# Past seminars

# 2023 & 2024

April 6 (Th) 11 am (note: double-header!)

In person KIAS 1423

Zoom See above (on site participation is encouraged)

Homin Lee (Northwestern)

Higher rank lattice actions with positive entropy

We discuss smooth actions on manifolds by higher-rank lattices. We mainly focus on lattices in SL(n,R) (n is at least 3). Recently, Brown-Fisher-Hurtado and Brown-Rodriguez Hertz-Wang showed that if the manifold has dimension at most (n-1), the action is either isometric or projective. In both cases, we don’t have chaotic dynamics from the action (zero entropy). We focus on the case when one element of the action acts with positive (topological) entropy. These dynamical properties (positive entropy element) significantly constrain the action. Especially, we deduce that if there is a smooth action with a positive entropy element on a closed n-manifold by a lattice in SL(n,R) (n is at least 3) then the lattice should be commensurable with SL(n,Z). This is the work in progress with Aaron Brown.

April 6 (Th), 13 (Th), 18 (Tu), 24 (Mon) 2023, 10 am (a series of four talks)

In person KIAS 1423

Zoom See above (on site participation is encouraged)

Inhyeok Choi (KIAS)

Counting pseudo-Anosov mapping classes (Parts 1 - 4)

This series of talks concerns the problem of counting pseudo-Anosov mapping classes in the mapping class group. The first talk will be devoted to the geometry of mapping class groups and related objects. Then, I will explain recent developments in the theory of random walks on the curve complex. Finally, I will describe the proof of the main problem and list future directions.

May 11 (Thur), 10 am

Online http://shkim.org/vz

In person KIAS 1423

Joonhee Kim (KIAS)

Pre-independence relations and forking in classification theory (expository)

In this talk, we introduce model theory and some examples of its application to other fields of mathematics. We also introduce the idea of classification theory, one of the main research areas of model theory, and its central concepts. We report on the latest research in classification theory and discuss what we can expect from these results.

May 16, 17, 19 (Tu, We, Fr), 10 am

Online http://shkim.org/vz

In person KIAS (Bldg.1, Room 1424)

Thomas Koberda (UVA)

First order theory of homeomorphism groups of manifolds

Talk 1: I will survey some basics of model theory, and discuss the first order rigidity of homeomorphism group of a compact manifold, in the context of it being a generalization of a classical result of Whittaker. The main result is joint with S. Kim and J. de la Nuez Gonzalez.

Talk 2: I will discuss some of the details of the proof of the result in the first talk.

Talk 3: I will survey some actual and conjectural applications of the main result, including definability of various natural topological and group theoretic properties, as well as applications to critical regularity in higher dimensions.

May 17 (W), 3:45 - 6 pm

[KAI-X Mathematics Distinguished Lecture Series]

Francois Labourie (Universite Cote d'Azur)

Organizer: Hyungryul Baik

Zoom [Link]

15:45~16:45 (Talk 1) Some work of Maryam Mirzakhani

17:00~18:00 (Talk 2) Representations of surface groups: Positivity

May 30, June 1, 10 - 11:15 am

Online http://shkim.org/vz

In person KIAS (1423)

Masato Mimura (Tohoku University)

Talk 1: An introduction to the theory of invariant quasimorphisms

Fix a group. A real valued function on it is called a quasimorphism if this satisfies the identity of being a group homomorphism up to uniformly bounded error. There are the following two types of quasimorphisms that might be considered as 'non-interesting' ones: genuine group homomorphisms and uniformly bounded maps. It is well known that the quotient vector space of that of all quasimorphisms over that of sums of functions of these two types is naturally isomorphic to the kernel of the comparison map from second bounded cohomology to second ordinary cohomology of the group. In many cases, this vector space is either the zero space or infinite dimensional.

In this talk, we take a pair (G,N) of a group G and its normal subgroup N. In this setting, we can define a notion of G-invariant quasimorphisms on N. We will present an introduction to this relatively new object. In particular, we can define a certain vector space related to invariant quasimorphisms; this can be finite-dimensional under a mild condition. For example, if G is the surface group of genus at least two and N is the commutator subgroup of G, then this vector space is one-dimensional. We will provide examples and motivations of quasimorphisms and invariant quasimorphisms in this first talk.

