Exchange Seminar
Organizers: Dohyeong Kim, Gye-Seon Lee, Seonhee Lim, and Junho Peter Whang (Seoul National University)
Schedule
2025
21 January 2025 (Tuesday) 13:30 - 14:30 / 129-301
David Xu (Korea Institute for Advanced Study)
Title: Convex-cocompact representations into the isometry group of the infinite-dimensional hyperbolic space
Abstract: Similarly to Euclidean spaces, there is an infinite-dimensional analog for the (algebraic) hyperbolic spaces. This space enjoys all the "geometric" properties of its finite-dimensional siblings. However, the topological aspects of this space and its group of isometries are more involved than in finite dimension. In particular, even the notion of discrete isometry groups needs to be specified in this context. In this talk, I will present the infinite-dimensional hyperbolic space and describe some of its properties, emphasizing some differences with finite dimension. Then, I will discuss a generalization of the classic stability result of convex-cocompact representations of finitely generated groups in hyperbolic spaces.
28 April 2025 (Monday) 15:00 - 17:00 / 27-220
Eungyu Jang (Seoul National University)
Title: Convex projective structures on double covers of Coxeter polytopes
Abstract: Let P be a compact Coxeter n-polytope in the hyperbolic n-space. Let Q be the double cover of P formed by taking two copies of P and gluing all the facets to each other. Let C(P) and C(Q) be the spaces of all convex projective structures on P and Q. If n=2, then C(P) and C(Q) are homeomorphic to R^N and R^2N for some N (Choi and Goldman, 2001). On the other hand, if n>2, then C(P) and C(Q) seem to have the same dimension. I will talk about partial results about this and possible approaches for proving it in general.
Kwansoo Kim (Seoul National University)
Title: Analogies of the Riemann-Hurwitz formula in number theory
Abstract: The Riemann-Hurwitz formula gives a relation between the genuses of two compact Riemann surfaces, when there is a nonconstant holomorphic map between them. In number theory, there are analogous formulas, such as Kida's formula, which give relations between the Iwasawa lambda-invariants of certain Iwasawa modules with Iwasawa mu-invariants 0, when there is an underlying finite p-extension between Z_p-fields. In this talk, I will introduce Kida's formula, and sketch Kataoka's approach to Kida's formula and its analogies using Selmer complexes.
20 June 2025 (Friday) 11:00 - 12:00 / 129-301
Insung Park (Stony Brook University)
Title: Zelditch's trace formula and effective equidistribution of closed geodesics in hyperbolic surfaces
Abstract: In the early 1990s, Zelditch adapted the Selberg trace formula to prove an effective version of Bowen's equidistribution theorem for closed geodesics on hyperbolic surfaces. Building on his approach, in joint work with Junehyuk Jung and Peter Zenz, we refined Zelditch's idea to achieve the optimal error term in the equidistribution of closed geodesics on compact hyperbolic surfaces. In this talk, we begin by reviewing the basic ideas of trace formulas, followed by a discussion of the new contributions. No prior knowledge of trace formulas is required.
8 July 2025 (Monday) 09:30 - 10:30 / Online
Reila Zheng (University of Toronto)
Title: Sharkovsky Ordering on the Mandelbrot set and the Kneading Determinant of Veins
Abstract: This talk will consist of two results on the symbolic dynamics of the veins along the Mandelbrot set.
Sharkovsky’s Theorem is a classical result on the forcing of periodic orbits of interval maps. In this talk, we will consider n-Sharkovsky orderings which describe orbit forcing of tree maps. I will describe n-Sharkovsky orderings on the Mandelbrot set along principal and a family of non principal veins, and the dynamics of some coordinating parameters along these veins.
Milnor-Thurston Kneading Theory is a classical theory on piecewise monotone interval maps. The kneading determinant is an invariant that captures many dynamical properties of such a map. Using the kneading theory of Tan Lei on piecewise continuous interval maps, I will generalize the work of Lindsey-Tiozzo-Wu to define kneading determinants of along non-principal veins, and relate it to the Markov polynomial.
9 July 2025 (Tuesday) 09:30 - 10:30 / Online
Michael Bersudsky (Ohio State University)
Title: Limiting distributions of k-dimensional lattices of the n-dimensional space
Abstract: The study of limiting distributions of k-dimensional lattices of Rn is motivated by classical lattice point counting problems and by the study of orbits of discrete subgroups of SL(n,R) acting on homogeneous spaces.
