SNU Geometry and Topology Seminar

SNU Geometry, Topology

& Dynamics Seminar

Cheol Hyun Cho, Jungsoo Kang, Dano Kim, Hong-Jong Kim, Hyuk Kim, Otto van Koert, Woong Kook, Seonhee Lim, Jongil Park

snutop@googlegroups.com

https://groups.google.com/forum/#!forum/snutop

December 14 (Fri) 11 am - 12 pm, Bldg 27-220.

Junehyuk Jung (Texas A&M)

Ergodicity and nodal counting of eigenfunctions on 3-manifolds

It was unclear until recently whether the ergodicity of the geodesic flow on a given Riemannian manifold has any significant impact on the growth of the number of nodal domains of eigenfunctions of Laplace-Beltrami operator , as the eigenvalue . In this talk, I'm going to explain my recent work with Steve Zelditch, where we prove that, when is a principle -bundle equipped with a generic Kaluza-Klein metric, the nodal counting of eigenfunctions is typically 2, independent of the eigenvalues. Note that principle -bundle equipped with a Kaluza-Klein metric never admits ergodic geodesic flow. This, for instance, contrasts the case when is a surface with non-empty boundary with ergodic geodesic flow (billiard flow), in which case the number of nodal domains of typical eigenfunctions tends to (proven in my paper with Seung Uk Jang).

I will also present an orthonormal eigenbasis of Laplacian on a flat 3-torus, where every non-constant eigenfunction has exactly two nodal domains. This provides a negative answer to the question raised by Thomas Hoffmann-Ostenhof:

``For any given orthonormal eigenbasis of the Laplace--Beltrami operator, can we always find a subsequence where the number of nodal domains tends to ?''

November 28 (Wed) 2 pm, Bldg 129-301

Kathryn Mann (Brown University)

Geometric structures from dynamical rigidity

A major theme in dynamics is that rigidity (of a group action on a space, or of a representation) often comes from some underlying geometric phenomenon. In this talk, I will explain a new result with Maxime Wolff for surface group actions, along these lines. We show that the only source of topological rigidity for surface groups acting on the circle is an underlying geometric structure: if an action is rigid, then it comes from an embedding of the group as a lattice in PSL(2,R) or one of its finite extensions, acting in the standard way.

The talk will introduce and motivate the broad theme of our work, and explain the philosophy of how to “reconstruct” a hyperbolic surface out of basic dynamical information.

October 30 (Tues), 16:00-17:00, Room 129-406

Sangbum Cho (Hanyang University)

The mapping class groups of Heegaard splittings

The mapping class group of a Heegaard splitting for a 3-manifold is the group of isotopy classes of self-homeomorphisms of the manifold preserving the splitting. It is natural to try to understand the structure of each of those groups and to find a reasonable generating set or a presentation of it. In this talk, we review the history of this problem and introduce some recent progress on it. In particular, we will look at finite presentations of the groups of reducible, genus two Heegaard splittings, and the key ideas to obtain them. This is a joint work with Yuya Koda.

Yat-Hin Suen (Marco) from IBS-CGP

Wednesday (Oct 24) 16:00~18:00

Thursday (Oct 25) 11:00~13:00

Place: 25-103

Two introductory lectures on SYZ (Strominger-Yau-Zaslow) mirror symmetry.

Abstract: Mirror symmetry is a duality between complex geometry and

symplectic geometry. In 1994, Kontsevich suggested an algebraic way to

understand mirror symmetry, which is now known as the homological

mirror symmetry. Two years later, Strominger, Yau and Zaslow proposed

an entire geometric way to understand mirror symmetry. Roughly

speaking, the SYZ conjecture suggests that mirror symmetry can be

understood as a duality between two special Lagrangian torus

fibrations over an integral affine manifold B with singularities. In

this lecture series, I will give an introduction to SYZ mirror

symmetry. I will start by reviewing Floer theory and its Morse

theoretic approach. Then I will discuss the SYZ construction in

details. By coupling with the Witten-Morse theory, I will show that

how SYZ approach can be used to understand homological mirror symmetry

in the semi-flat case (no singular fibers). The non-semi-flat case

will be discussed mainly in dimension 2. We will see the effect of

holomorphic disks contribution from the smooth Lagrangian fibers,

whose locus on B form a set of geometric objects called walls. I will

then discuss the wall crossing and scattering phenomena in the

reconstruction problem. If time are allowed, I will discuss some

recent progress on theta functions and geometric quantization in

various situations.

