Date/Time : 13 Dec 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Xin Sun, Peking University and University of Pennsylvania

Smirnov’s proof of Cardy’s formula for percolation on the triangular lattice leads to a discrete approximation of conformal maps, which we call the Cardy-Smirnov embedding. Under this embedding, Holden and I proved that the uniform triangulation converge to a continuum random geometry called pure Liouville quantum gravity. There is a variant of the Gaussian free field governing the random geometry, which is an important example of conformal field theory called Liouville CFT. A key motivation for understanding Liouville quantum gravity rigorously is its application to the evaluation of scaling exponents and dimensions for 2D critical systems such as percolation. Recently, with Nolin, Qian and Zhuang, we used this idea and the integrable structure of Liouville CFT to derive a scaling exponent for planar percolation called the backbone exponent, which was unknown for several decades.

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Date/Time : 18 Oct 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Ruobing Zhang, Princeton University

This lecture concerns the metric Riemannian geometry of Einstein manifolds, which is a central theme in modern differential geometry and is deeply connected to a large variety of fundamental problems in algebraic geometry, geometric topology, analysis of nonlinear PDEs, and mathematical physics. We will exhibit the rich geometric/topological structures of Einstein manifolds and specifically focus on the structure theory of moduli spaces of Einstein metrics.

My recent works center around the intriguing problems regarding the compactification of the moduli space of Einstein metrics, which tells us how Einstein manifolds can degenerate. Such problems constitute the most challenging part in the metric geometry of Einstein manifolds. We will introduce recent major progress and propose several open questions in the field.

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Date/Time : 23 Aug 2023 15:00
Venue : S17 #05-12 (SR4)
Speaker : William Johnson, Fudan University

Model theory is the study of “definable sets” in mathematical structures like posets, graphs, groups, rings, and fields. In some structures, the definable sets are very tame and well-behaved, while in other structures they are wild. Many of the model-theoretically tame structures are “finite dimensional”, in the sense that there is a good dimension theory for definable sets. There are several model-theoretic ways to define “dimension”. In this talk, we will mostly be interested in Morley rank and dp-rank. These notions of “dimension” or “rank” are intuitively analogous to notions like topological dimension and Hausdorff dimension in geometry, but they are defined purely based on the combinatorics of definable sets.

The driving question of this talk is the following: Which structures from algebra (e.g., groups, rings, and fields) are “finite-dimensional” (e.g., finite Morley rank, finite dp-rank)? In this talk, we will survey the classic work of Macintyre on fields and rings of finite Morley rank, the conjecture of Cherlin and Zilber on
groups of finite Morley rank, and my own work in recent years on fields and rings of finite dp-rank, including a recent unpublished result, joint with Yatir Halevi and Christian d’Elbee on rings of dp-rank 1.

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Date/Time : 10 Mar 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Jie Ma, University of Science and Technology of China

A fundamental theorem of Rademacher from 1941 led to the study of supersaturation problems of graphs, which aim to count the minimum number of copies of a given graph F among all graphs with n vertices and m edges. This is closely related to a central concept in Extremal Graph Theory — the Turán number of F, which denotes the maximum number of edges in an n-vertex graph which does not contain F as a subgraph. Famous results of Erdős, and Lovász and Simonovits determine the minimum number of cliques K_r in graphs whose number of edges exceed the Turán number of K_r. Subsequent works of Mubayi as well as Pikhurko and Yilma extend these classical results from cliques to color-critical graphs, a rich family playing important roles in extremal problems. In this talk, we will discuss supersaturation problems beyond color-critical graphs and investigate natural enumerative parameters.

Our results go beyond the previous results and show that supersaturation problems for general graphs can be rather complicated. Among others, we disprove a conjecture of Mubayi. Joint work with Long-Tu Yuan.

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Date/Time : 28 Oct 2022 15:00
Venue : S17 #04-06 (SR1)
Speaker : Ting-Kam Leonard Wong, University of Toronto

Abstract: Macroscopic properties of equity markets present interesting problems in portfolio management and stochastic modeling. If we think of stocks as stars in a galaxy, a macroscopic property would be how the galaxy spirals. We begin by presenting some empirical observations about the capital distribution curve, market diversity and intrinsic volatility, and discuss why these macroscopic properties are difficult to capture probabilistically. One way to exploit these properties is to use functionally generated portfolios in stochastic portfolio theory.

Financially, they capture market volatility by systematic rebalancing. Mathematically, they are generalized gradient fields of exponentially concave functions, i.e., functions whose exponentials are concave. We show that these portfolios solve an optimal transport problem whose cost function is related to the Dirichlet distribution. Finally, motivated by online portfolio selection, we consider construction of prior (probability) distributions on spaces of concave functions and present some interesting limit theorems.

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Date/Time : 30 Sep 2022 14:00
Venue : S17 #04-05 (SR2)
Speaker : Mengdi Wang, Princeton University

Abstract:  Consider the estimation of policy gradient (PG) using data that maybe collected in off-policy manners. The talk presents a Policy Gradient Bellman Equation and shows how to leverage Q function approximator for policy gradient estimation via a double fitted iteration. We show that the double fitted iteration is equivalent to a plug-in estimator. Further, the PG estimation error bound is determined by a restricted chi-square divergence that quantifies the interplay between distribution shift and function approximation. A matching Cramer Rao lower bound is also provided.

Next, we show how to extend policy gradient method to reinforcement learning beyond cumulative rewards. In particular, consider the objective that is a general concave utility function of the state-action occupancy measure, containing as special cases maximal exploration, imitation and safety constrained RL. Such generality invalidates the Bellman equation and Sutton’s Policy Gradient Theorem. We derive a new Variational Policy Gradient Theorem for RL with general utilities, which establishes that the parametrized policy gradient may be obtained as the solution of a stochastic saddle point problem. We also exploit the hidden convexity of the Markov decision process and prove that the variational policy gradient scheme converges globally to the optimal policy for the general objective, though the optimization problem is nonconvex.

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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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