I will survey my recent results, with particular emphasis on my recent work with Nimish Shah that settles a conjecture of U. Shapira and O. Sargent on the orthogonal lattices of integral vectors on hyperboloids.
2024
27 February 2024 (Tuesday) 13:30 - 15:30 / 129-406
Seongmin Kim (Seoul National University)
Title: On values of binary quadratic forms at integer points
Abstract: Let Q be an indefinite binary quadratic form. The goal of this talk is to introduce estimates for the number of integral solutions in large balls, of the form |Q(x,y)| < ε, obtained by Choudhuri and Dani (2015). The proof consists of dynamics, hyperbolic geometry and negative Hurwitz continued fraction.
Young Kyun Kim (Seoul National University)
Title: Finiteness on preperiodic and periodic points in arithmetic dynamics
Abstract: We show various results about finiteness of preperiodic points and finite orbits in arithmetic dynamics. First, we introduce results on dynamics on the projective space, showing that there are only finitely many preperiodic points for endomorphisms of projective space. Then we recall some results on the affine space, that there is a universal bound on the size of periodic lengths.
19 March 2024 (Tuesday) 15:30 - 17:30 / 129-104
Donsung Lee (Seoul National University)
Title: Salter's question on the image of the Burau representation of B3
Abstract: In 1974, Birman posed the question of under what conditions a matrix with Laurent polynomial entries is in the image of the Burau representation. In 1984, Squier observed that the matrices in the image are contained in a unitary group. In 2020, Salter formulated a specific question: whether the central quotient of the Burau image group is the central quotient of a certain subgroup of the unitary group. We solve this question negatively in the simplest nontrivial case, n = 3, algorithmically constructing a counterexample. In addition, we investigate analogous questions by changing the base ring from ℤ to 𝔽p by taking modulo p. This is still meaningful, as the Burau representation modulo p is faithful when n = 3 for every prime p. We answer the questions affirmatively when p = 2, and negatively when p > 2.
Carl-Fredrik Nyberg-Brodda (Korea Institute for Advanced Study)
Title: Some decision problems in combinatorial (semi)group theory
Abstract: I will give an introductory overview to some algorithmic problems in combinatorial algebra, including its history, and particularly focussed on the word problem and related problems concerning automata over groups and semigroups. One of these problems in particular can be interpreted in terms of elementary symbolic dynamics and a problem analogous to the Collatz conjecture. The talk will include recent progress by myself, I. Foniqi, and R. D. Gray on the word problem for one-relation monoids, a problem which has remained open for over a century since it was first studied by Axel Thue in 1914.
2 April 2024 (Tuesday) 15:30 - 17:30 / 129-104
Donsung Lee (Seoul National University)
Title: Salter's question on the image of the Burau representation of B3
Abstract: In 1974, Birman posed the question of under what conditions a matrix with Laurent polynomial entries is in the image of the Burau representation. In 1984, Squier observed that the matrices in the image are contained in a unitary group. In 2020, Salter formulated a specific question: whether the central quotient of the Burau image group is the central quotient of a certain subgroup of the unitary group. We solve this question negatively in the simplest nontrivial case, n = 3, algorithmically constructing a counterexample. In addition, we investigate analogous questions by changing the base ring from ℤ to 𝔽p by taking modulo p. This is still meaningful, as the Burau representation modulo p is faithful when n = 3 for every prime p. We answer the questions affirmatively when p = 2, and negatively when p > 2.
Seongyoun Kim (Seoul National University)
Title: A Survey on the integer solutions of Mordell’s equation X^3+Y^3-XYZ+1=0
Abstract: In 1952, Mordell proved that the equation X^3+Y^3-XYZ+1=0 has infinitely many integer solutions. In 1969 Mohanty discovered that the equation has a peculiar nonlinear symmetry. Recently, integer solutions of the equation naturally have arisen from the SL(3)-character variety of free group of two generators, providing integral points in the Hitchin locus. On the other hand, the integer solutions of this equation are closely connected with special totally real cubic fields. This connection leads us to parametrize a family of periodic torus orbits and analyze their distribution in the locally symmetric space over SL(2,R) or SL(3,R). In this seminar, we will explore how all of these are intertwined and pose some furtherer questions.