October 23 (Tues), 14:00-15:00, Room 129-406.

Takuya KATAYAMA (Hiroshima University)

On virtual embeddability between the mapping class groups of some surfaces

By the Birman-Hilden theory, the braid group on 2g strands is embedded in the mapping class group of the closed surface of genus 2g possibly with marked points.

In this talk, using some right-anlged Artin groups in the mapping class groups, we show that any finite index subgroup of the braid group on 2g+1 strands cannot be embedded in the mapping class group of the closed surface of genus g with at most one marked point.

Teruaki Kitano (Soka University)

Riley polynomials of 2-bridge links and epimorphisms

Date Oct 04, 2018

Room 129-310

Time 10:00-13:00

Riley polynomial is defined for some kind of a presentation of 2-bridge link groups. Any zero of this polynomial is corresponding to a parabolic representation in SL(2;C). If there exists epimorphism between link groups, then Riley polynomial of the source can be divided by the one of the target. In this talk we prove that there exists an epimorphism if the above relation between Riley polynomials holds.

Twisted Alexander polynomial for a torus link

http://www.math.snu.ac.kr/board/seminars/7877512018.10.01 13:31:31 12

Date Oct 03, 2018

Room 129-310

Time 14:00-15:00

Twisted Alexander polynomial is one generalization of the Alexander polynomial for a link with linear representation. In this talk we consider this invariant for a SL(2;C)-representation and discuss the behavior of this for a torus link. This is part of joint works with Anh Tran and Takayuki Morifuji.

Reidemeister torsion of a homology 3-sphere

Date Oct 03, 2018

Room 129-310

Time 10:30-13:30

Reidemeister torsion is a numerical invariant for a finite cell complex with a linear representation of the fundamental group. In this talk we consider this invariant for a homology 3-sphere M with an SL(2;C)-irreducible representation. We assume that the SL(2;C)-character variety of M is finite set. Then we can define a polynomial whose zeros are values of Reidemeister torsions. For Brieskorn homology 3-spheres and the manifolds obtained by 1/n-surgery along the figure eight knot, we can give explicit formulas by using Chebyshev polynomials of the first and second types. Further we discuss the relation between this polynomial and the SL(2;C)-Casson invariant. This is part of joint works with Anh Tran.

July 10 (Tues), 16:00-17:00, Bldg 129-406

Yash Lodha (EPFL)

Nonamenable groups of piecewise projective homeomorphisms

Groups of piecewise projective homeomorphisms provide elegant examples of groups that are non amenable, yet do not contain non abelian free subgroups. In this talk I will present a survey of these groups and discuss their striking properties. I will discuss properties such as (non)amenability, finiteness properties, normal subgroup structure, actions by various degrees of regularity and Tarski numbers.

July 5 (Th), 11:00-12:00, Bldg 129-406

Sungkyung Kang (Oxford)

A transverse knot invariant from Z/2-equivariant Heegaard Floer cohomology

The Z/2-equivariant Heegaard Floer cohomlogy of the double cover of along a based link, defined by Lipshitz, Hendricks, and Sarkar, is an isomorphism class of -modules. In this talk, we show that, in the case of knots, this invariant is natural, and is functorial under based cobordisms. Then, as a topological application, given a transverse knot K in the standard contact 3-sphere, we will define an element of the Z/2-equivariant Heegaard Floer cohomology that depends only on the transverse isotopy class of K, and show that it satisfies some interesting properties.

