Date/Time : 26 Nov 2024 14:00
Venue : S16 #05-18
Speaker : Patrick Lutz, University of California, Berkeley

Recently, there has been a spate of work applying tools from computability theory to prove theorems about Hausdorff dimension and packing dimension. Central to these applications are computability-theoretic analogues of Hausdorff dimension and packing dimension, known respectively as effective Hausdorff dimension and effective packing dimension. A fact which is useful for many applications is that for sufficiently simple sets—in particular, for Pi-0-1 sets—Hausdorff dimension and effective Hausdorff dimension agree. Unfortunately, the corresponding statement for packing dimension is known to fail. In particular, an example due to Conidis shows that there is a Pi-0-1 set of packing dimension 0 and effective packing dimension 1. In this talk, I will consider the question of exactly how bad this failure is. In particular, given a Pi-0-1 set E, what is the minimum complexity of an oracle A for which the packing dimension of E is equal to the effective packing dimension of E relative to A? Surprisingly, it turns out that there is not always even a hyperarithmetic oracle. I will also discuss to what extent this affects potential applications of effective packing dimension.

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Date/Time : 25 Nov 2024 14:00
Venue : S17 #04-06 (SR1)
Speaker : Tomasz Przebinda, University of Oklahoma

We describe an integral kernel operator which should map a Harish-Chandra character of a representation of one member of a dual pair to a distribution on the other member of the dual pair. A well founded hope is that this distribution coincides with the character of the big theta lift. In part this is a joint work with Hung Yean Loke.

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Date/Time : 21 Nov 2024 15:00
Venue : S17 #04-04 (SR3)
Speaker : Gunther Uhlmann, University of Washington

We will consider the question on whether we can determine the structure of space time by making measurements near the worldline of an observer. We will consider both active and passive measurements. For the case of passive measurements one measures the fronts of light sources near the observer. For the case of active measurements we couple Einstein equations with matter or electromagnetic fields and formulate the question of determining the structure of space time as the problem of recovering the metric from observations of waves near the observer, The method applies to several other inverse problems for nonlinear equations, for example, the Westervelt equation arising in nonlinear acoustics.

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Date/Time : 14 Nov 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Finn Wiersig, National University of Singapore

The Riemann-Hilbert correspondence establishes a profound connection between certain differential equations on a given complex analytic manifold and its topology. This result had numerous interesting applications; for example, it was applied by Beilinson-Bernstein and Brylinski-Kashiwara in the 1980s to study the representation theory of semisimple algebraic groups. This talk explores ongoing research leveraging these concepts to investigate representation theory appearing in the p-adic local Langlands program, focusing on progress towards a p-adic analytic Riemann-Hilbert correspondence.

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Date/Time : 13 Nov 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Duc-Bao Nguyen, National University of Singapore

We generalize previous diameter estimates and local non-vanishing of volumes for Kähler metrics to the case of big cohomology classes. In our proof, among other things, we will prove a uniform diameter estimate for a family of smooth Kähler metrics only involving an integrability condition. We also have to use fine stability properties of complex Monge-Ampere equations with prescribed singularities. This is joint work with Duc-Viet Vu

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Date/Time : 13 Nov 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Dilip Raghavan, National University of Singapore

I will present some recent consistency results on nowhere dense ultrafilters and a weakening of selectivity.

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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 : 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 14:00
Venue : S17 #04-05 (SR2)
Speaker : Tuan Tran, University of Science and Technology of China

Consider a multipartite graph G with maximum degree at most n-o(n), parts V1,…,Vk of size n each, and every vertex has at most o(n) neighbors in any part Vi. Loh and Sudakov proved that any such G has an independent transversal. They further conjectured that the vertex set of G can be decomposed into pairwise disjoint independent transversals. We resolve this conjecture approximately by showing that G contains n-o(n) pairwise disjoint independent transversals. As applications, we give approximate answers to questions of Yuster, and of Fischer, Kühn, and Osthus.

Joint work with Debsoumya Chakraborti.

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