K-Garage G++ Math Seminar

Younghun Jo (SNU)

Title: The Eyring–Kramers law for extinction time of contact process on stars

Abstract: Metastability is a ubiquitous phenomenon that manifests a first-order phase transition of a stochastic system. The metastable behavior of contact process—the susceptible-infected-susceptible (SIS) model on networks—has been studied for a large class of graphs. However, the precise exponent of the mean extinction time has remained unknown, except for some trivial cases. In this talk, we establish the sharp asymptotic behavior for the mean extinction time of contact process on star graphs using the potential theoretic approach for nonreversible systems. This talk is based on work in progress supervised by Prof. Insuk Seo.

Jinseo Park (Chicago) (via zoom)

Title:Information as Physical Object

Abstract: In Physics, we often cannot draw a fine line between theory and matter. Particles and Fields were once subjects of such discussion; now information is. We examine recent attempts at trying to "materialize" the information. Landauer's Principle and Szilard's Engine are revisited. The caveats of using macroscopic thermodynamical variables are highlighted, and proposed solutions are presented.

Woobin Yang (SNU)

Title: Discriminant bound

Abstract: One corollary of Minkowski's theorem is that it gives lower bound for absolute value of discriminant of number fields of prescribed degree and sign, and the lower bound grows exponentially. Much progress has been made to improve the lower bound. Classical aspect is to use geometry of numbers, which has led to Rogers and Mulholland’s work for totally real and totally complex fields, respectively. In 1970’s Stark introduced the formula estimating the discriminant in terms of zeros of Dedekind zeta function. Odlyzko and Serre improved the lower bound using Stark’s method, and also proved that if one assumes the generalized Riemann hypothesis(GRH) then far better result is obtained. In this talk, we introduce two methods for proving the discriminant bound: the geometry of numbers, and Stark’s analytic method.

Changki Yun (SNU)

Title: First-Price Auction is 1 - e^{-2} efficient 

Abstract: Mechanism Design is a field that explores the outcomes (such as Social Welfare, Revenue, etc.) that emerge when individuals make optimal decisions. Among the well-known problems in this field is auctions. Let's look into several fundamental results regarding auctions and then delve into recent findings by Jin and Lu (2023, 2024).

Joonsoo Ha (SNU)

Title: Big Picture of Physical Mathematics

Abstract:  For a long time, physics has developed by the language of mathematics. But in the late 80s, the idea of physics, namely mirror symmetry, helped to find an answer to the unsolved problem in algebraic geometry. It turned out that this mysterious phenomenon was a tiny part of what a remarkable relationship between mathematics and physics has predicted. It created a new field in mathematics, which is so broad and relatively rough that it is difficult to understand the big picture. In this talk, we give a big picture of physical mathematics and introduce related concepts.

Sangwook Tae (SNU)

Title: Introduction to mathematical fluid dynamics

Abstract: Fluid dynamics is an important subject in physics and engineering, since it is applicable in various situations, such as weather forcast, airplane simulation, and so on. On the other hand, fluid dynamics is an interesting subject also in mathematics, since there are lots of challenging mathematical problems, such as singularity formation of Euler equations and Navier Stokes equations. Thus, in this talk, we will give an introduction to this subject. 

Deukjae Yun (SNU)

Title: A Brief Introduction to Gromov-Witten Theory

Abstract: For the quintic threefold, its mirror symmetry is given by the equality of the A-model and B-model correlation function via the mirror map.  To define the A-model correlation function, we use the instanton number of degree d. Naively, this is the number of degree d rational curves contained in the quintic threefold. However, we should define instanton numbers more carefully to make the mirror symmetry work well. For this, we define Gromov-Witten invariants. In this talk, we will look at how Gromov-Witten theory is built with the moduli space of stable maps and its virtual fundamental class. 

