Date/Time : 16 Sep 2022 09:00
Venue : via Zoom
Speaker : Yusheng Luo, Stony Brook University

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https://nus-sg.zoom.us/j/88075038959?pwd=cDAyUWMwMDVrd1lwZDQwdXJkbzk3QT09

Meeting ID: 880 7503 8959
Passcode: 384618

Abstract: In 1980s, D. Sullivan formulated a dictionary between discrete subgroups of PSL_2(\C), known as Kleinian groups, and rational maps of \hat \C. The Sullivan dictionary provides a conceptual framework for understanding the connections between these two branches of conformal dynamics.

Around the same time, W. Thurston’s 3-dimensional insights revolutionized the theory of Kleinian groups. He formulated the geometrization conjecture for 3-manifolds, and proved the hyperbolization theorem which implies the conjecture for a large class of 3-manifolds. The keys to the hyperbolization theorem are two bounded results for certain deformation spaces of Kleinian groups.

In 1990s, motivated by Thurston’s boundedness theorem and the Sullivan dictionary, C. McMullen conjectured that certain deformation spaces of hyperbolic rational maps are bounded.

In this talk, I will start with a survey of historical backgrounds, and discuss a general strategy of the proof of Thurston’s boundedness theorem. I will then explain how a similar strategy could work for rational maps, and discuss some recent breakthrough towards McMullen’s boundedness conjecture.

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Date/Time : 26 Aug 2022 09:00
Venue : via Zoom
Speaker : Jinyoung Park, Stanford University

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https://nus-sg.zoom.us/j/84821387838?pwd=QnRaVGtNVFBscWNreG4yVjJ2WGM3QT09

Meeting ID: 848 2138 7838
Passcode: 969669

Abstract: Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is never far from its “expectation-threshold,” which is a natural (and often easy to calculate) lower bound on the threshold. In this talk, I will present recent progress on this topic. Based on joint work with Huy Tuan Pham.

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Date/Time : 30 Jun 2022 16:00
Venue : S17 #04-06 (SR1)
Speaker : Shirshendu Ganguly, University of California, Berkeley

Abstract: The Kardar-Parisi-Zhang (KPZ) universality class refers to a broad family of models of random growth which are believed to exhibit certain common features, such as universal scaling exponents and limiting distributions.

Planar first passage percolation models (FPP), where one considers a distortion of the planar Euclidean metric by random noise are canonical examples of random planar metrics expected to lie in this class. A key observable of interest is the geodesic, the (random) shortest path between specified endpoints.

These models can also be viewed as examples of disordered systems admitting complex energy landscapes. In this formulation, the geodesic is the ground state, the configuration with the lowest energy, lying at the base of the deepest valley.

In this talk we will report recent progress in understanding various geometric features of such energy landscapes for certain variants of FPP known as last passage percolation (LPP) admitting certain key algebraic features, and their manifestations in intricate fractal behavior as well as sensitivity to dynamical perturbations of the underlying noise (noise-sensitivity and chaos).

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Date/Time : 27 May 2022 09:00
Venue : via Zoom
Speaker : Alan Chang, Princeton University

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https://nus-sg.zoom.us/j/85408531758?pwd=VDVXRXVFQ09RNjQ3QWFNeC9wUDlLdz09

Meeting ID: 854 0853 1758
Passcode: 633851

Abstract: The Kakeya needle problem asks the following: What is the smallest area in the plane in which a unit line segment can be rotated 180 degrees? This question, which can be solved with elementary geometry, has surprising connections to many other topics (e.g., harmonic analysis, PDEs, analytic number theory, theoretical computer science). In this talk, we will give a survey of this classical problem, and then we will discuss a variant where the line segment has been replaced by a C^1 curve (or more generally, a 1-rectifiable set). This is joint work with Marianna Csörnyei.

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Date/Time : 22 Apr 2022 09:00
Venue : via Zoom
Speaker : Yuansi Chen, Duke University

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https://nus-sg.zoom.us/j/85706793159?pwd=bTlhVVZJeVE3Z2dDU0w1bDZmUGxXQT09

Meeting ID: 857 0679 3159
Passcode: 533414

Abstract: Kannan, Lovász, and Simonovits (KLS) conjectured in 1995 that the Cheeger isoperimetric coefficient of any log-concave density or any convex body is achieved by half-spaces up to a universal constant factor. This conjecture now plays a central role in the field of convex geometry, unifying or implying older conjectures. In particular, it implies Bourgain’s slicing conjecture (1986) and the thin-shell conjecture (2003). The previous best bound with dimension dependency $d^{1/4}$ was established by Lee and Vempala in 2017, which also matches the best dimension dependency Klartag obtained in 2006 for Bourgain’s slicing conjecture.

