Geometric group theory seminar@SNU (Past)
For current seminar, please visit http://snutop.cayley.kr
This is an archive page of SNU Geometry and Topology Seminar.
2017
2016
Keivan Mallahi-Karai (Jacobs University)
- December 12th (Monday), 4:00 pm - 5:00 pm, Bldg. 129-301
- On the chromatic number of structured Cayley graphs
- The famous Hadwider-Nielson problem asks for the smallest number $n$ such that the points of the plane can be partitioned into $n$ sets such that no two points in the same set are at distance $1$. This problem, among many similar problems, can be reformulated in terms of the chromatic number of certain Cayley graphs. In this talk, I will discuss two recent results in this direction: one pertains to the Borel chromatic number of the unit-distance graph associated to quadratic forms over local fields. The other result is a lower bound on the chromatic number of Cayley graphs of finite groups of Lie type.
Lim Seonhee (Seoul National University)
- December 6th (Tuesday), 4:00 pm - 5:00 pm, Bldg. 129-406
- Around quantitative Oppenheim conjecture
- We will explain properties of geometry and dynamics on homogeneous space that are used to prove a quantitative version of Oppenheim conjecture, which was proved by Eskin-Margulis-Mozes for the real case. We will introduce analogous statement in S-arithmetic case, which we proved in a joint work with K. Mallahi-Karai and J. Han. We will report preliminary observations for the case of signature (2,2).
Lei Zhao (Chern Institute of Mathematics)
- November 30 (Wednesday), 4:00 pm - 5:00 pm, Bldg. 129-406 (to be confirmed)
- Relative equilibrium motions and real moment map geometry
- Chenciner and Jimenez Perez showed that the range of the spectra of the angular momenta of all the rigid motions of a fixed central configuration in a general Euclidean space form a convex polytope. In this note we explain how this result follows from a general real convexity theorem of O Shea and Sjamaar in symplectic geometry. Finally, we provide a representation theoretic description of the pushforward of the normalized measure under the real moment map for Riemannian symmetric pairs.
Hyowon Park (Seoul National University)
- November 22 (Tuesday), 4:00 pm - 5:00 pm, Bldg. 129-104
- Finite index subgroups of right-angled Artin groups
- Right-angled Artin groups are the graph product whose vertex groups are infinite cyclic groups, which are defined by finite simple graphs. A finite simple graph is called thin-chordal if it has no induced subgraphs that are isomorphic to either the cycle with 4 vertices or the path with 4 vertices. We will discuss group properties related to right-angled Artin groups from thin-chordal graphs. We show that a right-angled Artin group is defined by a thin-chordal graph if and only if every finite index subgroup of the group is a right-angled Artin group.
Sang-hyun Kim (Seoul National University)
- November 16 (Wednesday), 4:30 pm - 5:30 pm, Bldg. 27-116
- Flexibility of projective representations
- For which countable group G, does the moduli space
- X(G) = Aut(G) \ Hom(G,PSL(2,R)) / Inn(PSL(2,R))
- contain (uncountably many distinct equivalence classes of) dense faithful representations? Groups with such properties are called flexible. We prove combination theorems for flexible groups, and show that most Fuchsian groups and all limit groups (possibly with torsion) are flexible. The diversity of quasi-morphisms on those groups will follow. Implications for word-hyperbolic groups and mapping class groups will also be discussed. Joint with Thomas Koberda and Mahan Mj.
Paul Jung (KAIST) (Topology/dynamics joint seminar)
- November 8 (Tuesday), 4 pm - 5 pm, Bldg. 129-104
- An alpha-stable limit theorem for Sinai billiards with cusps
- We consider the dynamical system of Sinai billiards with a cusp where two walls of the billiard table meet at the vertex of a cusp and have zero one-sided curvature, thus forming a "flat point" at the vertex. For Holder continuous observables (random variables), we show that the properly normalized Birkhoff sums of stationary variables, with respect to the so-called ergodic billiard map, converge in distribution to a totally skewed alpha-stable law, for some alpha between 1 and 2.
Yoshifumi Matsuda (Aoyama Gakuin University)
- October 24 (Monday), 4 pm - 5 pm, Bldg. 27-325
- Bounded Euler number of actions of 2-orbifold groups on the circle
- Burger, Iozzi and Wienhard defined bounded Euler number for actions of the fundamental group of connected oriented surfaces of finite type possibly with punctures on the circle by orientation preserving homeomorphisms. Bounded Euler number can be extended to actions of 2-orbifold groups. We discuss Milnor-wood type inequality and rigidity phenomenon involving bounded Euler number for actions of certain 2-orbifold groups, such as the modular group.
