Speaker: Dain Kim
Time: January 8th, 10:00 - 11:15
Title: Minimal Surfaces
Abstract: TBA
Speaker: Seongjae Park
Time: January 8th, 11:30 - 12:45
Title: Partition functions and gap probabilities of 2D Coulomb gases
Abstract: In this talk, I introduce recent studies of two-dimensional Coulomb gases, a generalization of complex Ginibre ensemble. I start from basic non-Hermitian random matrix theory and logarithmic potential theory. Then I focus on the conjecture and results on asymptotics of partition functions and gap probabilities.
Speaker: Jeonghyun Ahn
Time: January 8th, 14:00 - 15:15
Title: Metastability of the three-state Potts model with asymmetrical external field
Abstract: In this talk, we explore the metastability of the three-state Potts model with an asymmetrical external field in a low-temperature regime. First, we analyze the energy landscape of the model and its analogy to the original Ising model. This analysis leads to the identification of two metastable states and a unique stable state. Due to the presence of two metastable states with different stability levels, the model exhibits a non-metastable transition. We elaborate this behavior by showing the Eyring-Kramers law. Finally, we illustrate the Markov chain model reduction, which indicates the convergence of the model as the temperature goes to zero. We establish this convergence for two different speed-up scales.
Speaker: Seongyoun Kim
Time: January 8th, 15:30- 16:45
Title : Counting SL(2)-representations of compact 3-manifold groups
Abstract : Given a generic compact 3-manifold, it is known that the SL2-character variety is 0-dimensional. I will give a naive explanation to answer two key questions : 1) why counting the points counts, 2) how one counts the points. For the first part, the Diophantine equational viewpoint will be useful and the significance of Casson invariant will also be considered. For the second part, we will softly touch things by examples such as Dehn fillings of basic knot complements. The coverage of full material may be subject to the time constraints.
Speaker: Kyuhyeon Choi
Time: January 9th, 10:00 - 11:15
Title : A brief sketch on the LDP
Abstract : In this talk, we introduce a brief exploration of the Large Deviation Principle (LDP), focusing on its crucial applications in probability theory and statistical mechanics. We begin by outlining the foundational aspects of LDP, particularly in understanding rare events. The exposition highlights Cramer's Theorem, detailing its relevance to the sum of random variables, and Sanov's Theorem, with its applications in empirical measures. Additionally, we illustrates these concepts through the Curie-Weiss Model, demonstrating LDP's role in predicting phase transitions in physical systems.
Speaker: Woobin Yang
Time: January 9th, 11:30 - 12:45
Title: Shtuka
Abstract: TBA
Speaker: Younghun Jo
Time: January 9th, 14:00 - 16:45
Title: Equality of Field of Definition and Field of Moduli
Abstract: The dynamics of rational maps on $\mathbb{P}^1$ is a central focus in the theory of arithmetic dynamics. In this talk, we address the problem of establishing equality between the field of moduli and a field of definition for a dynamical system on $\mathbb{P}^1$. The approach to this problem incorporates a range of terminology from various mathematical domains, including Galois cohomology, algebraic curves, and finite subgroups of the projective linear group. This exploration is based on Silverman's result in 1995.
Speaker: Kangrae Park
Time: January 10th, 10:00 - 11:15
Title: An introduction to Hausdorff Dimension and its Application to Homogeneous Dynamics
Abstract: In this talk, we simplify the concepts of Hausdorff measure and dimension, starting with the Cantor set to illustrate these ideas. We highlight the Hausdorff dimension results when a real number is represented in n-ary form with fixed digit proportions. Additionally, we introduce the dimensions of various 'badly approximable sets' in continued fractions, focusing on their upper bounds. We also discuss the upper bound of the dimension for the exceptional set in Littlewood's conjecture.
Speaker: Minchan Kang
Time: January 10th, 11:30 - 12:45
Title: A Modern Viewpoint of Modular Forms
Abstract: We explore the connections between the classical theory of modular forms and the theory of automorphic representation on GL_2 over the field of rational numbers. We introduce modular forms, automorphic forms, and adelic automorphic forms. Then we observe their connections. If time permits, we study admissible representations, define automorphic representations, and claim that each cuspidal Hecke eigenform lies in a unique automorphic representation.
Speaker: Jaehwan Kim
Time: January 10th, 14:00 - 16:45
Title: Holonomic $D$-modules and Riemann-Hilbert correspondence
Abstract: In this article, we introduce the Riemann-Hilbert correspondence between regular holonomic $D$-modules and perverse sheaves. The classical Riemann-Hilbert correspondence addresses flat connections of $\mathcal{O}$ vector bundles on a Riemann surface. $D$-modules represents a generalization of flat connections on vector bundles to an algebraic framework. However, if we regard only vector bundles in $D$-modules, they do not exhibit desirable behavior in a functorial way. For instance, the pushforward of a vector bundle is not a vector bundle in general. To address this, we introduce the category of holonomic $D$-modules with Kashiwara’s constructibility theorem after dealing with some basic properties of $D$-modules. Subsequently, we will define the de Rham functor (or a solution functor), which is a functor from the derived category of holonomic $D$-modules to the category of constructible sheaves on $X$. We will then state the Riemann-Hilbert correspondence between regular holonomic $D$-modules(complex concentrated in degree $0$) and perverse sheaves via this functor.