Date/Time : 20 Aug 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Sung-Soo Byun and Seong-Mi Seo , Seoul National University and Chungnam University

This is a lecture series on Coulomb gases on Riemann surfaces. We will in particular focus on partition functions of two-dimensional Coulomb gases on Riemann surfaces and correlations of determinantal point processes on complex manifolds. We will discuss the macroscopic properties of Coulomb gases, which can be effectively described using logarithmic potential theory. A key topic in Coulomb gas theory is free energy expansion as the system size tends to infinity. This asymptotic expansion plays a crucial role in establishing the law of large numbers and central limit theorems, and it also finds applications in several large deviation probabilities, such as hole probabilities. Additionally, the coefficients in this expansion encapsulate potential theoretic, topological, and conformal geometric aspects of the model. Throughout these lectures, we will explore predictions made by the physics community, as well as recent advancements within the mathematics community, with a focus on the planar case. On the other hand, in random matrix theory, the eigenvalues of a normal matrix form a determinantal point process that can be described as a system of Coulomb gases on the complex plane. This lecture series will introduce a generalization of this random normal matrix ensemble to complex manifolds, while preserving its determinantal structure. We will explore the global and local behaviour of the particle system in the limit of many particles, with a focus on the scaling limits of correlation functions and their universality.

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Date/Time : 19 Aug 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Sung-Soo Byun and Seong-Mi Seo , Seoul National University and Chungnam University

This is a lecture series on Coulomb gases on Riemann surfaces. We will in particular focus on partition functions of two-dimensional Coulomb gases on Riemann surfaces and correlations of determinantal point processes on complex manifolds. We will discuss the macroscopic properties of Coulomb gases, which can be effectively described using logarithmic potential theory. A key topic in Coulomb gas theory is free energy expansion as the system size tends to infinity. This asymptotic expansion plays a crucial role in establishing the law of large numbers and central limit theorems, and it also finds applications in several large deviation probabilities, such as hole probabilities. Additionally, the coefficients in this expansion encapsulate potential theoretic, topological, and conformal geometric aspects of the model. Throughout these lectures, we will explore predictions made by the physics community, as well as recent advancements within the mathematics community, with a focus on the planar case. On the other hand, in random matrix theory, the eigenvalues of a normal matrix form a determinantal point process that can be described as a system of Coulomb gases on the complex plane. This lecture series will introduce a generalization of this random normal matrix ensemble to complex manifolds, while preserving its determinantal structure. We will explore the global and local behaviour of the particle system in the limit of many particles, with a focus on the scaling limits of correlation functions and their universality.

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Date/Time : 31 Jul 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Jan M. Swart, The Czech Academy of Sciences

In a classical game two players, Alice and Bob, take turns to play n moves each. Alice starts. For each move each player has two options, 1 and 2. The outcome is determined by the exact sequences of moves played by each player. Prior to the game, a winner is assigned to each of the 2^{2n} possible outcomes in an i.i.d. fashion, where p is the probability that Bob is the winner for a given outcome. Then it is known that there exists a value 0<p_c<1 such that the probability that Bob has a winning strategy for large n tends to one if p>p_c and to zero if p<p_c. We study a modification of this game for which the outcome is determined by the exact sequence of moves played by Alice as before, but in the case of Bob all that matters is how often he has played move 1. We show that also in this case, there exists a sharp threshold p’_c that determines which player has with large probability a winning strategy in the limit as n tends to infinity.

Joint work with Anja Sturm and Natalia Cardona-Tob\’on (G\”ottingen).

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Date/Time : 12 Jul 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Nishant Chandgotia, TIFR CAM (Bangalore)

The dimer model, also referred to as domino tilings or perfect matching, are tilings of the Z^d lattice by boxes exactly one of whose sides has length 2 and the rest have length 1. This is a very well-studied statistical physics model in two dimensions with many tools like height functions and Kasteleyn determinant representation coming to its aid. The higher dimensional picture is a little daunting because most of these tools are limited to two dimensions. In these series of talks I will describe some recent progress in 3 dimensions where we show a large deviations principle for dimer tilings analogous to those by Cohn, Kenyon and Propp (2000). This is joint work with Scott Sheffield and Catherine Wolfram.