JaeChoon Cha (Postech)
- October 6 (Thursday), 4 pm - 5 pm, Bldg. 129-101
- 4-manifold topology and disk embedding (Colloquium)
- TBA
BoGwang Jeon (Columbia University)
- August 8 (Monday), 11 am - 12 pm, Bldg. 129-301
- The Unlikely Intersection Theory and the Cosmetic Surgery Conjecture
- The main result of this talk is the following theorem:
- Let \(M\) be a 1-cusped hyperbolic 3-manifold whose cusp shape is not quadratic, and \(M(p/q)\) be its \(p/q\)-Dehn filled manifold. If \(p/q\) is not equal to \(p'/q'\) for sufficiently large \(|p|+|q|\) and \(|p'|+|q'|\), there is no orientation preserving isometry between \(M(p/q)\) and \(M(p'/q')\).
- This resolves the conjecture of C. Gordon, which is so called the Cosmetic Surgery Conjecture, for hyperbolic 3-manifolds belonging to the aforementioned class except for possibly finitely many exceptions for each manifold. We also consider its generalization to more cusped manifolds. The key ingredient of the proof is the unlikely intersection theory developed by E. Bombieri, D. Masser, and U. Zannier.
Sejong Park (National University of Ireland Galway)
- August 2 (Tuesday), 4 - 5 pm, Bldg. 129-301
- Toward a structure theorem for double Burnside algebras
- Given a finite group \(G\), the \((G, G)\)- bisets form the double Burnside ring \(B(G, G)\) with multiplication given by "tensor product over \(G\)". Unlike the Burnside ring \(B(G)\), not much is known about the ring structure of \(B(G, G)\). Boltje and Danz (2013) gave a "ghost map" for \(B(G, G)\) over rationals when \(G\) is cyclic. I will describe their result in a more conceptual form and show how this approach can be applied to some noncyclic cases. This is a joint work with Goetz Pfeiffer and Brendan Masterson.
Insuk Seo (UC Berkeley)
- July 20 (Wednesday), 4 - 5 pm, Bldg. 129-301
- Metastable behavior of the dynamics perturbed by a small random noise
- We consider a class of stochastic processes, which can be regarded as the perturbation of deterministic dynamics in a potential field. These processes exhibit a phenomenon known as metastable behavior if the potential field has several local minima. Metastable behavior is the phenomenon in which a process starting from one of local minima arrives at the neighborhood of the global minimum after a sufficiently long time scale. The precise asymptotic analysis of this transition time has been known only for the reversible dynamics, based on the potential theory of reversible Markov processes. In this presentation, we review this metastability theory for reversible dynamics, and introduce our recent generalization of this theory for non-reversible processes. (joint work with C. Landim)
Urs Frauenfelder (University of Augsburg)
- July 15 (Friday), 11am - 12 pm, Bldg. 129-301
- Periodic orbits in the restricted three body problem and Arnold's J^+ invariant.
- This is joint work with Kai Cieliebak. After regularization periodic orbits in the restricted three body problem become knots in RP^3. Their projection to position space is a curve in the plane. We show that Arnold's J^+ invariant is invariant under homotopies of periodic orbits and gives an additional invariant to their knot type in RP^3.
Thomas Koberda (University of Virginia)
- July 14 (Thursday), 4 - 5 pm, Bldg. 129-301
- Group actions on low dimensions I
- July 28 (Thursday), 4 - 5 pm, Bldg. 129-406
- Group actions on low dimensions II
Kasra Rafi (University of Toronto)
- July 12 (Tuesday), 4 - 5 pm, Bldg. 129-301
- The Teichmüller diameter of the thick part of moduli space
- We study the shape of the moduli space of a surface of finite type. In particular, we compute the asymptotic behavior of the Teichmüller diameter of the thick part of the moduli space. This turns out to be closely related to diameter of the space of trivalent graphs equipped with simultaneous whitehead move metric.
Jaesuk Park
- May 20 (Friday), 2 - 4 pm, Bldg 25-103
- Non-commutative quantum field theory and Chen’s iterated path integrals
- Iterated path integrals, generalising the familiar line integrals, are functions in the path space of smooth manifold which notion has introduced and used to extend de Rham cohomology theory to a homotopy theory on the fundamental group level by by K.-T. Chen. The theory has found many interesting applications, beyond algebraic topology, in algebraic geometry and number theory. Theory of homotopy category of homotopy QFT algebras is this speaker’s attempts to understand quantum field theory mathematically. In this lecture I will show that iterated integral(s) is quantum expectation of a (0+0)-dimensional non-commutative QFT obtained by certain quantisation of the algebra of differential forms on a manifold such that Chen’s homotopy functionals are equivalentto quantum correlation functions. Both Chen’s theory and NC QFT theory will be introduced beforehand.
Mounir Nisse
- May 19 (Thursday), 4 - 6 pm, Bldg 25-103
- Amoebas and Coamoebas
- Amoebas (resp. coamoebas) are the image under the logarithmic (resp. argument) map of algebraic (or analytic) varieties of the complex algebraic torus.They inherit some algebraic, geometric, and topological properties of the variety itself. In this introductory talk, I will define these two objects and tell you some of their nice properties and focalize myself on coamoebas. I will give several examples in the case of plane algebraic curves.
