Date/Time : 09 Oct 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Pierre Alquier, ESSEC

The PAC-Bayesian theory provides tools to understand the accuracy of Bayes-inspired algorithms that learn probability distributions on parameters. This theory was initially developed by McAllester about 20 years ago, and applied successfully to various machine learning algorithms in various problems. Recently, it led to tight generalization bounds for deep neural networks, a task that could not be achieved by standard “worst-case” generalization bounds such as Vapnik-Chervonenkis bounds. In this talk, I will provide a brief introduction to PAC-Bayes bounds, and explain the core ideas of the theory. I will also provide an overview of the recent research directions. In particular, I will highlight the application of PAC-Bayes bounds to derive minimax-optimal rates of convergence in classification and in regression, and the connection to mutual-information bounds.

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Date/Time : 23 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 : 22 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 : 21 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 : 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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