The talks should be accessible to a general audience but it will help to have working knowledge of probability and basic ergodic theory.

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Date/Time : 10 Jul 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Nishant Chandgotia, TIFR CAM (Bangalore)

The dimer model, also referred to as domino tilings or perfect matching, are tilings of the Z^d lattice by boxes exactly one of whose sides has length 2 and the rest have length 1. This is a very well-studied statistical physics model in two dimensions with many tools like height functions and Kasteleyn determinant representation coming to its aid. The higher dimensional picture is a little daunting because most of these tools are limited to two dimensions. In these series of talks I will describe some recent progress in 3 dimensions where we show a large deviations principle for dimer tilings analogous to those by Cohn, Kenyon and Propp (2000). This is joint work with Scott Sheffield and Catherine Wolfram.

The talks should be accessible to a general audience but it will help to have working knowledge of probability and basic ergodic theory.

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Date/Time : 18 Apr 2024 16:30
Venue : S14 #06-19
Speaker : Ching Wei Ho, Academia Sinica

It is a classical problem to study the evolution of roots of polynomials under application of a differential operator. In this talk, I will discuss the heat evolution of random polynomials with a rotationally invariant root distribution on the complex plane. The limiting root distribution of the heat-evolved random polynomial can be completely determined in terms of its log potential. For example, when a Weyl polynomial, whose root distribution converges to the uniform distribution on the unit disk, undergoes heat flow, the limiting root distribution is uniform on some ellipse until time 1 at which it becomes exactly the semicircle law. This is joint work with Brian Hall, Jonas Jalowy, and Zakhar Kabluchko.

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Date/Time : 18 Mar 2024 16:00
Venue : S16 #03-04
Speaker : Stefan Junk, Gakushuin University

We consider the directed polymer model, which describes paths affected by a random space-time environment in spatial dimension d>2. The model undergoes a phase transition between a high-temperature, weak disorder phase and a low-temperature, strong disorder phase, which is characterized by whether the associated (normalized) partition function converges to zero. From the physical point of view, it is more natural to consider a very-strong-disorder regime, characterized by whether the partition function converges to zero exponentially fast. It has been a long-standing conjecture that these two notions are equivalent and we now give a proof of this. Moreover, our proof reveals that weak disorder holds at the critical value. Joint work with Hubert Lacoin.

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Date/Time : 13 Mar 2024 16:00
Venue : S16 #03-09
Speaker : Hampus Nyberg, KTH Stockholm

Deep zero problems consider holomorphic functions satisfying local properties at a small number of points. We take a look at one such problem about uniqueness in the Bargmann-Fock space and show that it reduces down to a question about products such as $\frac{\sin{1}}{\sin{2}}\frac{\sin{3}}{\sin{4}}\frac{\sin{5}}{\sin{6}}…$.

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Date/Time : 25 Jan 2024 15:00
Venue : S16-06-118, Seminar Room
Speaker : Zhou Fan, Yale University

In statistical and probabilistic models, “mean-field” behavior arises when the aggregate effect of many variables is well-approximated by an average or effective field. Ideas from the analysis of mean-field models in statistical physics have been influential in our understanding of various problems of high-dimensional statistics in recent years. In this talk, I will argue that statistical applications can conversely drive the basic study of mean-field phenomena.

As a motivating example, I will discuss an application of high-dimensional mean-field theory to a nonparametric empirical Bayes procedure for PCA. I will then describe several developments of the theory beyond i.i.d. models, including extensions of Approximate Message Passing algorithms and analyses of spin glass models where the couplings matrices are orthogonally invariant in law.

Joint work w/ Yufan Li, Subhabrata Sen, Chang Su, Tianhao Wang, Yihong Wu, Xinyi Zhong.

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