Talk 2: Applications of invariant quasimorphisms and stable mixed commutator length

In this second talk, we will present some applications of the theory of invariant quasimorphisms. Some examples are related to symplectic geometry and group actions on the circle. Also, for a group pair (G,N) in our setting, we can define the mixed commutator length on the mixed commutator subgroup [G,N], which is the word length with respect to the set of simple (G,N)-commutators. The stabilization of the mixed commutator length is called the mixed scl; this equals the scl when N=G. We study large scale behavior of the mixed scl; more precisely, we study it in the aspect of coarse groups, the concept recently developed by Leitner and Vigolo.

June 20, 22 (10 - 11:30), 2023

Christian Rosendal (UMCP)

Online http://shkim.org/vz

In person KIAS (1423)

KIAS--Springer Lectures Geometric group theory beyond locally compact groups [poster]

These lectures will provide an introduction to the geometrisation of topological groups, in particular, the large scale geometric aspects of topological transformation groups, such as homeomorphism groups of compact manifolds, mapping class groups of infinite-type surfaces, and automorphism groups of countable structures. We will show how general considerations lead to a canonical large scale geometric structure on every topological group and provide criteria for its metrisability and for when the structure can be further improved to a quasi-metric structure. We apply the framework to a few significant examples including homeomorphism groups and automorphism groups of graphs. Finally, we address the interplay between model theory of countable structures and the geometry of their automorphism groups.

July 4, 2023, 2023 (10 am)

Online http://shkim.org/vz

In person KIAS (1423)

Seung Uk Jang (Rennes)

Vieta Actions on Tropicalized Markov Cubics

We discuss the algebraic dynamics on Markov cubics generated by Vieta involutions, in the tropicalized setting. It turns out that there is an invariant subset of the tropicalized Markov cubic where the action by Vieta involutions can be modeled by that of $(\infty,\infty,\infty)$-triangle reflection group on the hyperbolic plane.

This understanding of the tropicalized algebraic dynamics produces some results on Markov cubics over non-archimedean fields, including the existence of the Fatou domain and finitude of orbits with rational points having prime power denominators. Furthermore, if the time permits, we will introduce more families of varieties where tropical perspectives can give more nice descriptions.

July 6, 2023 (10 am)

Online only http://cayley.kr/vz

Carl-fredrik Nyberg Brodda (Universite Gustave Eiffel)

The Dehn functions of a class of one-relation monoids

The Dehn function for a semigroup or group M is an asymptotic measure of how bad the "naive solution" to the word problem in M may be. The word problem in M is decidable if and only if the Dehn function of M is a recursive function, but frequently the Dehn function is significantly more poorly behaved than the complexity of the word problem. On the other hand, the word problem for one-relation monoids is one of the most intriguing and important open problems in semigroup theory. For that reason, it makes sense to ask: how bad can the Dehn function of a one-relation monoid be? I will present the history of the problem, which has a natural starting point in S. I. Adian's classical theory of left cycle-free monoids. I will then present some of my own recent progress on this topic for the class of one-relation monoids where the relation is of the form bUa=a, for a word U. In particular, I will exhibit monoids with Dehn functions of exponential growth in this class, answering a question posed by Cain & Maltcev in 2013.

July 11, 2023 (10 am)

Online http://shkim.org/vz

In person KIAS 1423

Seung-Yeon Ryoo (Caltech)

Embedding finitely generated groups of polynomial growth into Euclidean space

Embedding finitely generated groups into some Banach space is a commonly studied question, with applications in geometric group theory and theoretical computer science. We ask: in the case where the group is of polynomial growth and we look at bi-Lipschitz or bi-Hölder embeddings, what is the role of the dimension of the target Banach space? It turns out that, if the target Banach space has sufficiently large dimension, we can obtain embeddings with the same asymptotic profile. We will discuss the implications of this to some questions in metric embedding theory.

July 20, 2023 (10 am)

Online only http://cayley.kr/vz

Carl-fredrik Nyberg Brodda (Universite Gustave Eiffel/KIAS)

Membership problems for one-relator groups

One-relator groups are fundamental to combinatorial and geometric group theory. In this talk, I will give an overview of some basic properties of one-relator groups. I will introduce the main membership problems for groups (the subgroup and submonoid membership problems), its relation to right-angled Artin groups (RAAGs) arising as subgroups of one-relator group, and talk about some recent progress by myself, Gray, and Foniqi, Gray & myself on this subject. Using results by Wilton & Louder, as well as Kim & Koberda, this includes a classification of the right-angled Artin subgroups of one-relator groups. It also includes a new family of RAAG-like groups I introduced, called right-angled Artin-Baumslag-Solitar groups (RABSAGs), which contain many interesting subgroups and which are commensurable to many one-relator groups. Finally, time permitting, I will discuss results by Cadilhac, Chistikov & Zetsche on certain membership problems in solvable Baumslag-Solitar groups, their relation to GL(2, Q), and one-relation monoids.

