Date/Time : 02 Dec 2024 16:00
Venue : S17 #04-04 (SR3)
Speaker : Changguang Dong, Nankai University

We will discuss a dynamically defined zeta function for partially hyperbolic systems. This is a generalization of the classical Ruelle zeta function for Anosov (more generally Axiom A) diffeomorphisms/flows, which may reflect dynamical invariants such as entropy and pressure, as well as the geometry/topology of the manifolds. Based on an ongoing project with Qiujie Qiao.

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Date/Time : 28 Nov 2024 13:30
Venue : S17 #04-05 (SR2)
Speaker : Peter Buhlmann, ETH Zurich

Nonparametric function estimation and prediction with moderate or large dimension of the covariates are particularly susceptible to extrapolation, because data points are typically far apart from each other in such moderate or higher dimension. Thus, there is a need to have machine learning methods that are extrapolation-aware, i.e. that automatically perform well (in a sense) when extrapolation occurs. Without such extrapolation-aware techniques, inference from standard machine learning and nonparametric procedures may be poor or invalid. We introduce a novel conceptual framework and introduce *Xtrapolation* which allows for extrapolation-aware inference with any ML algorithm.

This is joint work with Niklas Pfister (University of Copenhagen)

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Date/Time : 27 Nov 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Marco Vergamini, Scuola Normale Superiore

Let f be a complex Hénon map and µ its unique measure of maximal entropy. Recently, Bianchi-Dinh proved that µ is exponentially mixing of all orders for all Hölder observables, and that all such observables satisfy the central limit theorem with respect to µ. De Thélin-Vigny generalized these results for a certain class of bounded plurisubharmonic observables. We prove that these properties hold for all, not necessarily bounded, plurisubharmonic observables. This is a joint work with Hao Wu.

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Date/Time : 27 Nov 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Ruiyi Yang, Princeton University

Gaussian processes (GPs) are important random function models with desirable analytic properties that have found wide applications in inverse problems, spatial statistics, and machine learning. In this talk, we shall investigate a generalization of the popular Matérn GP to the manifold setting. In the first part, we formalize its definition and introduce a graph-based approximation that is computable given only a point cloud of samples from the manifold. The resulting graph Matérn GP enjoys a sparsity structure that facilitates computation and is widely applicable in many aspects of Bayesian methodologies. In the second part, we discuss in detail its application in a manifold optimization problem where gradients are intractable to obtain. Exploiting tools from Bayesian optimization, we propose an efficient algorithm with provable guarantees, whose effectiveness is further demonstrated by numerical examples.

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Date/Time : 27 Nov 2024 14:00
Venue : S17 #04-06 (SR1)
Speaker : Patrick Lutz, University of California, Berkeley

The field of computability theory studies the complexity of uncomputable problems (where a “problem” is usually formally defined as either a set of natural numbers or a set of binary strings). In this study, a special role is played by the Halting Problem—i.e. the problem of determining whether a given program stops after a finite number of steps or runs forever. Not only is it the first problem proved to be uncomputable, it also seems to be the simplest “natural” uncomputable problem.

Martin’s Conjecture is a long-standing open question in computability theory which partially explains why the Halting Problem plays such a special role. A key idea behind Martin’s Conjecture is to view the Halting Problem not just as an individual problem, but as an operator on problems, which takes any problem to a strictly harder one. Martin’s Conjecture consists of a classification of such operators, which says, in part, that the Halting Problem is the minimal non-trivial operator. I will discuss the background and motivation for Martin’s Conjecture, as well as recent progress by Benjamin Siskind and myself which essentially completes a proof of the conjecture for a special class of operators called “order-preserving” (i.e. those which preserve the relative complexity of problems).

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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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