July 3 (Tu), 4 (W) 129-406, 15:00 ~ 17:00

Nicolo Sibilla (University of Kent)

Lec 1:

The topological Fukaya category and mirror symmetry for 3d toric LG models

Abstract:

The Fukaya category of open symplectic manifolds is expected to have

good local-to-global properties. Based on this idea several people

have developed sheaf-theoretic models for the Fukaya category of

punctured Riemann surfaces: the name topological Fukaya category

appearing in the title refers to the (equivalent) constructions due to

Dyckerhoff-Kapranov, Nadler and Sibilla-Treumann-Zaslow. In this talk

I will introduce the topological Fukaya category and explain

applications to Homological Mirror Symmetry for 3-dimensional toric

LG models . This is joint work with James Pascaleff.

Lec 2:

Log schemes, root stacks and parabolic bundles

Log schemes are an enlargement of the category of schemes that was

introduced by Deligne, Faltings, Illlusie and Kato, and has

applications to moduli theory and deformation problems. Log schemes

play a central role in the Gross-Siebert program in mirror symmetry.

In this talk I will introduce log schemes and then explain recent work

joint with D. Carchedi, S. Scherotzke, and M. Talpo on various aspects

of their geometry. I will discuss a comparison result between two

different ways of associating to a log scheme its etale homotopy type,

respectively via root stacks and the Kato-Nakayama space. Our main

result is a new categorified excision result for parabolic sheaves,

which relies on the technology of root stacks.

June 14 (Th), 15:00-16:00, Bldg 129-301

* Note the change of the time/place

Thomas Koberda (Univ. Virginia)

Title

Topology of hyperplane arrangements

Abstract

I will discuss some new methods for analyzing the topology of the complement of a hyperplane arrangement in complex projective space. As a consequence, we can show that the complement of a hyperplane arrangement has torsion-free fundamental group, which resolves a conjecture of P. Orlik. I will give some applications to the study of Artin-Tits groups.

May 28 (Mon), 13:30 - 15:30, Bldg 27-325

May 29 (Tue), 16:00 - 17:30, Bldg 129-104

Lino Amorim (Kansas State University)

Title

Perversely categorified Lagrangian correspondences (Mon)

Categorical Gromov-Witten invariants (Tue)

Abstract

May 15 (Tue), 10:30 - 12:00, Bldg 129-406

Chaya Norton (Concordia University)

Title

The monodromy map and Darboux coordinates on the SL(2,C)-character variety

Abstract

The monodromy map sends a projective structure to a representation of fundamental group of the Riemann surface into PSL(2,C) defined up to overall conjugation. In other words we have a map from T*Mg to the SL(2,C)-character variety which depends on the origin section chosen for the moduli space of projective structures. We will discuss previous work regarding this monodromy map and present joint work with Bertola and Korotkin which proves the map is a symplectomorphism with base Bergman, Schottky, or Wirtinger projective connection when the character variety is equipped with the Goldman bracket. Comparing our results with Kawai 96 (and more recently Loustau 15, Takhtajan 17) we propose a generating function (describing the change of Darboux coordinates) for the equivalence between the symplectic structures induced from the base Bergman projective connection versus the base Bers projective connection. We hope to discuss some open questions resulting from this work.

April 19 (Th) 16:00-17:00, Bldg 27-116

April 20 (F) 16:00-17:00, Bldg 129-104 (note the change!)

Jessica Purcell (Monash University)

Title Decompositions of 3-manifolds and hyperbolic geometry

Abstract A highly useful technique for studying a 3-manifold is to decompose it into simpler pieces, such as tetrahedra, and to examine normal surfaces within the pieces. If the pieces admit additional data, e.g. an angle structure, then there are concrete geometric consequences for the manifold and the surfaces it contains. For example, we may determine conditions that guarantee the manifold is hyperbolic, estimate its volume, and identify quasifuchsian surfaces embedded within it.