23 April 2024 (Tuesday) 15:30 - 17:30 / 129-104
Balthazar Fléchelles (Institut des Hautes Études Scientifiques)
Title: Cubulated hyperbolic groups admit Anosov representations
Abstract: Anosov representations of hyperbolic groups are a generalization of convex-cocompact representations that satisfy very interesting dynamical and geometric properties. Unfortunately, it is not always easy to build Anosov representations for a given hyperbolic group. In this work in collaboration with Sami Douba, Theodore Weisman and Feng Zhu, we prove using Vinberg theoretic tools that any hyperbolic group acting properly discontinuously and cocompactly on a CAT(0) cube complex (such a group is said to be cubulated and hyperbolic) admits explicit Anosov representations. Among these groups, some were not previously known to admit Anosov representations. In fact, random groups (of density < 1/6) are cubulated and hyperbolic, so our result applies to a large variety of groups.
Carl-Fredrik Nyberg-Brodda (Korea Institute for Advanced Study)
Title: The freeness problem for parabolic subgroups of SL(2,Q)
Abstract: I will discuss some recent progress on the freeness problem for groups of 2x2 rational matrices generated by two parabolic matrices. In particular, I will discuss recent progress on determining the structural properties of such groups (beyond freeness) and when they have finite index in the finitely presented group SL(2,Z[1/m]), for appropriately chosen m.
28 May 2024(Tuesday) 15:30 - 17:30 / 129-104
Hanbin Kang (Seoul National University)
Title: Iwasawa invariant μ=0 theorem on the certain class of knots
Abstract: Iwasawa theory is one of the important interest of number theory, which was used to prove Fermat's last theorem by Wiles. On the other hand, there is an analogy between number theory and knot theory, called arithmetic topology. Recently, Tange and Ueki proved the Iwasawa μ=0 conjecture on the certain class of knots. In this talk, I will introduce the arithmetic topology dictionary and the result of Tange and Ueki.
Eugen Rogozinnikov (Max Planck Institute for Mathematics in the Sciences)
Title: Positive representations
Abstract: Higher Techmüller theory deals with spaces of representations of the fundamental group of a surface into a reductive Lie group G, modulo the conjugation, especially with the connected components (called higher Teichmüller spaces) that consist entirely of injective representations with discrete image.
In the last two decades in works of Fock, Goncharov, Burger, Iozzi, Guichard, Wienhard, and others researchers, it was discovered that the most interesting higher Teichmüller spaces are emerging from the groups G having a positive structure, i.e. certain submonoid G+ with no invertible non-unit elements. Some of these submonoids have been known since 1930’s as totally positive matrices and then generalized by Lustzig for split real Lie groups. However, it left out a large class of non-split reductive Lie groups such as SO(p,q). O. Guichard and A. Wienhard filled this gap in 2018 by introducing the Theta-positivity, which also includes submonoids SO(p,q)+ sitting in unipotent group of SO(p,q) and Sp(2n,R)+ which is the set of upper uni-triangular block 2x2-matrices with a symmetric positive definite matrix in the upper right corner.
In my talk, I introduce the Theta-positivity for Lie groups and explain how the spaces of positive representations of the fundamental group of a punctured surface into a Lie group with a positive structure can be parametrized, and how we can describe the topology of these spaces using this parametrization. This is a joint work with O. Guichard and A. Wienhard.
20 June 2024 (Thursday) 15:30 - 17:30 / 129-301
Sanghoon Kwon (Catholic Kwandong University)
Title: Geodesic flats, Ramanujan complex and Zeta functions
Abstract: We discuss the geometry and arithmetic of the space of lattices, exploring its intricate structure and the rich mathematical concepts it embodies. We will begin by examining geodesic flats within this space, and we will investigate the diagonal action on this space, focusing on the compact orbits of this action. These orbits play a crucial role in understanding the dynamic behavior and geometric characteristics of the space. Furthermore, we will explore the profound connections between these geometric concepts and zeta functions, which encode valuable information about the spaces.
