KIAS Topology Seminar
About
The KIAS Topology seminar is one of the regular group seminars at Korea Institute for Advanced Study (KIAS). The goal is to invite researchers working on topology to give seminar talks and share their expertise. The topics include low-dimensional topology, symplectic and contact topology, algebraic topology, algebraic K-theory, topological quantum field theory, category theory and any other related subjects. For more information about the seminar, or to add a name to the seminar mailing list, contact the webmaster.
The Organizers: Hakho Choi, Joontae Kim, Sangwook Lee, and Hsuan-Yi Liao, (Previously) Min Hoon Kim and Byungdo Park.
Schedule, Year 2019
26 December 2019 14:00--15:00 at 1424
Twisted differential K-theory, geometric models, and sheaves of spectra
Prof. Byungdo Park (Chungbuk National University, Korea)
A twisted vector bundle is a weaker notion of an ordinary vector bundle whose cocycle condition is off by a U(1)-valued Cech 2-cocycle (the cycle data of a U(1)-gerbe) called a topological twist. There is a nice twisted analogue of Chern-Weil theory of twisted vector bundles which have been found useful as cocycles of twisted differential K-theory. I shall explain how the theory of secondary characteristic classes can be used to provide a geometric model of twisted differential K-theory along the vein that I am thinking about this topic recently, especially in connection to a joint work with Daniel Grady.
24 December 2019 11:00--12:00 at 1423
Contact topology of singularities (ii)
23 December 2019 14:00--15:00 at 1423
Contact topology of singularities (i)
Dr. Myeonggi Kwon (Justus-Liebig-Universität Gießen in Gießen, Germany)
In this talk, we survey basic notions on links of isolated hypersurface singularities. We in particular introduce interesting examples in contact and symplectic topology coming from singularity theory, in terms of symplectic fillings and Floer homology.
19 December 2019 16:00--17:00 at 1424
Topology and Combinatorics of Richardson varieties
Dr. Seonjeong Park (KAIST, Korea)
The flag variety $\mathcal{F}\ell_n$ is a smooth projective variety consisting of chains $(\{0\}\subset V_1\subset\cdots\subset V_n=\mathbb{C}^n)$ of subspaces of $\mathbb{C}^n$ with $\dim_{\mathbb{C}} V_i=i$. Then the torus $\mathbb{T}=(\mathbb{C}^\ast)^n$ naturally acts on $\mathcal{F}\ell_n$. For $v$ and $w$ in the symmetric group $\mathfrak{S}_n$ with $v\leq w$ in Bruhat order, the Richardson variety $X^v_w$ is defined to be the intersection of the Schubert variety $X_w$ and the opposite Schubert variety $w_0X_{w_0v}$, and it is an irreducible $\mathbb{T}$-invariant subvariety of $\mathcal{F}\ell_n$. A $\mathbb{T}$-orbit closure $\mathcal{O}$ of $X^v_w$ is \emph{generic} if $(\overline{\mathcal{O}})^\mathbb{T}=(X^v_w)^\mathbb{T}$. We define $c(v,w)=\dim X^v_w -\dim \overline{\mathcal{O}}$ for a generic $\mathbb{T}$-orbit $\mathcal{O}$ in $X^v_w$. In this talk, we discuss topology and combinatorics of Richardson varieties with small $c(v,w)$. This talk is based on joint work with Eunjeong Lee and Mikiya Masuda.
18 December 2019 14:00--15:00 at 1423
Lefschetz-Bott fibrations and strong symplectic fillings
Dr. Takahiro Oba (Research Institutes for Mathematical Sciences, Kyoto University, Japan)
A Lefschetz--Bott fibration on a symplectic manifold is a smooth map to a surface with only Lefschetz-Bott critical points, which are modeled on Morse-Bott critical points. As Lefschetz fibrations have played an important role in the study of Stein fillings of contact manifolds, we expect Lefschetz-Bott fibrations to help us understand symplectic fillings of contact manifolds. In this talk, I will explain a relation between Lefschetz--Bott fibrations and strong symplectic fillings of contact manifolds. Moreover, by using this, I will provide some examples of strong symplectic fillings of links of isolated singularities.
17 December 2019 14:00--15:00 at 1423
Freely slice links
Dr. Min Hoon Kim (POSTECH, Korea)
The still open topological surgery conjecture for 4-manifolds is equivalent to the statement that all good boundary links are freely slice. In this talk, I will show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two or more components, and provides new freely slice links. This is joint work with Jae Choon Cha and Mark Powell.