July 27, 2023 (Thursday), 2 pm

Note the time and the zoom link!

In Person KIAS 1423

Zoom [Link]

Meeting ID: 863 9455 0969

Passcode: 554797

Jaelin Kim (Rényi Institute)

Central limit theorem of Brownian motions on manifolds of pinched negative curvature with non-uniform lattices.

On manifolds with negative curvature, Brownian motions are studied from the dynamical point of view, but most of the results were either for cocompact manifolds or for all Cartan-Hadamard manifolds with pinched negative curvature. In this talk, we prove the central limit theorem of the Brownian motion on a manifold with a non-uniform lattice, which generalizes the result for the cocompact manifolds by F. Ledrappier. We will see the relation between Brownian motions and asymptotic harmonic manifolds from the ergodic theoretic viewpoint.

Aug 9, 2023 (Wed, 2 pm)

In-person KIAS (1423)

Online http://cayley.kr/vz

Sungkyung Kang (IBS-CGP)

Involution and symmetries in Heegaard Floer homology

Heegaard Floer homology of 3-manifolds have various types of symmetries, and studying them have recently lead to resolutions of several long-standing open questions in low-dimensional topology. In this talk, we will review its construction, as originally predicted from the Seiberg-Witten side, as well as their computations and topological applications.

Aug 31, 2023 (Thur, 10 - 11 am, 11:15 am - 12:15 pm)

In-person KIAS (1423)

Online http://kimsh.kr/vz

Thomas Scanlon (Berkeley)

Lecture 1: Functional transcendence for Mahler functions from the model theory of difference fields

Starting in the 1930s, Mahler studied functions $f(x)$ which satisfy functional relations of the form $R(x,f(x),f(x^q)) = 0$ for some rational function $R$ and positive rational number $q \neq 1$, showing, for example, that under certain explicit conditions algebraic relations involving values of these functions are controlled by functional equations. When the equation take the form $f(x^q) = P(f(x))$ where $P \in \mathbb{C}(x)[Y]$ is a polynomial over the field of rational functions we show $f$ is algebraically independent from functions satisfying difference equations of the form $R(x,g(x), g(x^r), \ldots, g(x^{r^m}) = 0$ where $R$ is a rational function and $r$ is multiplicatively independent from $q$. We address this problem by giving an exhaustive account of a class of definable sets relative to the model companion of the theory of difference fields. (This is an account of joint work with Alice Medvedev and Khoa Dang Nguyen.)

Lecture 2: Definable quotient spaces and complex geometry

Various quotients of homogeneous spaces by the action of discrete groups play important roles in such subjects as the theory of quadratic forms, the study of modular functions, Hodge theory, and homogeneous dynamics, amongst others. Strictly speaking the map from such a homogeneous space to the quotient cannot be definable in an o-minimal structure, but as various other authors (e.g. Bakker, Klingler, Tsimerman, Peterzil, Starchenko, etc.) have observed, by making suitable restrictions it can be fruitfully analyzed using o-minimality and the quotient spaces themselves may be treated as definable objects. I will discuss how to develop this theory and some basic open questions and will then show how this formalism may be used to prove some general theorems around the Zilber-Pink conjectures. (This is an account of joint work with Jonathan Pila and separately with Sebastian Eterović.)

September 5, 7 (Tue/Thur), 2023, at 10 - 11:30 am

(Joint with KIAS--Springer Lecture Series)

Poster [Link]

Online http://kimsh.kr/vz

In person KIAS (1423)

Alan Reid (Rice)

Profinite Rigidity

In these two lectures I will discuss recent progress on the question of when a f.g. residually finite group is profinitely rigid; i.e. if H and G are two such groups and their profinite completions are isomorphic then H is isomorphic to G. This will largely focus on groups occurring in low-dimensional topology and geometry. The first lecture will be largely introductory, setting up what is needed to discuss more recent progress.