- May 20 (Friday), 4 - 6 pm, Bldg 25-103
- On Coamoebas of Algebraic Hypersurfaces
- I will describe coamoebas of algebraic complex hypersurfaces using their decomposition as pairs-of-pants shown by Mikhalkin. This talk will be focus on plane algebraic curves.
Koji Fujiwara (Kyoto University)
- May 11 (Wednesday), 5 - 6 pm, Bldg 129-104
- Handlebody subgroups in mapping class groups
- May 12 (Thursday), 4 - 5 pm (colloquium), Bldg 129-101
- Quasi-homomorphisms into non-commutative groups
- A function from a group \(G\) to integers \(\mathbb{Z}\) is called a quasi-morphism if there is a constant \(C\) such that for all \(g\) and \(h\) in \(G\), \(|f(gh)-f(g)-f(h)| < C\). Surprisingly, this idea has been useful. I will overview the theory of quasi-morphisms including applications.
Then we discuss a recent work with M. Kapovich when we replace the target group from \(\mathbb{Z}\) to a non-commutative group, for example, a free group.
Min Lee (University of Bristol)
- May 4 (Wednesday) 2-3pm, 129-406
- Effective equidistribution of primitive rational points on expanding horospheres
- The limit distribution of primitive rational points on expanding horospheres on SL(n,Z)\SL(n, R) has been derived in the recent work by M. Einsiedler, S. Mozes, N. Shah and U. Shapira. For n=3, in our joint project with Jens Marklof, we prove the effective equidistribution of q-primitive points on expanding horospheres as q→∞. Our proof relies on Fourier analysis and Weil's bound on Kloosterman sums. We will also discuss applications in number theory.
Jae Choon Cha (Postech)
- Apr 27 (Wednesday), 5 - 6 pm, Bldg. 129-104
- Quantitative topology and Cheeger-Gromov universal bounds
- I will begin with a quick introduction to the Cheeger-Gromov rho invariants from a topological viewpoint, and then present recent quantitative results on how they are related to triangulations and Heegaard splittings of 3-manifolds. I will also discuss quantitative bordism theory and an algebraic notion of controlled chain homotopy, which are the key ingredients of the proofs. Applications to topology of dimension 3 and 4 will be discussed if time permits.
Richard M. Weiss (Tufts University)
- Apr 5 (Tuesday), 5 - 6 pm, , Bldg. 129-104.
- Simple groups and buildings
We will describe the role of buildings in the study of simple groups, both finite and algebraic. We will focus on buildings of rank two, which can be easily defined in graph theoretical terms, and on groups of small rank. In particular, we will describe how the Suzuki groups, the Ree groups and a third family in this sequence arise in this context.
Masato Mimura (Tohoku University)
- Mar 22 (Tuesday), 3:30 - 4:45 pm, Bldg. 24-207
- Expanders and Margulis' construction
- Expanders (or, a family of expander graphs) are "sparse but high-connected" finite graphs. They have been showed to serve as key objects in many fields of pure/applied mathematics: for instance, geometric group theory, coarse geometry, low-dimensional topology, rigidity, random walks, and computer sciences. In the first talk, we introduce this concept. The first concrete example of expanders was provided by G. A. Margulis in 1973, and his construction is based on Cayley graphs of (finite) groups and Kazhdan's property (T) for finitely generated groups, which is equivalent to the fixed point property (F_{Hilb}) relative to Hilbert spaces. We explain his argument.
- Mar 23 (Wednesday), 2 - 3 pm, , Bldg. 129-406
- Strong algebraization of fixed point properties
- Since Shalom's breakthrough of Bounded generations and Kazhdan's property (T) in 1999, it had been a big problem to remove any form of "bounded generation" condition from algebraization processes in proving fixed point property. In this talk, we provide the affirmative resolution of this problem (arXiv:1505.06728).
Applications of our main theorem ("strong algebraization") include non-commutative universal lattices, and (Kac-Moody-)Steinberg groups over finitely generated unital ring.
- Mar 24 (Thursday), 3:30 - 4:45 pm, Bldg. 24-207
- Non-commutative universal lattices and unbounded rank expanders
- From Margulis' construction of examples above, the following natural problem arises: "Can special linear groups over finite fields of unbounded rank form expanders?" This was a long-standing problem, and M. Kassabov resolved this problem in the affirmative in 2005. Later, M. Ershov and A. Jaikin in 2010, furthermore, obtained the ideal answer to the problem above is, in fact, in the scope of Margulis' argument. They did it by establishing property (T) for the groups so-called non-commutative universal lattices (NCUL).
In the second talk, we focus on the NCUL, and its relation on the problem above. We discuss some difficulty in establishing property (T) for that group.
Ilya Gekhtman (Yale Univ. and Univ. of Bonn)
- March 18 (Friday) 2-3pm, 129-301. Dynamics of convex cocompact subgroups of mapping class groups
Giulio Tiozzo (Yale Univ.)