In the first talk, I will briefly describe some history of these decompositions, including work of Thurston, Menasco, and Lackenby, and then describe how to generalise their work to extend results to broader families of 3-manifolds. For example, we may allow pieces that are not simply connected, glued along faces that are not disks. We give examples of manifolds with these structures, particularly families of knot and link complements.

In the second talk, using a generalisation of normal surfaces, angle structures, and combinatorial area, I will describe geometric consequences of these decompositions and further applications.

This is joint work with Josh Howie.

April 11 (W) 15:00-17:00, April 12 (Th) 10:00-12:00

Bldg 129-406

Speaker

Oh, Yong-Geun (IBS-CGP & Postech)

Title

Analysis of pseudoholomorphic curves and Hofer's displacement energy

Abstract

In this series of lectures, I will start with basic analysis of Hamiltonian perturbed Cauchy-Riemann equations and then explain how one can use them to study the problem of disjuncting a Lagrangian submanifold from an open subset of symplectic manifolds. Such a problem is relatively easy when some form of Floer homolgy theory is available and the homology is not zero. The main focus of the lecture is the case when the latter machinery is not available such as the case of a pair of Lagrangian submanifold and an open subset of symplectic manifolds.

March 29 (Thursday), 27-220, 15:00-16:00

Bernhard Hanke (University of Augsburg)

Title: The space of positive scalar curvature metrics

Abstract: The construction and classification of Riemannian metrics satisfying certain

curvature bounds are of fundamental interest in differential geometry. In this talk we

specialize to metrics of positive scalar curvature, which are characterized

by a simple volume growth condition of balls with small radii.

Research during the last couple of years, based on a variety of methods,

revealed the topological complexity of the space of all such metrics on a fixed manifold.

We will give an overview of this development.

March 28 (W), 27-325, 16:00-17:00

Andrei Pajitnov (University of Nantes),

Title: Massey products in mapping tori

Abstract: Let f be a diffeomorphism of a compact manifold M onto itself and X its mapping torus; then X is fibred over a circle, and f can be interpreted as the monodromy map of this fibration.

In this talk we show a relation between non-zero higher Massey products in cohomology

of X and Jordan blocks of size > 1 of the homomorphism in the homology induced by f.

We prove in particular that if X is a Kaehler manifold, then the monodromy homomorphism in the homology is diagonalizable.

March 19 (M), 21 (W), 22 (Th)

KATO Tsuyoshi (Kyoto University)

Title: Twisted Donaldson Invariant

Abstract: In my three talks, I will explain the construction of a twisted Donaldson invariant. We also include basic subjects on both gauge theory and non commutative geometry. Below we describe the details of three talks. This is joint work with H. Sasahira and H.Wang.

Talk1: 3/19(Mon) 11:00~12:10, 24-112

Subtitle: Rough description of basic gauge theory

Abstract: We quickly review Yang-Mills gauge theory and Donaldson theory.

Talk2: 3/21(Wed) 11:00~12:10, 24-112

Subtitle: Twisted Donaldson invariant 1

Abstract: We define a twisted Donaldson’s invariant using the Dirac operator twisted by flat connections

when the fundamental group of a four manifold is free abelian. We also verify non triviality of the invariant

by presenting some exotic pairs which are obtained from them.

Talk 3: 3/22(Thur) 14:00~15:10, 25-103

Subtitle: Twisted Donaldson invariant 2

Abstract: Using Connes-Moscovici’s index theorem, we introduce the construction of twisted Donaldson’s invariant

when fundamental group is non-abelian. Non commutative geometry is a useful tool in the study of topology of Riemannian manifolds. Taking into account of the fundamental group in the formulation of a topological invariant, one can obtain refined topological invariants involving the C^*-algebra of the fundamental group. For example, the Novikov conjecture on homotopy invariance of higher signatures has been developed extensively using non-commutative geometry. We apply the method of non commutative geometry to the construction of twisted Donaldson’s invariant.