September 19, 20 (Tue/Wed), 2023 at 10 - 11:30 am

(Joint with KIAS--Springer Lecture Series)

Online http://kimsh.kr/vz

In person KIAS (1423)

Andrés Navas (Universidad de Santiago de Chile)

(Arc)-connectedness of the space of Z^d actions on 1-manifolds.

We will elaborate on two recent results obtained in collaboration with Hélène Eynard-Bontemps (Inst. Fourier, Grenoble):

- The space of Z^d actions by C^{1+ac} diffeomorphisms of a compact 1-manifold is path-connected;

- The space of Z^d actions by C^2 diffeomorphisms of the interval is connected.

Here, "ac" stands for absolutely continuous. I will deeply comment on several technical problems when dealing with these properties: Mather's homomorphisms, failure of Sternberg's linearization theorem, etc. Several questions will be addressed along the two lectures.

September 26, 2023 (Tue), 2023 at 10 am

Online http://kimsh.kr/vz

In person KIAS (1423)

Cristóbal Rivas (Universidad de Chile)

Regular orders on groups

I will discuss some recent results regarding the language complexity of (left) orders on groups. I'll focus on examples of groups having/not having positive cones that can/cannot be described by a regular language.

October 12, 2023 (Th) at 11 am

Online http://kimsh.kr/vz

In person KIAS (1423)

Nicholas Vlamis (CUNY Queens College)

Big mapping class groups that are small

Big mapping class groups refer to the mapping class groups of infinite-type surfaces, i.e. those surfaces whose fundamental group is not finitely generated. The adjective “big” refers to the underlying surface, but big mapping class groups are also bigger than their finite-type counterparts in other ways; for instance, they are uncountable groups, and they are also non-locally compact as topological groups. Despite this, and somewhat surprisingly, these groups can be geometrically small. We will discuss several ways in which uncountable groups can be small, and go over recent results placing various big mapping class groups into these categories. Part of the work discussed is joint with Justin Lanier.

October 24, 2023 (Tu) at 11 am

Online http://kimsh.kr/vz

In person KIAS (1423)

Thomas Koberda (U Virginia)

The cohomology basis graph for a right-angled Artin group

I will describe a certain class of graphs associated to the cohomology of a right-angled Artin group, and how these can be used to formulate spectral graph theory within the context of the group theory of right-angled Artin groups. I will also describe some connections to complexity theory. This represents joint work with Flores, Kahrobaei, and Le Coz.

October 31, 2023 (Tu) at 11 am

Zoom 861 2351 2846 Password 7998

In person KIAS (1423)

Seokbum Yoon (SUSTech)

Reidemeister torsions from (super-)Ptolemy coordinates

Ptolemy coordinates parameterize (a certain type of) SL(2,C)-representations of ideally triangulated 2- or 3-manifolds. The 1-loop conjecture, motivated by the generalized volume conjecture, predicts that the Jacobian of Ptolemy coordinates is essentially equal to the C^3-torsion, also known as the adjoint Reidemeister torsion. In this talk, we introduce super-Ptolemy coordinates for ideally triangulated 3-manifolds that parameterize OSp(2|1)-representations. Then we propose a conjectural formula for the C^2-torsion, analogous to the 1-loop conjecture. If time permits, I will sketch a proof of the conjecture for fibered 3-manifolds. This is joint work with Stavros Garoufalidis.

Nov 2, 2023 (Th), 2023 at 11 am

Zoom 861 2351 2846 Password 7998

In person KIAS (1424)

Robert Tang (Xi’an Jiaotong-Liverpool University)

Large-scale geometry of the Rips filtration

Given a metric space $X$ and a scale parameter $\sigma \geq 0$, the Rips graph $Rips^\sigma X$ has $X$ as its vertex set, with two vertices declared adjacent whenever their distance is at most $\sigma$. A classical fact is that $X$ is a quasigeodesic space precisely if it is quasi-isometric to its Rips graph at sufficiently large scale.

By considering all possible scales, we obtain a directed system of graphs known as the Rips filtration. How does the large-scale geometry of $Rips^\sigma X$ evolve as $\sigma \to \infty$? Is there a meaningful notion of limit and, if so, does it satisfy any nice properties? In this talk, I will discuss some work in progress inspired by these questions.