- March 16 (Wednesday) 2-3pm, 129-406. Random walks on weakly hyperbolic groups
- Mar 17 (Thursday) 4-5pm, 129-101 auditorium (Colloquium talk)
Random walks on weakly hyperbolic groups
Given a group of isometries of a metric space, one can draw a random sequence of group elements, and look at its action on the space. What are the asymptotic properties of such a random walk?
The answer depends on the geometry of the space. Starting from Furstenberg, people considered random walks of this type, and in particular they focused on the case of spaces of negative curvature.
On the other hand, several groups of interest in geometry and topology act on spaces which are not quite negatively curved (e.g., Teichmuller space) or on spaces which are hyperbolic, but not proper (such as the complex of curves).
We shall explore some results on the geometric properties of such random walks. For instance, we shall see a multiplicative ergodic theorem for mapping classes (which proves a conjecture of Kaimanovich), as well as convergence and positive drift for random walks on general Gromov hyperbolic spaces. This also yields the identification of the measure-theoretic boundary with the topological boundary.
Junehyuk Jung (KAIST)
- Mar 16 (Wednesday), 5 - 6 pm, Bldg. 129-104. Nodal domains of eigenfunctions on chaotic billiards
In this talk I’ll first go over some problems and related results in quantum chaos. Then I’ll explain how one can combine Quantum Ergodicity and Bochner’s theorem to prove that the number of nodal domains of quantum ergodic sequence of even eigenfunctions tends to infinity as the eigenvalue λ → +∞. In particular, this implies that the number of nodal domains of Maass-Hecke eigenforms grows with the eigenparameter. This talk is based on the joint works with S. Zelditch and with S. Jang.
Hyungryul Baik (MPI Bonn)
- Mar 3, 3:30 - 4:45, Room 24-207, Orderability
- Mar 8, 3:30 - 4:45, Room 24-207, Laminarity
- Mar 8, 5:00 - 6:00, Room 129-301, Convergence group
Abstract
Thursday, March 3, 3:30 - 4:45, Room 24-207 : Orderability
- Dynamical criterion - application of Kurosh's theorem to free product - Burns-Hale theorem - application to 3-manifold groups via Scott's core theorem
Tuesday, March 3, 3:30 - 4:45, Room 24-207 : Laminarity
- Geodesic laminations on the surface - characterization of geodesics laminations as saturated subsets of PTS - characterization of geodesic laminations as subset of Mobius band - constructing many very full laminations on surfaces - constructing virtual actions of fibered 3-manifold groups
Tuesday, March 3, 5:00 - 6:00, Room 129-301 : Convergence group
- Convergence group theorem - Convergence property from laminarity - Moore's theorem and Abstract Cannon-Thurston map - Convergence property on S^2 and the connection to Cannon's conjecture
강정수
- February 16 - 18, 4:30 - 6pm, Room 129-301.
- Symplectic capacities, Old and New I, II, III
Abstract
We give introduction to symplectic capacities and its recent developments.
Thomas Koberda (University of Virginia)
- January 7, 4 pm - 6 pm, 129-301, Exotic quotients of surface groups
Abstract
I will explain how to use TQFT representations of mapping class groups to produce linear representations of surface groups in which every simple closed curve has finite order, but which have infinite image. As a corollary, I will show how to produce covers of surfaces where the integral homology is not generated by pullbacks of simple closed curves on the base. This talk represents joint work with R. Santharoubane.
HyunKyu Kim (KIAS)
- Topological approach to seed-trivial mutation sequences in cluster algebras
- January 4, 2 pm, 129-301.
2015
At SNU.
November 25, Hyun-shik Shin (KAIST)
- Pseudo-Anosov mapping classes not arising from Penner's construction
- In this talk, we will discuss one property that is shared by all pseudo-Anosov mapping classes from Penner's construction. That is, all Galois conjugates of stretch factors of pseudo-Anosov mapping classes arising from Penner’s construction lie off the unit circle. As a consequence, we show that for all but a few exceptional surfaces, there are examples of pseudo-Anosov mapping classes so that no power of them arises from Penner’s construction. This resolves a conjecture of Penner. This is a joint work with Balazs Strenner.
- 4 pm, 129-104.
August 9–13, Geometric Topology Fair in Korea
July 9, Seung-wook Jang (University of Chicago)
January 16–21, Thomas Koberda (Yale)
- Lecture I. The relationship between right-angled Artin groups and mapping class groups
- In this general talk, we will give an overview of work joint with S. Kim concerning the relationship between right-angled Artin groups and mapping class groups. We will consider the broad question of, what right-angled Artin subgroups does a mapping class group or a right-angled Artin group admit?
- Lecture II. The curve complex for a right-angled Artin group
- In this talk we will give a more in depth discussion of the analogy between right-angled Artin groups and mapping class groups, through the comparison of the geometry of the extension graph and the curve complex.
