Abstracts

• Beomjun Choi (Postech)
  Curve shortening flow and geometric heat equations
  The curve shortening flow (CSF) refers the deformation of a given curve in the direction of its curvature. Simply put, the position vector \vec{x} changes according to the equation \partial_t \vec{x} = \kappa \vec{n}, where \vec{n} is the unit normal vector and \kappa is the curvature of the curve. As its name, the flow deceases the length of curve in the most efficient way.
  The flow shares similarities with the heat equation, \partial_{t}u= u_{xx}, which illustrates how heat diffuses over time in space. Both equations involve an evolution towards a more uniform and homogeneous state. In the case of CSF, it allows the curve to even out its bends and turns. This feature turned out to be an important tool in differential geometry such as the proof of Poincare conjecture by G. Perelman using the Ricci flow.    
  In this series of lectures, I will give a brief overview of this subject and cover some basic techniques (maximum principle, monotonicity formula), results (circularity theorem by Hamilton-Gage-Grayson) and their application such as the isoperimetric inequality. 

• Yukari Ito (IPMU-U.Tokyo)
  Introduction on group theory and algebraic geometry toward McKay Correspondence
  In these lectures, I will introduce McKay correspondence after learning group representation theory and basic algebraic geometry.
  1. Group and Geometry
  2. Representation of finite groups
  3. Introduction to Algebraic Geometry
  4. Singularity
  5. McKay correspondence

• Meiyue Jiang (Peking U.)
  Lectures on 2-dimensional geometric inequalities
  1. Preliminaries
  2. Isoperimetric Inequality
  3. The Brunn-Minkowski Inequality and the Minkowski Inequality for the Mixed Volume
  4. The log Brunn-Minkowski Inequality
  5. Introduction to the Higher Dimensional Theory (if time permits)

• Seonhee Lim (SNU)
  Number theory and dynamical systems

This lecture series will show examples of interaction between number theory and dynamical systems.

In dynamics, one of the main interests is to find properties of an orbit in a space under iterations of a function (or a flow) or, more generally, a group of functions. Properties of orbits of specific groups are closely related to some number theoretic questions related to quadratic forms, Diophantine approximations, primes, etc.

The lecture will roughly follow the following theme:

Lecture 1. Introduction to dynamical systems and ergodic theory, examples of number theoretic questions related to dynamics

Lecture 2. Continued fraction and geodesic flow on a hyperbolic surface

Lecture 3. Diophantine approximation and homogeneous dynamics I: unipotent flows

Lecture 4. Diophantine approximation and homogeneous dynamics II: diagonal flows

Lecture 5.  Oppenheim type conjectures

• Igor Sheipak  (Moscow State U)
  Spectral Properties of the Singular String Equation and Related Problems
  The course will talk about the spectral properties of a singular string with a self-similar (fractal) mass distribution. As problems related to this equation, a brief introduction to the theory of affine self-similar functions will be given. Their application to spectral problems will be disclosed.  The second direction related to the string equation with a weight-distribution is the theory of embeddings in Sobolev spaces. Exact embedding constants will be discussed. It is also planned to discuss the connection between the problem of exact estimates of intermediate derivatives in Sobolev spaces and approximation theory in L_p spaces.

• Urs Frauenfelder (Ausburg)
  Introduction to Rabinowitz Floer homology
  Rabinowitz Floer homology is the Morse homology for Rabinowitz action functional. This action functional is a Lagrange multiplier functional whose critical points are periodic orbits of fixed energy but arbitrary period. The period corresponds to the Lagrange multiplier. Since the Lagrange multiplier can be negative as well, periodic orbits can also be traversed backwards in time. Traversing backwards in time has a lot of connections to Tate homology. After explaining the construction of Rabinowitz Floer homology I plan to discuss its relations to Tate homology and the applications of this connection to space mission design.

• Shuai Guo (Peking U.)
  An introduction to enumerative geometry and mirror symmetry
  The topic of my talk is modern enumerative geometry and its connections with mirror symmetry. I will give a brief overview of the main concepts and techniques in this field, as well as some of the open problems and recent developments.

• Andrei I. Shafarevich (Moscow State U)
  Semi-classical quantization of invariant manifolds (in memory of V.P. Maslov).
  On August, 3, 2023, died Victor P. Maslov, great mathematician, author of several theories and dicoverer of several mathematical objects. In my lecture I will give a survey of one of theories, initiated by Maslov - theory of semi-classical quantization. Main idea of the theory is to study links between geometry of invariant manifolds of classical Hamiltonian systems and properties of the corresponding quantum operators.  

• Insuk Seo (SNU)

Universality in the mathematical models

In this lecture, we investigate several universal phenomenon appearing in mathematical models such as self-avoiding walks, percolation, and random cluster models. Most of these questions are still open. 

• Naoki Imai (U. Tokyo)
  Local Langlands correspondence and geometric realization
  The Langlands correspondence is a correspondence between Galois representations and automorphic representations for a number field. In this lecture we discuss the analogue of the correspondence for a p-adic local field and its realization using geometry.