11 December 2019 11:30--12:30 at 1424
M. Kontsevich’s graph complexes and universal structures on graded symplectic manifolds
Dr. Kevin Morand (Sogang University, Korea)
In the formulation of his celebrated Formality conjecture, M. Kontsevich introduced a universal version of the deformation theory for the Schouten algebra of polyvector fields on affine manifolds. This construction is reviewed and generalised to graded symplectic manifolds of arbitrary degree n ≥ 1. The corresponding graph model is given by the full Kontsevich graph complex fGCd where d=n+1 stands for the dimension of the associated AKSZ type σ-model. This generalisation is instrumental to classify universal structures on graded symplectic manifolds. We conclude by discussing the possible role played by this new deformation theory regarding the quantization problem for Courant algebroids and higher symplectic Lie-n algebroids.
20 November 2019 11:30--12:30 at 1424
Homotopy theory of linear cogebras
Dr. Damien Lejay (IBS Center for Geometry and Physics in Pohang, Korea)
The homotopy theory of linear algebras over operads is now well known. There is a model structure and the homotopy category can be fully described using the Bar adjunction. In this talk, I will talk about the ‘co’ side of this theory. Indeed operads can also be used to encode cogebras and we shall see how to endow the category of cogebras over a linear operad with a model structure and how to use the Cobar functor to describe its homotopy category. This is joint work with Brice Le Grignou.
31 October 2019 16:00--17:00 at 8309
Higher symplectic capacities by Kyler Siegel
Dr. Taekgyu Hwang (Ajou University in Suwon, Korea)
The goal of this talk is to review the recent construction of the symplectic capacities by Kyler Siegel (https://arxiv.org/abs/1902.01490). Symplectic capacities are ways of measuring the sizes of symplectic manifolds, which are often used as obstructions to symplectic embeddings. Kyler Siegel used the L-infinity structure on the linearized contact homology to define his symplectic capacities that are well-suited for the stabilized embedding problems.
31 October 2019 14:30--15:30 at 8309
Symplectic fillings of unit cotangent bundles
Dr. Myeonggi Kwon (Justus-Liebig-Universität Gießen in Gießen, Germany)
Given a contact manifold, it is a fundamental question in symplectic topology in how many ways the contact manifold can be written as the boundary of a symplectic filling. In this talk, we introduce uniqueness results to this question up to diffeomorphism type for unit cotangent bundles with the standard contact structure. This talk will be based on joint work with Hansjoerg Geiges and Kai Zehmisch.
28 October 2019 15:30--17:00 at 1423
Reconstructing $T_{\mathbb{P}^2}$ via tropical Lagrangian multi-section
Dr. Yat-Hin Suen (IBS Center for Geometry and Physics in Pohang, Korea)
In this talk, we study the reconstruction problem of the holomorphic tangent bundle $T_{\mathbb{P}^2}$ of the complex projective plane $\mathbb{P}^2$. We introduce the notion of tropical Lagrangian multi-section and cook up one from a family of Hermitian metrics defined on $T_{\mathbb{P}^2}$. Then we perform the reconstruction of $T_{\mathbb{P}^2}$ from this tropical Lagrangian multi-section. I will also discuss that relation with walling-crossing phenomenon in symplectic geometry.
09 August 2019 14:00--15:30 at 1424
Deformation quantization of Poisson manifold and path integral quantization of Poisson sigma model II
07 August 2019 14:00--15:30 at 1424
Deformation quantization of Poisson manifold and path integral quantization of Poisson sigma model I
Prof. Dr. Noriaki Ikeda (Ritsumeikan University in Kyoto, Japan)
I review Kontsevich's deformation quantization formula on a Poisson manifold and the formality theorem. We mainly discuss relations of the formality theorem with Cattaneo-Felder's derivation of the formula using the path integral quantization of the Poisson sigma model. This gives an interesting relation with deformation quantization and path integral quantization, and mathematical topic and physical techniques.
31 July 2019 11:00--12:00 at 1424
GT-shadows and their action on child's drawings
Prof. Dr. Vasily Dolgushev (Temple University in Philadelphia PA, USA)
The absolute Galois group of the field of rational numbers and the Grothendieck-Teichmueller group introduced by V. Drinfeld in 1990 are among the most mysterious objects in mathematics. My talk will be devoted to GT-shadows. These tantalizing objects may be thought of as ``approximations'' to elements of the mysterious Grothendieck-Teichmueller group. They form a groupoid and act on Grothendieck's child's drawings. Currently, the most amazing discovery related to GT-shadows is that the orbits of child's drawings with respect to the action of the absolute Galois group (when they can be computed) and the orbits of child's drawings with respect to the action of GT-shadows coincide! If time will permit, I show how to work with the software package for GT-shadows and their action on child's drawings. My talk is partially based on a joint paper (in preparation) with Khanh Q. Le and Aidan Lorenz.
