allows us to quantify the asymmetry of the edge transport. The evaluation of such an observable is challenging in practice. We then present a bulk-edge correspondence, a pillar of topological phases of matter in the physical literature, which shows that the interface current observable is in fact equal to the aforementioned simple topological invariant.

The theoretical findings are illustrated with examples ranging from electronics applications to geophysical fluid flows.

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Date/Time : 13 Nov 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Melvin Leok, University of California, San Diego

Geometric mechanics describes Lagrangian and Hamiltonian mechanics geometrically, and information geometry formulates statistical estimation, inference, and machine learning in terms of geometry. A divergence function is an asymmetric distance between two probability densities that induces differential geometric structures and yields efficient machine learning algorithms that minimize the duality gap. The connection between information geometry and geometric mechanics will yield a unified treatment of machine learning and structure-preserving discretizations. In particular, the divergence function of information geometry can be viewed as a discrete Lagrangian, which is a generating function of a symplectic map, that arise in discrete variational mechanics. This identification allows the methods of backward error analysis to be applied, and the symplectic map generated by a divergence function can be associated with the exact time-h flow map of a Hamiltonian system on the space of probability distributions. We will also discuss how time-adaptive Hamiltonian variational integrators can be used to discretize the Bregman Hamiltonian, whose flow generalizes the differential equation that describes the dynamics of the Nesterov accelerated gradient descent method.

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Date/Time : 11 Sep 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Binyong Sun, IASM, Zhejiang University

After introducing some basic notions in the representation theory of classical Lie groups, I will explain three results in this theory: the multiplicity one theorem for classical groups, the conservation relation for local theta correspondence, and the non-vanishing hypothesis at infinity for Rankin-Selberg convolutions. Some relevant or consequent results will also be explained.

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Date/Time : 21 Aug 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Shahar Mendelson, Australian National University

Structure recovery is at the heart of modern Data Science. Roughly put, the goal in structure recovery problems is to identify (or at least approximate) an unknown object using limited, random information – e.g., a random sample of the unknown object.

As it happens, key questions on recovery are fundamental (open) problems in Asymptotic Geometric Analysis and High Dimensional Probability. In this talk I will give one example (out of many) that exhibits the rather surprising ties between those seemingly unrelated areas.

I will explain why noise-free recovery is dictated by the geometry of the natural function class extension of a random projection of a subset of 𝑅^𝑑 : for a class of functions 𝐹 and n i.i.d random variables 𝜎 = (𝑋_1,…,𝑋_n), the corresponding coordinate projection of 𝐹 is the set 𝑃_𝜎 (𝐹) = { (𝑓(𝑋_1),….,𝑓(𝑋_n)) : 𝑓 ∈ 𝐹 } ⊂ 𝑅^n.

I will outline a (sharp) estimate on the structure of a typical 𝑃_𝜎 (𝐹) that leads to the solution of the noise-free recovery problem under minimal assumptions. I will explain why the same estimate resolves various questions in high dimensional probability (e.g., the smallest singular values of certain random matrices) and high dimensional geometry (e.g., the Gelfand width of a convex body).

The optimality of the solution is implied by a exposing a “hidden extremal structure” contained in 𝑃_𝜎 (𝐹), which in turn is based on a complete answer to Talagrand’s celebrated entropy problem.

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Date/Time : 22 Apr 2024 13:30
Venue : S17 #04-06 (SR1)
Speaker : Kevin Buzzard, Imperial College London

Computer programs which understand the axioms of mathematics have existed for decades. However, it is only in the last few years that they have captured the imagination of mathematicians. In this talk, I will give an overview of where we are in 2024, what these systems can currently do, and what they cannot yet do. I will discuss how these systems have helped mathematicians, and ways in which they may help mathematicians in the future. I will be careful to distinguish facts from science fiction. The talk will be suitable for a general mathematical audience and will not assume any expertise in AI, computer science, or interactive theorem provers.

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Date/Time : 03 Apr 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Tomoyuki Arakawa, Kyoto University

The 4D/2D duality is a physical theory discovered rather recently by Beem et al, which gives a vertex algebra to any 4D N=2 superconformal field theory. The resulting vertex algebra is an important invariant of a 4-dimensional theory, which may be regarded as a refinement of the Higgs branch.

In this talk I will discuss the representation theory of vertex algebras arising from 4-dimensional theories.

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Date/Time : 14 Feb 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Zhiming Chen, Chinese Academy of Sciences

We consider high-order unfitted finite element methods on Cartesian meshes with hanging nodes for elliptic interface problems, which release the work of body-fitted mesh generation and allow us to design adaptive finite element methods for solving curved geometric singularities. We introduce new concepts of the large element and interface deviation to solve the small cut cell problem of unfitted finite element methods. We construct reliable algorithms to merge small interface elements with their surrounding elements to automatically generate the finite element mesh whose elements are large with respect to both domains. Numerical examples are presented to illustrate the competitive performance of the method. This talk is based on joint works with Ke Li, Yong Liu and Xueshuang Xiang.

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Date/Time : 08 Feb 2024 15:00
Venue : S17-05-11 (SR 5)
Speaker : Tan Bui-Thanh, The University of Texas at Austin

Digital twins (DTs) are digital replicas of systems and processes. At the core of a DT is a  physical/mathematical model which captures the behavior of the real system across temporal and spatial scales. One of the key roles of DTs is enabling “what if” scenario testing of hypothetical simulations to understand the implications at any point throughout the life cycle of the process, to monitor the process, and to calibrate parameters to match the actual process. In this talk we will present various (faster than) real time Scientific Deep Learning (SciDL) approaches for forward, inverse, and UQ problems. Both theoretical and numerical results for various problems including transport, heat, Burgers, Euler, and Navier-Stokes equations will be presented.

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Date/Time : 06 Feb 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Dong Bin, Peking University

This presentation delves into the recent advancements in applying artificial intelligence (AI) to mathematical exploration. Initially, we will outline the current state of integrating AI methodologies, including machine learning, deep learning, and neural networks, into mathematical research, thereby fostering enhanced approaches and innovative concepts. Subsequently, we will highlight some preliminary results from ongoing research in this intersecting domain. Concluding the discussion, we will anticipate the future trajectory of research at the intersection of AI and mathematics, focusing on the potential of AI techniques to tackle new challenges and uncover novel problems in mathematical studies.

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