Wooyeon Kim (ETH Zurich)
Title: Moments of Margulis functions and Quantitative Oppenheim conjecture
Abstract: The Oppenheim conjecture, proved by Margulis in 1986, states that for a non-degenerate indefinite irrational quadratic form Q in n ≥ 3 variables, the image set Q(Zn) of integral vectors is a dense subset of the real line. Determining the distribution of values of an indefinite quadratic form at integral points asymptotically is referred to as quantitative Oppenheim conjecture. The quantitative Oppenheim conjecture was established by Eskin, Margulis, and Mozes for quadratic forms in n ≥ 4 variables. In this talk, we discuss the quantitative Oppenheim conjecture for ternary quadratic forms (n=3). The main ingredient of the proof is a uniform boundedness result for the moments of the Margulis function over expanding translates of a unipotent orbit in the space of 3-dimensional lattices, under suitable Diophantine conditions of the initial unipotent orbit.
9 July 2024 (Tuesday) 10:00 - 11:00 / Online
Yongquan Zhang (Stony Brook University)
Title: Existence and rigidity of infinite circle packings
Abstract: Circle packings on the Riemann sphere play an important role in many problems in geometry, dynamics and number theory. For example, the Apollonian gasket is the limit set of a Kleinian group, homeomorphic to the Julia set of some rational map, and satisfies many interesting number-theoretic properties.
Given a circle packing, we can record its combinatorics by its contact graph, whose vertices and edges correspond to circles and touching points between two circles respectively. It is interesting to understand the following questions in the opposite direction: given a graph in the sphere, does there exist a circle packing with the same combinatorics? If so, is the packing unique up to Möbius transformation? These questions are answered for finite graphs by Kobe-Andreev-Thurston, and have been studied extensively for some infinite graphs (e.g. the infinite hexagonal graph).
In this talk, I will discuss some recent joint work with Yusheng Luo and give a complete answer to these questions for graphs generated by subdivision rules, using renormalization theory and iterations on Teichmüller spaces. Examples include the Apollonian gasket, and in fact all circle packings appearing as limit sets of Kleinian groups. If time permits, I will also discuss some applications to rigidity of Kleinian groups, and relatively hyperbolic groups in general.
28 August 2024 (Wednesday) 13:30 - 15:30 / 129-301
Heecheol Kwon (Seoul National University)
Title: The strongly dense representations in the Hitchin components
Abstract: In this presentation, we explore a mysterious property called strongly dense which is defined as follows: Let G be an algebraic group, and let H be its subgroup. H is strongly dense if for each two non-commuting elements of H, the subgroup generated by them is Zariski dense. A representation into an algebraic group is strongly dense if its image is strongly dense. We will see two examples of deformation spaces of real projective structures of orbifolds (Δ(3,4,4) and Vol3) and some results to see why this property is mysterious. In one of our examples, rawly speaking, there is a fact that almost all representations in a Hitchin component are strongly dense. But even giving an example of the strongly dense representation looks challenging.
Young Kyun Kim (Seoul National University)
Title: Preperiodic orbits of unramified morphisms
Abstract: We review results on the bound of the size of finite orbits in arithmetic dynamical systems. Then we show that if an integral algebraic dynamical system has a unramified reduction then eventually fixed points are controlled by the dynamical system modulo that reduction. With this we show there exists a upper bound for finite orbits of integral dynamical systems with unramified morphisms.
15 October 2024 (Tuesday) 15:30 - 17:30 / 129-301
Hanbin Kang (Seoul National University)
Title: Twisted homology of knot complements
Abstract: One of the most important knot invariant is the Alexander polynomial, and there are various methods to compute this polynomial. In this talk, I will introduce the twisted homology and the explicit calculation of the twisted Alexander polynomial.
Jaeyoung Kim (Seoul National University)
Title: Markov chain of diagonal action on the standard quotient of the building B_3(Fq((t−1)))
Abstract: Kwon described the local transition probability of the diagonal action on the standard non-uniform quotient of PGL3 associated to type 1 geodesic flow. Meanwhile, Ciobotaru, Finkelshtein and Sert demonstrated the non-escape of mass of the horospherical action on trees via Markov chain. Extending these ideas, we first estabilsh the Markov chain of diagonal action and apply the idea to induce an effective statement that there is no escape of mass of a horospherical action on the building B3(Fq((t−1))), hence on PGL(3,Fq[t])∖PGL(3,Fq((t−1))).