February 19 - 23

Workshop on symplectic dynamics and celestial mechanics,

#27-129

Info: http://www.math.snu.ac.kr/~okoert/workshop2018

February 2 (F)

15:00 - 16:00 Lecture 1

16:30 - 17:00 Lecture 2

#27-325

- Jae-suk Park (Postech)

- Title: Coalgebraic Principles of Quantum Field Theory

- Abstract: We consider formal quantum field theory as study of homotopy category hoQFT of QFT-infinity algebras. This allows us not only to define a formal quantum field theory as an object together with a morphism to an initial object in the category but also to characterizes the whole tower of quantum correlation functions of the theory, including quantum correlation functions in the usual sense as well as their higher homotopy generalizations, as invariants attached to a quasi-isomorphism from the classical cohomology of the QFT-infinity-algebra, or the classical equation of motion space, such that two theories based on quasi-isomorphic QFT-infinity-algebras have isomorphic towers of quantum correlation functions. It also provide us a concrete method for determining the tower of quantum correlations up to certain ambiguity order by order in $hbar$ by an infinite sequence of classical cohomological computations. Our approach is COalgebraic, but there emerges an equivalent geometrical picture, provided that we consider a quantum field theory with finite dimensional classical cohomology, involving certain homotopy equivariant flat formal $kbar$-superconnection, a generalization of the formal version of Deligne's flat $lambda$-connection introduced in the context of non-Abelian Hodge theory, on the tangent bundle to the based formal moduli space of the theory. For an anomaly-free theory with finite dimensional classical cohomology we have a pencil of flat superconnection with some additional condition such that its 1-form part is torsion-free, leading to a structure of formal super F-manifold on the moduli space.

Nov 20, 2017 (Mon)

4 - 6 pm

#27-326

Hyungryul Baik (KAIST)

Title : Dynamics on Surfaces

Abstract : Thurston classified surface homeomorphisms up to isotopy. Most surface homeomorphisms are so-called pseudo-Anosov. For each pseudo-Anosov homeomorphism, there is an associated number called the stretch factor which tells us how the iterations of the homeomorphism change the length of simple closed curves on the surface (with respect to an arbitrary metric of constant curvature). We try to find a number-theoretic characterization of these numbers, and discuss the difficulty of the problem and recent partial results. This talk partially represents joint work with A. Rafiqu and C. Wu.

July 21, 2017 (F)

Speaker: Thomas Koberda

Time: 4 - 5 pm

Places: SNU Bldg. 27-116

Title : The Free Product Structure of Diffeomorphism Groups

Abstract : I will discuss some aspects of the algebraic structure of finitely generated groups of diffeomorphisms of compact one-manifolds. In particular, we show that if G is not virtually metabelian then (G x Z)*Z cannot act faithfully by C^2 diffeomorphisms on a compact one-manifold. Among the consequences of this result is a completion of the classification of right-angled Artin groups which admit faithful C^{\infty} actions on the circle, a program initiated together with H. Baik and S. Kim. This represents joint work with S. Kim.

July 14, 2017 (F)

Speaker: Thomas Koberda

Dates:Friday)

Time: 4 - 5 pm

Places: SNU Bldg. 129-301

Title : Square-roots of Thompson’s group

Abstract : I will discuss square roots of Thompson’s group F, which are certain two-generator subgroups of the homeomorphism group of the interval, the squares of which generate a copy of Thompson’s group F. We prove that these groups may contain nonabelian free groups, they can fail to be smoothable, and can fail to be finitely presented. This represents joint work with Y. Lodha.

June 27, 2017

Time: 4:30 pm

Places: SNU Bldg. 129-301

Speaker: Wouter van Limbeek

Title : Towers of regular self-covers and linear endomorphisms of tori

Abstract : Let M be a closed manifold that admits a nontrivial cover diffeomorphic to itself. Which manifolds have such a self-similar structure? Examples are provided by tori, in which case the covering is homotopic to a linear endomorphism. Under the assumption that all iterates of the covering of M are regular, we show that any self-cover is induced by a linear endomorphism of a torus on a quotient of the fundamental group. Under further hypotheses we show that a finite cover of M is a principal torus bundle. We use this to give an application to holomorphic self-covers of Kaehler manifolds.