Jeonghyun Ahn (SNU)

Title: 3 Manifolds

Abstract: In this talk, we explain the brief history of the studies on 3-manifolds. This will involve decomposition of 3-manifolds, Dehn surgery, and eight Thurston geometries. We wrap up by introducing the well-known Poincare conjecture, and geometric topologists' approaches regarding this conjecture. 

Minki Cho (SNU)

Title: Long-time Behavior of Logarithmic Spiral Vortex Sheets with Two Branches

Abstract: https://arxiv.org/abs/2312.02072 

Minchan Kang (SNU)

Title: A Brief Introduction to Langlands Reciprocity 

Abstract: The Langlands reciprocity conjecture predicts a correspondence between Galois representations and automorphic representations. We provide a concise historical introduction to the Langlands reciprocity conjecture, tracing its origins from Gauss’s quadratic reciprocity law. 

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Younghun Jo (SNU)

Title: Concepts in Stochastic Models and Processes

Abstract: Stochastic models and random processes are commonly used to study the behavior of various objects in physics, chemistry, biology, economics, combinatorics, and much more. There are notable phenomena that often arise in such models. We will introduce the notions of phase transition, rapid mixing, cutoff, and metastability, along with their appearance in various models and their Monte Carlo Markov Chains

Jaehwan Kim (SNU)

Title: Riemann-Hilbert Correspondence 

Abstract: Consider $M$ as a Riemann surface. The Riemann-Hilbert correspondence establishes an equivalence between the category of flat connections on meromorphic bundles over $M$ with regular singularities and the category of locally constant sheaves of vector spaces on $M$. In this talk, we will first revisit the equivalence between the category of locally constant sheaves of vector spaces on $M$ and the category of representations of the fundamental group of $M$ with easy examples. We then introduce the Riemann-Hilbert correspondence through the local solution functor $\textit{Sol}$. Following an exploration of a linear systems with regular singularities on a punctured disc $D^*$, we will sketch of the prove of RH-equivalence.

Simo Ryu (KAIST)

Title: Learning from human preferences and overoptimization 

Abstract: The major success of ChatGPT is based on two key methodologies: "reward model training" and "RLHF". Let's delve into what each of these methods are and how they've been evolving. 

Dain Kim (SNU)

Title: Parabolic frequency on Ricci flows

Abstract: [2201.05505] Parabolic frequency on Ricci flows (arxiv.org)

Kangrae Park (SNU)

Title: Size of Exceptional Sets in Weakly Mixing Systems

Abstract: [2301.09786] Size of exceptional sets in weakly mixing systems (arxiv.org) 

Changki Yun (SNU)

Title: Approximation algorithms using LP duality

Abstract: From the beginning, combinatorial optimization has evolved with the help of linear programming. This includes concepts such as total unimodularity, LP relaxation, and the primal-dual algorithm. Let's delve into these related concepts.

Jeonghyun Ahn (SNU)

Title: An Introduction to Hyperbolic Manifolds

Abstract: Isometric action on hyperbolic space $\mathbb{H}^n$, Margulis thick-thin decomposition, Mostow rigidity. 

Younghun Jo (SNU)

Title: Coffee Break: Academic Trip to America

Abstract: I attended the Random Structures & Algorithms (RS&A) as a member of Prof. Insuk Seo's lab. The RS&A is a conference that covers cutting-edge achievements in probablistic combinatorics and algorithms. Here I share my personal experiences and impressions, with notable academic results I encountered from the conference.

Changki Yun (SNU)

Title: LP rounding algorithm for s-t TSP 

Abstract: The Traveling Salesman Problem (TSP) is a famous NP-hard problem that seeks to find the least-cost Hamiltonian path or circuit in a graph. However, for cases where the cost meets the metric condition, a 1.5-approximation algorithm is known. Among these, there is a case called (s, t) TSP, where both the starting point 's' and ending point 't' are pre-determined. I will introduce a polynomial time approximation algorithm that uses Linear Programming Rounding to achieve a 1.618 approximation for the (s, t) TSP.