Date/Time : 09 Apr 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Dougal Davis, University of Melbourne

One of the oldest open problems in representation theory is to classify the irreducible unitary representations of a real reductive Lie group G_R. Such representations play a fundamental role in harmonic analysis and the Langlands program and arise in physics as the Hilbert spaces of states of quantum systems with G_R-symmetry. The orbit method, which goes back to the work of Kirillov in the 1960s, takes the physical perspective seriously and proposes to construct all unitary representations as quantisations of orbits of the group on the dual of its Lie algebra. This has proved difficult to implement for reductive groups, however: the best general results use deformation quantisation to construct interesting examples of representations but provide no tools to prove that they are unitary. In this talk, I will explain joint work in preparation with Lucas Mason-Brown overcoming this difficulty: we give a uniform proof that, under suitable hypotheses, the orbit method always produces unitary representations. The main tool is recent joint work of myself and Kari Vilonen, which gives a new characterisation of unitary representations in terms of Hodge theory on complex flag varieties.

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Date/Time : 27 Mar 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Xiaochuan Tian, University of California, San Diego

Meshfree and particle methods are widely used in computational studies of partial differential equations, offering many advantages compared to traditional mesh- or grid-based numerical methods. Nevertheless, many practical questions revolve around meshfree methods, encompassing concerns about stability and accuracy. We propose a new paradigm of designing meshfree methods for solving partial differential equations (PDEs) through nonlocal analysis, inspired by the recent development of nonlocal calculus and its applications. We propose that the development of stable, accurate, and efficient meshfree methods relies on two key factors: (1) the formulation of well-posed continuum nonlocal models to approximate the PDE models and (2) the development of asymptotically compatible schemes for robust discretization of nonlocal models that allow a flexible coupling of the modeling and discretization parameters. We will review several aspects of nonlocal calculus and asymptotically compatible schemes and demonstrate the idea with a monotone meshfree method for solving linear elliptic equations in non-divergence form.

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Date/Time : 17 Jan 2024 14:00
Venue : S17 #04-06 (SR1)
Speaker : Ruixiang Zhang, University of California, Berkeley

Fourier restriction type problems form a class of important problems in harmonic analysis. They are also related to PDE, analytic number theory, geometric measure theory and mathematical physics. In the past 15 years, much progress has been made on these problems. In this talk, we will introduce Fourier restriction type problems and survey some recent results, and then conclude by mentioning some personal favorite future directions.

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Date/Time : 11 Jan 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : Yair Shenfeld, Brown University

It is classical that the ball is the unique shape which minimizes surface area for a given fixed volume. But what if we want to minimize surface area, where in addition to fixing the volume, we also fix the average width of the shape? This question turns out to be a very special case of long-standing conjectures at the heart of convex geometry. In the first part of the talk I will explain these conjectures and the significant progress that we made towards their resolutions. In the second part of the talk I will explain how these geometric problems can provide information on problems in combinatorics, which led us to the discovery of some surprising results in the theory of partially ordered sets.

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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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