On Affine DG Group Scheme

Speaker: Jae-Suk Park (IBS–CGP)

Dates: May 12, 2017

Time: 5 pm

Places: SNU Bldg. 129-406

Abstract

An affine dg group scheme is a representable functor from the homotopy category of dg commutative algebras to the category of groups. I will talk about its representations and Riemann-Hilbert type correspondence.

Some symplectic properties of hypersurface cusp singularities

Speaker: Ailsa Keating (Columbia University)

Dates: April 17 - 19, 2017

Places: SNU Bldg. 129-406 and 129-301

Abstract

We will give a symplectic geometer's introduction to hypersurface cusp singularities, i.e. singularities of the form where c is a non-zero constant. Loosely speaking, these are the "next most complicated" singularities after the simple (i.e. ADE) singularities. We will focus on properties of the Milnor fibres of these singularities; these are the open complex surfaces obtained by smoothing them. We will first explain how to get explicit descriptions of these as total spaces of Lefschetz fibrations. We will then

1. construct some new examples of exact Lagrangian tori in these surfaces, and all Milnor fibres of non-ADE hypersurface singularities.

2. give a proof of homological mirror symmetry for the hypersurface cusp singularities, which ties into Gross, Hacking and Keel's proof of Looijenga's conjecture on cusp singularities.

No prior knowledge of singularity theory will be assumed.

Schedule

Lecture 1, April 17, 16:00 - 17:30, Sangsan 129-406

Lecture 2, April 18, 14:00 - 15:30, Sangsan 129-301

Lecture 3, April 19, 14:00 - 15:30, Sangsan 129-406

Intensive Lectures on Knot Floer homology

Speaker: Kyungbae Park (KIAS)

Place: Room 103, SNU Bldg. 25

Dates: March 9, 16, 24, April 6, 13, 20, 2017

Time: 14:00 - 15:30

Organizer: Jongil Park (SNU)

Abstract

Heegaard Floer homology is a package of invariants for objects in the low dimensional topology, in a framework of TQFT. In this lecture series, we introduce some extensions of Heegaard Floer invariants: the knot Floer homology for knots in 3-manifolds and the bordered Floer homology for 3-manifolds with boundary. In particular, we will focus on the relationship between these invariants and classical knot invariants. More precisely, the knot Floer homology is known as a categorification of the Alexander polynomial and contains strictly more topological information than it, such as the genera and fiberedness of knots. On the other hand, it is also known that the knot Floer homology cannot recover the Alexander module of a knot, a refinement of Alexander polynomial. Recently, Hom, Lidman and Watson showed that the Alexander module and the Seifert form of a knot can be recovered from the bordered Floer invariant of the complement of a Seifert surface of the knot.

Schedule

Lecture I (March 9): Review on Heegaard Floer homology

Lecture II (March 16): Introduction to the knot Floer homology

Lecture III (March 24; 13:00-14:30): Knot Floer homology as a categorification of the Alexander polynomial - Time changed!

Lecture IV (April 6): Bordered Floer homology I

Lecture V (April 13): Bordered Floer homology II

Lecture VI (April 20): Heegaard Floer homology and the Alexander module

Collar lemma for Hitchin representations

Speaker: Gye-seon Lee (Univ. of Heidelberg)

Date: March 27, 16:30 - 18:30

Place: Room 301, SNU Bldg. 129

Abstract

There is a classical result first due to Keen known as the collar lemma for hyperbolic surfaces. A consequence of the collar lemma is that if two closed curves A and B on a closed orientable hyperbolizable surface have non-zero geometric intersection number, then there is an explicit lower bound for the length of A in terms of the length of B, which holds for any hyperbolic structure on the surface. By slightly weakening this lower bound, we generalize this statement to hold for all Hitchin representations. This is a joint work with Tengren Zhang.