# Geometry and Topology Seminar

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Welcome to SNU Geometry and Topology Seminar http://seminar.cayley.kr

## 2016

### 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 (FHilb) 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
Abstract

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.