Date/Time : 18 Dec 2024 10:45
Venue : S17 #04-05 (SR2)
Speaker : Robin Zhang, MIT

In general, local converse theorems say that generic or cuspidal representations can be uniquely determined by their Rankin–Selberg gamma factors. After Jacquet and Langlands first proved such a statement for complex representations of local GL(2), an extensive series of works has established local converse theorems for complex representations of classical groups, some other local quasi-split reductive groups, and finite GL(n). A “naïve local converse theorem” fails for modular representations of finite GL(n) in characteristic ℓ > 0, but I will discuss how to fix this and prove a local converse theorem for finite GL(n) after defining a more general gamma factor with values in certain ℤ[1/p, ζp]-algebras. In the spirit of Henniart’s proof that the local Langlands correspondence for GL(n) is uniquely characterized by the compatibility of gamma factors, one may hope that this can give a “finite field local Langlands in families” determined by gamma factors. This is based on joint work with J. Bakeberg, M. Gerbelli-Gauthier, H. Goodson, A. Iyengar, and G. Moss.

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Date/Time : 18 Dec 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Jiepeng Fang, University of Hong Kong

By using Lusztig sheaves on the framed quiver, we give a categorical realization of the tensor product of integrable highest weight modules of the quantum group such that simple perverse sheaves in the category provide the canonical basis of the tensor product. As a consequence, we obtain the positivity of the canonical basis of the tensor product. This talk is based on a joint work with Yixin Lan.

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Date/Time : 18 Dec 2024 15:15
Venue : S17 #04-05 (SR2)
Speaker : Matteo Parisi, Harvard University

The amplituhedron is the image of the positive Grassmannian—the region of the Grassmannian where all Plücker coordinates are nonnegative—under a totally positive linear map. It is a far-reaching generalization of cyclic polytopes and hyperplane arrangements, and the positive Grassmannian itself. The “volume” of the amplituhedron encodes probabilities of particle interactions in the quantum field theory N=4 super Yang-Mills, and calculating this volume involves decomposing (or tiling) the amplituhedron into smaller pieces (or tiles) and summing their volumes. This talk will delve into the rich combinatorics of these tiles and tilings, presenting recent results on some of the central conjectures in this area, including the magic number, BCFW tiling, and cluster adjacency conjectures.

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Date/Time : 18 Dec 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Wenjie Fang, Université Gustave Eiffel

We consider the number of occurrences of subwords (non-consecutive sub-sequences) in a given word. We first introduce the notion of subword entropy that measures the maximal number of occurrences among all possible subwords. We then give upper and lower bounds of minimal subword entropy for words of fixed length in a fixed alphabet, and show that minimal subword entropy per letter has a limit value. A better upper bound of minimal subword entropy for a binary alphabet is then given by looking at certain families of periodic words. We also give some conjectures based on experimental observations.

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Date/Time : 18 Dec 2024 09:30
Venue : S17 #04-05 (SR2)
Speaker : Kazim Buyukboduk, University College Dublin

I will discuss the factorization of a certain triple product p-adic L-function whose interpolation range is empty. The said factorization reflects, but does not formally follow from due to the absence of an interpolation locus, the Artin formalism for the underlying family of motives (that decompose as the sum of 2 motives of respective degrees 2 and 6). I will explain how this factorization problem can be recast as the comparison of two families of Arithmetic GGP conjectures (and can be proved in some cases using this reduction).

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Date/Time : 18 Dec 2024 15:00
Venue : S17 #06-11 (SR6)
Speaker : Ding Lijun, University of California, San Diego

We revisit a formulation technique for inequality-constrained optimization problems that has been known for decades: the substitution of squared variables for nonnegative variables. Using this technique, inequality constraints are converted to equality constraints via the introduction of a squared-slack variable. Such formulations have the superficial advantage that inequality constraints can be dispensed with altogether. But there are clear disadvantages, not least being that first-order optimal points for the squared-variable reformulation may not correspond to first-order optimal points for the original problem, because the Lagrange multipliers may have the wrong sign. Extending previous results, this paper shows that points satisfying approximate second-order optimality conditions for the squared-variable reformulation also, under certain conditions, satisfy approximate second-order optimality conditions for the original formulation, and vice versa. Such results allow us to adapt complexity analysis of methods for equality-constrained optimization to account for inequality constraints. On the algorithmic side, we examine squared-variable formulations for several interesting problem classes, including bound-constrained quadratic programming, linear programming, and nonnegative matrix factorization. We show that algorithms built on these formulations are surprisingly competitive with standard methods.

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Date/Time : 18 Dec 2024 17:00
Venue : S17 #05-11 (SR5)
Speaker : Manlio Valenti, Swansea University

Recall that, in a partial order P, b is a minimal cover of b if the interval (a,b) is empty. A strengthening of this notion is the one of strong minimal cover, namely b is a strong minimal cover of a if c<b implies c\le a. In this talk, we provide a complete characterization of how minimal covers and strong minimal covers in the Weihrauch degrees look like. As a consequence, we obtain that the Weihrauch degree of the identity problem id can be defined as the highest top of a strong minimal cover, or the least degree whose upper cone is dense. In other words, “being (uniformly) computable” is a first-order definable property in (W,<). This implies that the first-order theory of the Weihrauch degrees (below id) is recursively isomorphic to the third-order theory of true arithmetic.

This is joint work with Steffen Lempp, Joe Miller, Arno Pauly, and Mariya Soskova.

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Date/Time : 12 Dec 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Yihong Du, University of New England, Australia

A new phenomenon in nonlocal diffusion models is that accelerated propagation may happen, that is, the propagation speed could be infinite, which never occurs in the corresponding local diffusion model with compactly supported initial data. In this talk, we will first briefly review the history of the KPP model used to describe the propagation of biological/chemical species, and then look at some very recent results on the KPP equation with nonlocal diffusion and free boundaries. For several natural classes of kernel functions appearing in the nonlocal diffusion term, we will show how the exact rate of acceleration can be determined. The talk is based on joint works with several collaborators.

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Date/Time : 12 Dec 2024 16:30
Venue : S17 #04-04 (SR3)
Speaker : Huang Jizu, Academy of Mathematics and Systems Science, Chinese Academy of Sciences

In this talk, we propose a novel accuracy-preserved tensor-train (APTT) method to solve the Boltzmann-BGK equation. We use the Crank-Nicolson Leap Frog (CNLF) scheme for temporal discretization, based on which the collision term can be evaluated only once at each time step. A second-order scheme based on the upwind rule is employed for spatial discretization to match the accuracy of the CNLF scheme. At each time step, the linear system is constructed with the tensor-train (TT) format, where the matrix, the right-hand side, and the collision term involved in the linear system are all represented using the low-rank TT format. Based on such a representation, an efficient and effective iterative solver can be implemented for solving the linear system, which can reduce two orders of magnitude in terms of both time and memory costs compared with classical methods. We present a theoretical analysis to show that the proposed APTT method can maintain accuracy and an extensive set of 3D3V Boltzmann-BGK test examples to demonstrate its effectiveness.

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