- Lecture III. Convex cocompactness for subgroups of right-angled Artin groups
- In this talk we will discuss a new result joint with J. Mangahas and S. Taylor which characterizes finitely generated subgroups of right-angled Artin groups which have quasi-isometric orbit maps on the extension graph. These are analogous to convex cocompact subgroups of mapping class groups as defined by B. Farb and L. Mosher, and which are still rather poorly understood despite having attracted so much attention in recent years.
- Jan 16 (2pm), 19 (4pm), 21 (4pm), Bldg 129-301
January 13, BoGwang Jeon (Columbia)
- Hyperbolic three manifolds of bounded volume and trace field degree
- In this talk, I present my recent proof of the conjecture that there are only a finite number of hyperbolic three manifolds of bounded volume and trace field degree.
- 4 pm, 129-301
2014
At SNU.
December 9, Piotr Przytycki (McGill University)
- Balanced wall for random groups
- Gromov showed that one way to obtain a word-hyperbolic group is to choose a presentation "at random". I will survey random group properties in Gromov's model at various values of the density parameter. We will then focus on Ollivier-Wise cubulation of random groups for density parameter <1/5. I will indicate how to construct new walls that work at higher densities. This is joint work with John Mackay.
- 5 pm.
Related classes at SNU
- 12/09, 12/10
- Systolic group I, II
Related talk at KAIST
- 2014-12-11 (15:00 - 16:00)
- Arcs intersecting at most once
- We prove that on a punctured oriented surface with Eulercharacteristic chi < 0, the maximal cardinality of a set of essential simple arcs that are pairwise non-homotopic and intersecting at most once is 2|chi|(|chi|+1). This gives a cubic estimate in |chi| for a set of curves pairwise intersecting at most once on a closed surface.
November 13, Alden Walker (University of Chicago)
- Random groups contain surface subgroups
- Gromov asked whether every one-ended hyperbolic group contains a surface subgroup. I'll explain this question and sketch the proof that a random group (an example of a one-ended hyperbolic group) contains a surface subgroup. I'll give all necessary background and motivation on random groups. This is joint work with Danny Calegari.
- 2PM, 129-104.
November 4, Jason Behrstock (CUNY)
- Higher dimensional filling and divergence for mapping class groups
- We will discuss filling and divergence functions. We will describe their behaviors for mapping class groups of surfaces and show that these functions exhibit phase transitions at the rank, in analogy to the corresponding result for symmetric spaces. This work is joint with Cornelia Drutu.
October 14, Michael Brandenbursky, (CRM, Univ. Montreal)
- Concordance group and stable commutator length in braid groups
- In this talk I will define quasi-homomorphisms from braid groups to the concordance group of knots and examine its properties and consequences of its existence. In particular, I will provide a relation between the stable four ball genus in the concordance group and the stable commutator length in braid groups, and produce examples of infinite families of concordance classes of knots with uniformly bounded four ball genus. I will also provide applications to the geometry of the infinite braid group. In particular, I will show that its commutator subgroup admits a stably unbounded conjugation invariant norm. This answers an open problem posed by Burago, Ivanov and Polterovich. If time permits I will describe an interesting connection between the concordance group of knots and number theory. This work is partially joint with Jarek Kedra.
- 129-104
October 7, Moon Duchin (Tufts)
- Geodesics in nilpotent groups
- Perhaps the simplest non-abelian infinite group to understand is the Heisenberg group H(Z). Given a generating set as our "alphabet," a geodesic in the group is an efficient "spelling" of a group element. It is quite challenging to understand these precisely for an arbitrary choice of generators, but the large-scale geometric structure of the group makes it possible.
- 129-104
October 2, Sang-jin Lee (Konkuk Univ)
- Embedding of RAAG into braid groups
- 27-220
September 30, Alessandro Sisto (ETH)
- Acylindrically hyperbolic groups
- Acylindrically hyperbolic groups form an extensive class of groups that contain, for example, non-elementary (relatively) hyperbolic groups, mapping class groups of hyperbolic surfaces and non-Abelian RAAGs. Their defining feature is that they admit a "non-trivial enough" action on some hyperbolic space. Despite the generality of the notion, many results can be proven about them, for example that they are SQ-universal, meaning that if G is acylindrically hyperbolic then any countable group embeds in some quotient of G (in particular, G has uncountably many pairwise non-isomorphic quotients). We will discuss geometric properties of acylindrically hyperbolic groups, focusing on ingredients that one can use to prove SQ-universality.
- 129-301
August 7–12, Geometric Topology Fair in Korea (ICM satellite)
2013
Geometric group seminar at KAIST.