November 7, 2023 (Tu) at 11 am

Zoom 861 2351 2846 Password 7998

In person KIAS (1423)

Shuhei Maruyama (Kanazawa University)

Non-descendible quasimorphisms and characteristic classes

This talk is based on joint work with Morimichi Kawasaki. In this talk, I will explain a relationship between the

non-descendibility of quasimorphisms and the boundedness of characteristic classes for foliated bundles. I will also present two specific characteristic classes for foliated symplectic and contact fibrations: one is a bounded class, and the other is an unbounded class.

November 9, 2023 (Th) at 11 am

Zoom 861 2351 2846 Password 7998

In person KIAS (1424)

Shuhei Maruyama (Kanazawa University)

Non-extendable quasimorphisms and characteristic classes

This talk is based on joint work with Morimichi Kawasaki, Mitsuaki Kimura, Takahiro Matsushita, and Masato Mimura.

In this talk, I will explain recent progress in the study of non-extendable invariant quasimorphisms from the viewpoint of characteristic classes for foliated bundles. Especially, I will provide a construction of non-extendable quasimorphisms using group actions on the circle and the Euler class for foliated circle bundles.

November 16, 2023 (Th) at 11 am

Zoom http://kimsh.kr/vz

Meeting ID: 822 3235 0014

Passcode: 7998

In person KIAS (1423)

Minkyu Kim (KIAS)

An obstruction problem associated with finite path integral

Finite path integral is a finite version of Feynman’s path integral, which is a mathematical methodology to construct TQFT’s (topological quantum field theories) from finite gauge theory. It was introudced by Dijkgraaf and Witten in 1990. We study finite path integral model by replacing finite gauge theory with homological algebra based on bicommutative Hopf algebras. It turns out that Mayer-Vietoris functors such as homology theories extend to TQFT which preserves compositions up to a scalar. This talk concerns the second cohomology class of cobordism (more generally, cospan) categories induced by such scalars. In particular, we will explain that the obstruction class is described purely by homological algebra, not via finite path integral.

November 17, 2023 (Friday) at 11 am

In person KIAS (1424)

Online http://kimsh.kr/vz

Geunho Lim (Einstein Institute of Mathematics, Hebrew University of Jerusalem)

Linear bounds on rho-invariants and simplicial complexity of manifolds

Using L^2 cohomology, Cheeger and Gromov define the L^2 rho-invariant on manifolds with arbitrary fundamental groups, as a generalization of the Atiyah-Singer rho-invariant. There are many interesting applications in geometry and topology. In this talk, we show linear bounds on the rho-invariants in terms of simplicial complexity of manifolds. First, we obtain linear bounds on Cheeger-Gromov invariants, using hyperbolizations. Next, we give linear bounds on Atiyah-Singer invariants, employing a combinatorial concept of G-colored polyhedra. As applications, we give new concrete examples in the complexity theory of high-dimensional (homotopy) lens spaces. This is a joint work with Shmuel Weinberger.

## 2023 Fall

All talks are in-person & online, unless noted otherwise.

Aug 9 (W at 2 pm), Sungkyung Kang (IBS-CGP)

Aug 31 (TH at 10 & 11:15 am), Thomas Scanlon (Berkeley)

Sep 5, 7 (Tu/Th at 10 - 11:30 am), Alan Reid (Rice)

Sep 19, 20 (Tu/W at 10 am), Andrés Navas (USACH)

Sep 26 (Tu at 10 am), Cristóbal Rivas (U Chile)

Oct 12 (Th at 11am, online only), Nicholas Vlamis (CUNY Queens)

Oct 24 (Tu at 11 am), Thomas Koberda (U Virginia)

Oct 28 (Sa, morning, in-person only), GGT Session, KMS Fall Meeting, Seoul National University

Oct 31 (Tu at 11 am), Seokbum Yoon (SUSTech)

Nov 2 (Th at 11 am), Robert Tang (Xi'an Jiaotong-Liverpool)

Nov 7, 9 (Tu, TH at 11 am), Shuhei Maruyama (Kanazawa University)

Nov 16 (Th at 11 am), Minkyu Kim (KIAS)

Nov 17 (F at 11 am), Geunho Lim (Einstein Institute of Mathematics, Hebrew University of Jerusalem)

Nov 20 (M at 11 am), Don-Sung Lee (SNU)