In recent work, we improve the Eldan’s stochastic localization proof technique deployed in Lee and Vempala (2007) to prove an almost constant Cheeger isoperimetric coefficient in the KLS conjecture with dimension dependency $d^{o(1)}$.

In this talk, first we briefly survey the origin and the main consequences of these conjectures. Then we present the development and the refinement of Eldan’s stochastic localization scheme. Finally, we explain a few proof details which result in the current best bound of the Cheeger isoperimetric coefficient in the KLS conjecture.

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Date/Time : 25 Mar 2022 09:00
Venue : via Zoom
Speaker : Chao Li, Columbia University

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https://nus-sg.zoom.us/j/87647484260?pwd=Zys0M211ekFkeE9OZlRCOHcvS0NLZz09

Meeting ID: 876 4748 4260
Passcode: 511096

Abstract: Can an integer n be represented as a sum of two squares n=x^2+y^2? If so, how many different representations are there? We begin with the answers to these classical questions due to Fermat and Jacobi and put it in the modern perspective of the Siegel-Weil formula. After illustrating Kudla’s influential program on geometric and arithmetic generalizations using the example of modular curves, we report on recent theorems establishing arithmetic Siegel-Weil formulas and discuss applications including BSD-type conjectures for higher dimensional varieties.

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Date/Time : 25 Feb 2022 09:00
Venue : via Zoom
Speaker : Daniel Russo, Columbia University

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https://nus-sg.zoom.us/j/86446962104?pwd=cFFTQzhtOU9OTFd5RUZhOHBPSjcwQT09

Meeting ID: 864 4696 2104
Passcode: 673331

Abstract: Multi-armed bandit algorithms minimize experimentation costs required to converge on optimal behavior. They do so by rapidly adapting experimentation effort away from poorly performing actions as feedback is observed. But this desirable feature makes them sensitive to confounding, which is the primary concern underlying classical randomized controlled trials. We highlight, for instance, that popular bandit algorithms cannot address the problem of identifying the best action when day-of-week effects may confound inferences. In response, this paper proposes deconfounded Thompson sampling, which makes simple, but critical, modifications to the way Thompson sampling is usually applied. Theoretical guarantees suggest the algorithm strikes a delicate balance between adaptivity and robustness to confounding. It attains asymptotic lower bounds on the number of samples required to confidently identify the best action — suggesting optimal adaptivity — but also satisfies strong performance guarantees in the presence of day-of-week effects and delayed observations — suggesting unusual robustness.

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Date/Time : 28 Jan 2022 16:00
Venue : via Zoom
Speaker : Riddhipratim Basu, ICTS-TIFR

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https://nus-sg.zoom.us/j/81804554884?pwd=MEx6eVptVmxYZjNPMjZXZE1IaWxUQT09

Meeting ID: 818 0455 4884
Passcode: 743763

Abstract: First and last passage percolation on the two dimensional Euclidean lattice are canonical models of stochastic planar growth that are generically believed to exhibit universal behaviour in the Kardar-Parisi-Zhang (KPZ) class. This, however, is rigorously known only for a handful of integrable models, e.g., the exponential LPP on the plane. In this talk, I shall describe some results on understanding of the geometry of the geodesics in these exactly solvable models using estimates from integrable probability and illustrate how this geometric understanding, in turn, can be used to obtain more refined information on the passage time landscape.

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Date/Time : 22 Dec 2021 09:00
Venue : via Zoom
Speaker : Jiaoyang Huang, New York University

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https://nus-sg.zoom.us/j/86201788855?pwd=ZW9WQ0pmM2RTOS9SbSsrakdIS2FRQT09

Meeting ID: 862 0178 8855
Passcode: 627300

Abstract: The success of random matrices in modeling physical systems lies in the universality phenomenon of their eigenvalue statistics. The general belief is for systems with many strongly interacting components, we expect to see random matrix statistics. The universality is well understood for Wigner matrices, which are dense. In this talk, I will first discuss some results concerning random matrix statistics in sparse systems—the sparse random graphs. Beyond matrix setting, random matrix statistics are conjectured to govern the asymptotic behavior of various random growth models and interacting particle systems. However, this was only proven for some exactly solvable models. I will discuss a general strategy to prove the universality of random matrix statistics, and our recent result proving that for random lozenge tilings of polygons, the scaling limit of the extreme path is given by random matrix statistics.

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