August 12- 16, KAIST Geometric Topology Fair
- The 11th KAIST Geometric Topology Fair
- Co-organizers: Sang-hyun Kim, Suhyoung Choi and Kihyoung Ko
July 9 - 11, Thomas Koberda (Yale)
- Two lectures on curve graphs for right-angled Artin groups
- Lecture I. Curve graphs for right-angled Artin groups I: right-angled Artin group actions on the extension graph
- I will discuss basics of extension graphs for right-angled Artin groups and the actions of right-angled Artin groups on their extension graphs. The main result in this lecture will be a version of the Nielsen--Thurston classification for right-angled Artin groups. Joint with S. Kim.
- Lecture II. Curve graphs for right-angled Artin groups II: the large and small scale geometry of the extension graph
- I will discuss various aspects of the geometry of the extension graph. I will discuss vertex link projections, the bounded geodesic image theorem, and a distance formula for right-angled Artin groups. Joint with S. Kim.
- July 9 and 11, 4-5 pm. Building E6-1, Room 4415.
May 20 - June 10, Igor Mineyev (UIUC)
- Ten lectures on Groups, Cell Complexes and l2-Homology
- One of the longest outstanding conjectures in group theory was Hanna Neumann Conjecture (1957):
- if H and K are subgroups of a free group, then rank(H∩K)-1 ≤ (rank(H)-1) (rank(K)-1).
- A long list of group theorists attempted to prove this conjecture, and also have contributed to related results. Mineyev first resolved this question completely in a paper at Annals of Mathematics (2012). In this series of ten lectures, he will give the main construction of the proof and its applications.
- Topics
- Cell Complexes and Group Actions.
- The L2 Homology and L2 Betti Numbers.
- The Hanna Neumann Conjecture.
- Orderability of Groups.
- Submultiplicativity and the Deep-Fall Property.
- The Atiyah Problem.
- May 20 - June 10. MWF 11 - 12:30 pm. Building E6-1, Room 2411.
- One-credit intensive course (MAS 583).
May 30, Irene Peng (POSTECH)
- 4:30 - 5:30 pm. Room 1501.
- Amenability and all that
- Amenability is one of those properties of group that has many different characterizations. I will discuss what it means in terms of invariant means, random walks and C* algebras. If time permits, I will also describe some related notions such as property rapid decay in the C* algebra setting.
June 4, Hyungryul Baik (Cornell)
- 4:30 pm - 5:30 pm. Room 4415.
- Circular-Orderability of Three-Manifold Groups and Laminations of the Circle
- We will discuss the connection between the circular-orderability of the fundamental group of a 3-manifold M and the existence of certain codimension-1 foliations on M via Thurston's universal circle theory. This theory provides a motivation to study group actions on the circle with dense invariant laminations. As an one lower dimensional example, we will give a complete characterization of Fuchsian groups in terms of its (topological) invariant laminations.
April 10 - 15, Jason Fox Manning (U of Buffalo)
- Four Lectures on Hyperbolic Dehn Fillings of Groups and Spaces
- Lecture 1: The Gromov-Thurston 2\pi Theorem.
- In the first lecture, I'll describe an explicit construction of negatively curved metrics on closed 3-manifolds obtained by Dehn filling of cusped hyperbolic manifolds. I also plan to sketch an application by Cooper and Long to finding surface subgroups of 3-manifolds. I'll talk about how to extend the 2\pi Theorem to cusped hyperbolic manifolds of dimension larger than 3.
- Lecture 2: Relatively hyperbolic groups.
- I'll define and give examples of relatively hyperbolic groups, and talk about what it means to do Dehn filling on a group pair.
- Lecture 3: The relatively hyperbolic Dehn filling theorem.
- I'll state the main theorem and sketch a proof.
- Lecture 4: Quasiconvex subgroups and Dehn filling.
- I'll define relatively quasiconvex subgroups, and talk about how to do Dehn filling while preserving quasiconvexity.
- April 10 (W), 11 (Th), 12 (Fr), 16 (T). 4 pm - 5:15 pm. Room 4415.
May 3 - 9, Cameron McA. Gordon (U Texas at Austin)
- Four Lectures on Dehn Surgery and Three-Manifold Groups
- Seminar: Dehn Surgery
- The lectures will be an introduction to Dehn surgery. This is a construction, going back to Dehn in 1910, for producing closed 3-manifolds from knots. A natural generalization is Dehn filling, in which some torus boundary component $T$ of a 3-manifold $M$ is capped off with a solid torus $V$. If $\alpha$ is the isotopy class of the loop on $T$ that bounds a disk in $V$, the resulting filled manifold is denoted by $M(\alpha)$. Generically, the topological and geometric properties of $M$ persist in $M(\alpha)$; in particular if $M$ is hyperbolic then $M(\alpha)$ is usually also hyperbolic. If this fails then the filling is said to be {\it exceptional}. We will outline a program to classify the triples $(M;\alpha,\beta)$ with $M(\alpha)$ and $M(\beta)$ exceptional, describing what is known in this direction and what remains to be done.