Nov 20 (M at 3 pm), Heejoung Kim (KPU)

Nov 23 (Th at 11 am), James Rickards (Colorado)

Dec 5 (Tu at 11 am), Javier de le Nuez-Gonzalez (KIAS)

Dec 7 (Th at 2 pm), Tara Brendle (University of Glasgow)

November 20, 2023 (Tu) at 11 am

In person KIAS (1424)

Online http://kimsh.kr/vz

Refreshment KIAS lounge, 10:30 am

Don-Sung Lee (SNU)

Salter's question on the image of the Burau representation of B_3

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 \mathbb{Z} to \mathbb{F}_{p} 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.November 20, 2023 (M) at 3 pm

In person KAIST (E6-1, 4407)

Online http://kimsh.kr/vz

Heejoung Kim (KPU)

Mapping class groups of infinite-type surfaces and surface Houghton groups

The mapping class group Map(S) of a surface S is the group of isotopy classes of diffeomorphisms of S. When S is a finite-type surface, the classical mapping class group Map(S) has been well understood. On the other hand, there are recent developments on mapping class groups of infinite-type surfaces. In this talk, we discuss mapping class groups of finite-type and infinite-type surfaces and elements of these groups. Also, we define surface Houghton groups, which are subgroups of mapping class groups of certain infinite-type surfaces. Then we discuss finiteness properties of surface Houghton groups, which is a joint work with Aramayona, Bux, and Leininger.

November 23, 2023 (Thursday) at 11 am

In person KIAS (1424)

Online http://kimsh.kr/vz

James Rickards (Colorado)

Failure of the local-global conjecture in thin (semi)groups

The study of orbits of thin (semi)groups encapsulates many famous problems, including Zaremba's conjecture and Apollonian circle packings. The local-global conjecture states that as long as the (semi)group is big enough, every large enough integer that satisfies certain congruence restrictions will appear in an orbit. In a recent breakthrough, this conjecture was proven to be false for many Apollonian circle packings. We will discuss the history of the conjecture, this recent work, and give new examples of thin semigroups where the conjecture is false.

December 5, 2023 (Tuesday) at 11 am

In person KIAS (1423)

Online http://kimsh.kr/vz

Javier de la Nuez-González (KIAS)

Minimality of the compact-open topology on diffeomorphism and homeomorphism groups

We will talk about recent work in which we prove that the restriction of the compact-open topology to the diffeomorphism group of a manifold without boundary of dimension different from 3 is a minimal element of the lattice of Hausdorff group topologies on the group. If the dimension is also different from 4 it follows that the same holds for the compact-open topology on the homeomophism group, which combined with K. Mann's automatic continuity results implies the latter admits a unique separable Hausdorff group topology

December 7, 2023 (Thursday) at 2 pm

In person KIAS (1423)

Online http://kimsh.kr/vz

Tara Brendle (University of Glasgow)

Semi-direct product structures in mapping class groups of 3-manifolds

We will show that a certain short exact sequence associated with mapping class groups of 3-manifolds admits a splitting. One by-product of this result is that Out(F_n) arises as a certain stabilizer subgroup of the mapping class group of a connected sum of n copies of S2 x S1. The general case is slightly more complicated: using recent work of Chen-Tshishiku, we will describe the second factor in the semi-direct product structure in terms of the prime decomposition of the 3-manifold. This is joint work with Nathan Broaddus and Andrew Putman.

Jan 23 and 25, 2024 (Tu and Th) at 11 am - noon

KIAS 1503

Ken'ichi Ohshika (Gakushuin)

Thurston’s broken windows only theorem and his proof of the bounded image theorem (Parts 1, 2)

Both the broken windows only theorem and the bounded image theorem constitute important parts of Thurston’s proof of the unifomisation theorem for Haken manifolds dated back to the early 1980s. According to his plan of proofs, the proof for the bounded image theorem relies essentially on the broken windows only theorem, in particular its second statement. We show that this second statement has counter examples. Although we can fix it by weakening the result, this weak form cannot be used for the proof of the bounded image theorem. We will also explain how the bounded image theorem (whose proof was recently given by Cyril Lecuire and myself) would follow, if this second statement were true.