29 October 2024 (Tuesday) 15:30 - 17:30 / 129-301
Dong Hyun Lee (Seoul National University)
Title: Zariski dense 3-manifold groups in SL(4,Z)
Abstract: In 2011, Long, Reid, and Thistlethwaite presented an infinite family of Zariski-dense surface groups of genus 2 in SL(3,Z). This result naturally raises a similar question for 3-manifolds. In this talk, I will provide some historical context and propose an approach to addressing this question.
Hanbin Kang (Seoul National University)
Title: Twisted homology of knot complements
Abstract: One of the most important knot invariant is the Alexander polynomial, and there are various methods to compute this polynomial. In this talk, I will introduce the twisted homology and the explicit calculation of the twisted Alexander polynomial.
26 November 2024 (Tuesday) 15:30 - 17:30 / 129-301
Donghae Lee (Seoul National University)
Title: Surface quotients of right-angled hyperbolic buildings
Abstract: A Fuchsian building is a hyperbolic structure defined as the universal cover of a complex of groups with a single regular right-angled p-gon as its underlying polygonal complex, which exhibits significant symmetry. In 2012, Futer and Thomas established some sufficient conditions on p, v, and g for a surface quotient lattice of genus g to exist in Bourdon's building I_{p, v}, a special type of Fuchsian building. We identify sufficient conditions on the tuple representing the thickness at each edge of the chambers for a general Fuchsian building for existence of a surface quotient lattice when p is even. Furthermore, we show that this tuple must satisfy certain symmetries for such a lattice to exist.
Myeonggwan Hwang (Seoul National University)
Title: Application of Hida theory for modular symbols
Abstract: Hida studied the ordinary part of the space of the modular forms of level Γ(Np^r) and weight k. We can compare how such spaces change when r grows up to infinity or k grows up to infinity in Hida theory. In 1999, Emerton discussed similar approach with the modular symbols. Now we will have the further observation about modular symbols. In this talk, I will introduce Emerton's study.
17 December 2024 (Tuesday) 13:30 - 15:00 / 129-301
Frederic Paulin (Paris-Saclay University)
Title: Divergent geodesics in hyperbolic manifolds and arithmetic applications
Abstract: We give an asymptotic formula for the number of common perpendiculars of length tending to infinity between two divergent geodesics in finite volume real hyperbolic manifolds, presenting a surprising non-purely exponential growth. We apply this result to count ambiguous geodesics in the modular curve recovering results of Sarnak, and to prove a conjecture of Motohashi on the binary additive divisor problem in imaginary quadratic number fields. This is a joint work with Jouni Parkkonen.
2023
| Date | Time | Place | Speaker |
| 10 January 2023 | 13:30 - 15:30 | 129-301 | 심덕원, 김성윤 |
| 16 February 2023 | 13:30 - 17:00 | 129-301 | 심덕원, 김성윤 |
| 7 March 2023 | 15:30 - 18:30 | 129-301 | 임선희, 이계선 |
| 9 May 2023 | 15:30 - 18:00 | 129-309 | 황준호, 정홍택 |
| 23 June 2023 | 13:30 - 16:00 | 129-406 | 정홍택, Balthazar Fléchelles |
| 30 June 2023 | 14:00 - 15:30 | 27-325 | 장승욱 |
| 30 August 2023 | 13:30 - 16:00 | 129-301 | 김경로, 김재영 |
| 15 September 2023 | 13:30 - 16:00 | 129-301 | 김경로, 신은주 |
| 26 September 2023 | 15:30 - 16:30 | 129-301 | 고성환 |
| 10 October 2023 | 15:30 - 16:30 | 129-301 | 방국영 |
| 24 October 2023 | 15:30 - 16:30 | 129-301 | 김태형 |
| 7 November 2023 | 15:30 - 16:30 | 129-406 | 이영준 |
| 28 November 2023 | 15:30 - 16:30 | 129-104 | 황승훈 |
2022
| Date | Time | Place | Speaker |
| 11 October 2022 | 16:00 - 18:00 | 129-301 | 임선희, 황준호 |
| 29 November 2022 | 15:30 - 18:00 | 129-406 | 이계선 |