- Colloquium: Left-Orderability of Three-Manifold Groups
- We will discuss connections between three notions in 3-dimensional topology that are, roughly speaking, algebraic, topological, and analytic. These are: the left-orderability of the fundamental group of a 3-manifold M, the existence of certain codimension 1 foliations on M, and the Heegaard Floer homology of M.
- Reference
- Park City Lectures Dehn Surgery and 3-Manifolds, in Low Dimensional Topology, ed. T.S. Mrowka and P.S. Ozsvath, IAS/Park City Mathematics Series, Vol. 15, AMS 2009.
- Seminar May 3 (F), 7 (T), 8 (W). 4:30 - 5:30 pm. Room 4415
- Colloquium May 9 (Th). 4:30 - 5:30 pm. Building E6, 1501.
2012
At KAIST.
August 13 - 17, KAIST Geometric Topology Fair
- The 10th KAIST Geometric Topology Fair will focus on the recent developments in geometric group theory and three-manifold theory. The lecture series and talks will be aimed at graduate students and early career researchers.
- Lecture Series by: Michael Davis (Ohio State Univ), Koji Fujiwara (Tohoku Univ.), Alan Reid (Univ. Texas at Austin).
Research Talks by: Jinseok Cho (KIAS), Kanghyun Choi (KAIST), Stefan Friedl (Univ. of Cologne), Thilo Kuessner (KIAS), Sang-hyun Kim (KAIST), Taehee Kim (Konkuk Univ.), Sang-Jin Lee (Konkuk Univ.), Gye-Seon Lee (Seoul National Univ.), Seonhee Lim (Seoul National Univ.), Ken’ichi Ohshika (Osaka University).
August 21 - 30, Genevieve S. Walsh (Tufts)
- Four Lectures on Introduction to Hyperbolic Orbifolds and Knot Commensurability.
- Aug 21, 23, 28, 30. TTh 4 - 5 pm. Room 3433.
- Lecture 1: 2-dimensional orbifolds
- In this lecture we will define and describe orbifolds and set notation. In particular, we will discuss orbifold Euler characteristic, orbifold covers, good orbifolds, bad orbifolds, and the orbifold fundamental group. Explicit examples of spherical, Euclidean and hyperbolic 2-orbifolds will be given. We will also prove that there is a smallest closed hyperbolic 2-orbifold.
- Lecture 2: 3-dimensional orbifolds
- Here we will explore 3-dimensional orbifolds, restricting mainly to good orbifolds. Although we will give explicit examples of many different types of 3-dimensional orbifolds, the focus will be on hyperbolic 3-orbifolds. To this end, we will discuss hyperbolic isometries and the geometry of hyperbolic orbifolds and hyperbolic orbifolds. We will discuss how useful orbifolds are to the study of 3-manifolds, and give a statement of geometrization.
- Lecture 3: Commensurability
- Commensurability is an equivalence relation on manifolds and orbifolds which is a refinement of geometrization. Here we will describe the current study of commensurability of hyperbolic manifolds, focusing on commensurability of knot complements. We will describe hyperbolic knot complements and their symmetry groups, and discuss the commensurator group and the orbifold commensurator quotient of a hyperbolic non-arithmetic knot complement.
- Lecture 4: Some results on commensurability of knot complements
- A conjecture of Reid and Walsh asserts that there are at most 3 hyperbolic knot complements in any commensurability class. Here we discuss this conjecture, and give results under certain circumstances. The problem naturally divides itself into two cases, the case of hidden symmetries and the case of no hidden symmetries, and we discuss both. The new results presented here are joint with M. Boileau, S. Boyer, and R. Cebanu.
- References
- Not Knot Part 1 (video)
- Not Knot Part 2 (video)
- Thurston's course notes, Chapter 13
- Michael Kapovich, Hyperbolic manifolds and discrete groups, Chapter 6
- PDF "Orbifolds and commensurability"
September 5–7, Ian Agol (UC Berkeley)
- Three Lectures on Virtual Haken Conjecture.
- September 5, 6, 7 (WThF), 4 - 5 pm. Room 3433.
- Prequel Lecture September 5, Wednesday, 4 – 5 pm @ Room 3433
- An Invitation to Cube Complexes by Sang-hyun Kim (Slides)
- We survey basic facts on cube complexes and discuss how those facts are related to the study of subgroups of right-angled Artin groups.
- Lecture I
- We will discuss the proof of the virtual Haken conjecture and related questions. The first lecture will be an overview and an explanation of how to reduce the problem to a conjecture of Wise in geometric group theory.
- Lecture II
- The second lecture will be on the RFRS condition and virtual fibering for hyperbolic 3-manifolds.
- Lecture III
- The third lecture will be on the proof of Wise's conjecture, that cubulated hyperbolic groups are virtually special.
- The following video lectures @ KAIST by Alan Reid are not prerequisites, but provide valuable information on the background and a big picture surrounding this problem:
- 3-manifold groups, covering spaces and LERF
http://vod.mathnet.or.kr/sub2_2.php?no=2437 http://vod.mathnet.or.kr/sub2_2.php?no=2445 http://vod.mathnet.or.kr/sub2_2.php?no=2450
- Selected References
- F. Haglund and D. Wise, Special cube complexes, Geom. Funct. Anal. (2007) 1–69.