Feb 26 (Mon 3-4 pm), Feb 27 (Tue 2-3 pm)

KIAS 1423

Michele Triestino (Université de Bourgogne)

Nicolás Matte Bon (Institut Camille Jordan - Université Lyon 1)

Laminations and structure theorems for group actions on the line (Part 1,2)

A lamination of the real line is a closed collection of pairwise unliked, finite intervals. They appear naturally when studying certain classes of group actions on the line. More precisely, we will discuss actions of solvable groups, and of locally moving groups (these are subgroups of Homeo(R) such that for any open interval I, the subgroups of elements fixing the complement of I acts minimally on I). A famous exampke of a locally moving group is Thompson's group F. Both classes admit "standard models" of actions on the line: solvable groups act by affine transformations, whereas locally moving groups have their defining actions. We prove a structure theorem which says that any minimal faithful action of a finitely generated group in this class is either standard, or preserves a lamination. Moreover, the large scale dynamics of actions preserving laminations can be described in terms of the standard actions. We will briefly mention the results for solvable groups, and focus the discussion on locally moving groups. This is based on works with J. Brum and C. Rivas.

Feb 29 (Thur 10 am - 5 pm), 2024

Korea-France Workshop on Dynamical Group Theory (KIAS)

March 14 (Thu 2 pm - 3pm), 2024

KIAS 1423

Paolo Marimon (Vienna)

Minimal operations over permutation groups

Joint work with Michael Pinsker. Let B be a fixed relational structure (e.g. a graph). The Constraint Satisfaction Problem for B, CSP(B), is the computational problem of, given a finite structure A in the same language, deciding whether there is a homomorphism from A into B. Many natural problems in computational complexity can be framed in this fashion. When the structure B is finite, we know that CSP(B) is always in NP, and, assuming P \neq NP, we know that whether this problem is in P or NP-complete can be characterised in terms of identities satisfied by the polymorphisms of B (a higher arity generalisation of homomorphisms) (Bulatov 2017, Zhuk 2017).

We are interested in generalising the tools used to study constraint satisfaction problems for finite structures to some infinite structures with many symmetries (finitely bounded homogeneous structures). Since understanding the polymorphisms of B is essential to understand the computational complexity of CSP(B), it is often helpful to find polymorphisms of low arity behaving in some non-trivial way. For this reason, we study what are called the minimal polymorphisms above the automorphism group of B. We carry out a classification of the types of minimal operations which may appear over an arbitrary permutation group G\acts B, generalising the work of Rosenberg (1986) for the trivial group and of Bodirsky and Chen (2007) for oligomorphic permutation groups. This allows us to answer some open problems mentioned by Bodirsky (2021) in his book on infinite domain CSPs.

March 28, 2024. 10:30 am - 12 pm (Note the time!)

KIAS 1423

Nhat Minh Doan (VAST / NUS)

Optimal special polygon for congruence subgroup $\Gamma_0(p^n)$.

For a positive integer $n>1$ and a prime $p$, we show that $\Gamma_0(p^n)$ possesses a special fundamental domain (in the sense of Kulkarni) with its cusp set encoded using a Farey sequence of denominators less than or equal to $(f_{p+1}/2)p^{n-1}+f_pp^{n/2}$, where $f_m$ is the m-th Fibonacci number ($f_1=f_2=1, f_m=f_{m-1}+f_{m-2}$ for all $m\in \mathbb{N}$). In the special case where $p=2$, we establish a sharp upper bound of $2^{n-1}$. This is join work with Sang-hyun Kim, Mong Lung Lang, and Ser Peow Tan.

May 9, 2024. 11 am - 12 pm

KIAS 1423

Carl-Fredrik Nyberg-Brodda (KIAS)

Densities, shifts, and rational languages

I will go over one aspect of recent joint work on densities in shift spaces with V. Berthé, H. Goulet-Oullet, D. Perrin, and K. Petersen. Rational languages capture simple patterns (e.g. "the set of all binary strings containing an even number of 0s"), and are precisely the languages which are the pre-image of some subset under a morphism onto a finite semigroup. A particularly simple type of rational languages, called group languages, are those which arise when the recognising semigroup is a (finite) group. In the talk, I will show one way in which group languages can be used to study patterns appearing in subshifts in symbolic dynamics. In particular, when a certain skew product of the subshift with the finite group admits an ergodic measure, this leads to a (Cèsaro) density formula for the group language in the shift. For my examples, I will focus on examples arising from substitutive dendric subshifts, e.g. the Fibonacci shift.