Date/Time : 27 Dec 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Jingbo Wan, Columbia University

In this talk, we discuss the rigid behaviour of contracting maps between positively curved manifolds via the method of geometric heat flows. We establish rigidity results for distance and area non-increasing maps between spheres, complex projective spaces and other positively curved manifolds under various curvature conditions. The methods rely on energy monotonicity for harmonic map heat flow, and the Strong Maximum Principle for Mean Curvature Flow or the coupled Mean Curvature Flow-Ricci Flow system. These results demonstrate and extend an interesting application of geometric heat flows to the homotopy classes of maps, pioneered by Prof. Mao-Pei Tsui and Prof. Mu-Tao Wang in 2004. This is based on joint works with Man-Chun Lee and Luen-Fai Tam.

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Date/Time : 26 Dec 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Sang-hyun Kim, Korea Institute for Advanced Study

Given a topological (C0) action of a group on a manifold, one can ask the best possible regularity (Cr) with which the group may act smoothly. Estimation of such regularities involve interplays among group theory (nilpotence, solvability and so forth), analysis (Hölder regularity) and dynamics (tracing orbits). In this talk, We prove that if a countable group does not contain a finitely generated subgroup of exponential growth, then every topological action of the group on a compact connected one-manifold can be blown-up to a C1 action. The proof is based on a functional characterisation of such groups. Joint work with Mikael de la Salle, Nicolas Matte Bon and Michele Triestino.

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Date/Time : 20 Dec 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Shouhong Wang, Indiana University

The first part of the talk is to introduce the general framework of phase transition dynamics for deterministic dissipative systems, with various applications to statistical physics and fluid flows. This is based on joint work with Dr. Tian Ma. In the second part, we study transitions in stochastic non-equilibrium systems. As we know, a central challenge in physics is to describe non-equilibrium systems driven by randomness. For deterministic systems, the center manifold theory has shown a prodigious efficiency to often completely characterize how the onset of linear instability translates into the emergence of nonlinear patterns, associated with genuine physical regimes. In presence of random fluctuations, the underlying reduction principle to the center manifold is seriously challenged due to large excursions caused by the noise. We present an alternative framework to cope with these difficulties exploiting the approximation theory of stochastic invariant manifolds, on one hand, and energy estimates measuring the defect of parameterization of the high-modes, on the other. As a result, the approach enables us to predict, from reduced equations of the stochastic fluid problem, the occurrence in large probability of a stochastic analogue to the pitchfork bifurcation. This is joint work with Mickael Chekroun, Honghu Liu, and James McWilliams [JDE, 346(2023), 145-204].

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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 : 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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