- I. Agol, Criteria for virtual fibering, J. Topol. 1 (2008), no. 2, 269–284.
- D. Wise, Research announcement: the structure of groups with a quasiconvex hierarchy., Electron. Res. Announc. Math. Sci. 16 (2009), 44–55.
- CBMS lecture videos, notes and related preprints: http://comet.lehman.cuny.edu/behrstock/cbms/program.html
September 25–October 4, Thomas Koberda (Yale)
- Four Lectures on Mapping Class Groups and Right-Angled Artin Groups
- An Introduction to Right-Angled Artin Groups and Mapping Class Group In this lecture, we will begin with some basic facts about right-angled Artin groups and mapping class groups. The goal is to provide a foundation for various new results concerning the structure and geometry of right-angled Artin groups, mapping class groups, and their subgroups.
- An Introduction to Right-Angled Artin Groups and Mapping Class Groups In this lecture, we will discuss the primary result of [3], which roughly says that if we take any collection of mapping classes, say {f1,...,fk} and replace them by sufficiently high powers {f1^N,...,fk^N}, they generate a right-angled Artin subgroup of the mapping class group of the expected type. Unless otherwise noted, all examples and statements can be found with proof (or appropriate reference) in [3].
- Right-Angled Artin Subgroups of Right-Angled Artin Groups In this lecture, we will discuss the primary results of [2]. In that article, the authors develop a general theory for determining when there exists an embedding A(X) -> A(Y) for two graphs X and Y.
- A Dictionary Between Mapping Class Groups and Right-Angled Artin Groups Via Curve Complexes In this lecture, we will primarily be discussing the results of [1], together with appropriate background. The general principle we would like to explore is that right-angled Artin groups behave a lot like mapping class groups from the point of view of their actions on their extension graphs and curve complexes respectively.
- References
- Sang-hyun Kim and Thomas Koberda. Actions of right-angled Artin groups on quasi–trees. In preparation.
- Sang-hyun Kim and Thomas Koberda. Embedability of right-angled Artin groups. Preprint.
- Thomas Koberda. Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups. To appear in Geom. Funct. Anal.
- September 25, 27, October 2, 4. T 4 - 5 pm, Th 2:30 - 3:30. Room 3433
2011
At KAIST.
August 8–12
- The 9th KAIST Geometric Topology Fair===
- Lecture Series
- Random rigidity in free groups, Danny Calegari (Caltech)
- Recognizing low-dimensional manifold groups, Jason F. Manning (State University of New York at Buffalo)
- Embedding finitely generated groups into finitely presented groups, Mark V. Sapir (Vanderbilt)
- Asymptotic group theory - pro-p groups; property T and expansion, Efim Zelmanov (Univeristy of California at San Diego)
- Research Talks
- Simplicial volume and bounded group cohomology, Sungwoon Kim (KIAS)
- Embeddability between right-angled Artin groups, Sang-hyun Kim (KAIST)
- Graph braid groups: its 10 year history, Kihyoung Ko (KAIST)
- Combinatorial group theory applied to 2-bridge link groups, Donghi Lee (Pusan National University)
- Commensurizer group and its growth, Seonhee Lim (Seoul National University)
September 1 - 16, Jon McCammond (UC Santa Barbara)
- Ten Lectures on Coxeter Groups and Reflection Symmetry
- September 1 - 16, 2011 (except for Saturday, Sunday and 09/12, 09/13) MWF 4 - 5:30 pm, TTh 1 - 2:30 pm
- One-credit intensive course.
- Symmetry and Abstraction - Why do mathematicians see only 17 types of wallpaper?, a public Lecture
- Coxeter groups are a central object of study in many parts of mathematics. They include the groups of symmetries of the regular polytopes, the finite reflection groups and the Weyl groups at the core of the study of Lie groups and Lie algebras. They have many remarkable properties including the fact that they have faithful linear representations and a proper cocompact action by isometries on piecewise Euclidean space of nonpositive curvature. In this course I will focus on laying the foundations for the geometry, topology and combinatorics of Coxeter groups.
- Prerequisites
- The prerequisites are only Linear Algebra and Abstract Algebra (i.e. groups). Some familiarity with groups given by generators and relations, and fundamental groups and covering spaces would be nice but probably not absolutely necessary.
- Lecture Plan
- 1 & 2 Regular polytopes and spherical Coxeter groups
- 3 & 4 Lie groups and Euclidean Coxeter groups
- 5 & 6 Coxeter groups in general
- 7 & 8 Linear representations and basic facts
- 9 & 10 Non-positively curved spaces and geometric actionsGrading
- Letter grades are given based on (1) attendance / participation (2) a short paper due 09/01/2011.