Lecture Series

• Lecture H: Youngju Kim (Konkuk Univ.)
  • Title: Introduction to the Heisneberg group
  • Abstract: This talk provides an introduction to the Heisenberg group, focusing on its geometric role as the ideal boundary of the complex hyperbolic space. We will begin with various model of complex hyperbolic spaces. We will explore how the Heisenberg group acts as the boundary at infinity of ℍⁿ_ℂ , serving as a model for studying parabolic isometries. Designed for graduate students, this session aims to bridge Diophantine approximation with hyperbolic geometry.

• Lecture D: Seulbee Lee (SNU)
  • Title: Diophantine approximation
  • Abstract: Diophantine approximation studies how well real numbers can be approximated by rational numbers. A basic result in this direction is Dirichlet’s theorem, which guarantees good rational approximations for every real number. In the first lecture, we start with Dirichlet’s theorem and then introduce Hurwitz’s theorem and the Hurwitz constant. This leads to the Lagrange spectrum, which records the possible optimal constants for rational approximation. We also introduce the Markov spectrum, which arises from indefinite binary quadratic forms, and explain its relation to the Lagrange spectrum through continued fractions. In the second lecture, we discuss a geometric interpretation of the same objects. Continued fractions are closely related to the action of the modular group and to the coding of geodesics on the modular surface. Through this connection, approximation constants can be studied using hyperbolic geometry and dynamics.

Research Talks

• Talk 1: Dong-Han Kim (Dongguk Univ.)
  • Title : Markoff spectrum of Hecke groups
  • Abstract: In 1997, Vulakh characterized the Markoff spectra of Hecke groups of index q ≥ 3 up to their first accumulation points. In this talk, we develop an expansion of real numbers using the Hecke groups and completely characterize the real numbers corresponding the initial discrete parts of the Markoff spectra of the Hecke groups. This is joint work with Byungchul Cha.

• Talk 2: Byungchul Cha (Hongik Univ.)
  • Title: On the difference between the Markoff and Lagrange spectra
  • Abstract: LetMbe the Markoff spectrum and L be the Lagrange spectrum. It is known that L is a proper subset of M. In this talk, we will survey some recent results regarding the structure of M− L. Then we will discuss some generalization of this problem in the context of Diophantine approximation of the Hecke triangle group. This is joint work with Seul Bee Lee.

• Talk 3: Seonhee Lim (SNU)
  • Title: Complex continued fractions, Kleinian Circle Packings and Diophantine Approximations
  • Abstract: In this talk, we introduce the complex continued fraction of nearest integers and explain how we use it to show some mod p non-vanishing of twisted average Lvalues. We will then talk about Diophantine approximation on Kleinian circle packings as a generalization of complex continued fractions. For each result, I will mention what is known about Heisenberg groups. (This talk is based on a joint work with Dohyeong Kim, Jungwon Lee, and a joint work with Kangrae Park and Yonquan Zhang.)

• Talk 4: Jiyoung Han (Busan Univ.)
  • Title: Quantification of Khintchine-Groshev theorem
  • Abstract: Khintchine-Groshev theorem is one of the classical problems in Diophantine approximation, which tells us that for a given approximating function, the set of real numbers (or vectors, matrices) that are "approximated infinitely often to the precision the given (non-increasing) function allows" either occupies almost every point or almost none of them, in the sense of Lebesgue measure. And this is determined by whether the sum associated with the approximating function diverges or converges. In this talk, we quantify the KG theorem, when the approximating function diverges: we deduce an asymptotic formula of a sequence of functions counting approximating integer solutions as the norm parameter increases. To obtain counting results, one uses the moment methods for Siegel transforms, which are basic tools when we apply homogeneous dynamics to number-theoretic problems.

• Talk 5: Jaemin Park (SookmyungWomen’s Univ.)
  • Title: Singular vectors in simultaneous Diophantine approximation: results and methods
  • Abstract: Dirichlet’s theorem provides, for every x ∈ ℝⁿ, infinitely many simultaneous rational approximations of a guaranteed quality. A vector is singular when this quality can be improved by an arbitrarily small constant, uniformly at all large scales — the extreme opposite of being badly approximable. In dimension one only the rationals are singular, but from dimension two onward the singular set Sing(n) is a rich, Lebesgue-null fractal. This talk is an introduction to singular vectors for a broad audience. After defining singular vectors, I will survey the Hausdorff-dimension theory of the singular set, from Cheung’s original theorem that dim_H Sing(2) = ⁴⁄₃ to recent results of Taehyeong Kim and the speaker on weighted singular vectors. Along the way I will compare the principal methods of this circle of ideas: the Dani correspondence and divergent diagonal trajectories on the space of lattices, the parametric geometry of numbers, and selfsimilar coverings. Time permitting, I will end with a brief look at the intrinsic problem of singular vectors lying on a fractal — ongoing joint work with Taehyeong Kim — whose ultimate goal is a sharp dimension lower bound dim_H(Sing(2) ∩ K) ≥ ²⁄₃ dim_H K for a certain fractal K ⊂ ℝ².