09 July 2019 17:00--18:00 at 8101
Brown representability for directed graphs
Dr. Zachary McGuirk (Hebrew University of Jerusalem in Jerusalem, Israel)
Homotopy theory of directed graphs is an emerging topic in mathematics seeking invariants that are not too trivial, not too difficult to compute, sharing similar properties to those satisfied by invariants on manifolds, and corresponding to the invariants on manifolds if a graph approximates one. My talk is about digraph homotopy theory of Chen, Yau, and Yeh further developed later by Grigor'yan, Lin, Muranov, Yau. I shall report the research outcome of a joint work with Byungdo Park (KIAS) that any contravariant functor from the homotopy category of finite directed graphs to abelian groups satisfying the additivity axiom and the Mayer-Vietoris axiom is representable. I will give some motivating examples and explain necessary backgrounds so that non-experts in either algebraic topology or graph theory would be able to follow through.
05 July 2019 17:10--18:10 at 1423
Topological quantum field theories and higher algebraic structures (2)
05 July 2019 16:00--17:10 at 1423
Topological quantum field theories and higher algebraic structures (1)
Prof. Dr. Rune Haugseng (NTNU in Trondheim, Norway / IBS Center for Geometry and Physics in Pohang, Korea)
Topological quantum field theories (TQFTs) arose in physics as a particularly simple class of quantum field theories. I will introduce the modern mathematical formalization of TQFTs using homotopical higher categories, and discuss work in progress towards constructing the appropriate target needed to apply this formalism to the main class of TQFTs relevant to physics and representation theory.
We thank Dr. Sreedhar Bhamidi at School of Mathematics, KIAS for co-hosting Prof. Dr. Rune Haugseng.
05 July 2019 14:00--15:30 at 1424
Formality theorem and Kontsevich-Duflo theorem for Lie pairs
Prof. Dr. Ping Xu (Pennsylvania State University in State College PA, USA)
A Lie pair (L,A) consists of a Lie algebra (or more generally, a Lie algebroid) L together with a Lie subalgebra (or Lie subalgebroid) A. A wide range of geometric situations can be described in terms of Lie pairs including complex manifolds, foliations, and manifolds equipped with Lie group actions. To each Lie pair (L,A) are associated two L-infinity algebras, canonical up to isomorphisms, which play roles similar to the spaces of polyvector fields and polydifferential operators. We establish the formality theorem for Lie pairs. As an application, we obtain Kontsevich-Duflo type theorem for Lie pairs. Besides using Kontsevich formality theorem, our approach is based on the construction of a dg manifold (L[1] + L/A, Q) together with a dg foliation, called the Fedosov dg Lie algebroid. This is a joint work with Hsuan-Yi Liao and Mathieu Stienon.
02 July 2019 14:00--15:30 at 1423
Representable presheaves of groups on $\textbf{ccdgC}(\mathbb{k})$ and Tannakian reconstruction
Mr. Jaehyeok Lee (Pohang University of Science and Technology in Pohang, Korea)
Motivated by rational homotopy theory, we study a representable presheaf of groups on the category $\textbf{ccdgC}(\mathbb{k})$ of cocommutative dg-coalgebras over a field $\mathbb{k}$, and its Tannakian reconstruction. We begin with a review on Tannakian reconstruction of an affine group scheme. This is a joint work with Jae-Suk Park.
19 June 2019 14:00--15:30 at 1423
Asymptotic behavior of exotic Lagrangian tori
Mr. Weonmo Lee (Pohang University of Science and Technology in Pohang, Korea)
Vianna constructed an infinite family of monotone Lagrangian tori in complex projective plane. We prove that the Gromov capacity of the complement of each torus is greater than or equal to 1/3 of the area of the complex line, independent of the choice of torus. We also prove that these tori is not dense in complex projective plane. We prove that any of these tori can be embedded into k times blown-ups of complex projective plane, for k less than or equal to 5. In particular, this answers affirmatively to a question of Chekanov and Schlenk. This is a joint work with Yong-Geun Oh and Renato Vianna.
29 May 2019 14:00--15:00 at 1424
BPS invariants for Seifert manifolds from Chern-Simons matrix model
Dr. Hee-Joung Chung (Yau Mathematical Sciences Center, Tsinghua University in Beijing, P. R. China)
In this talk, I first provide an introduction to the homological block of the Witten-Reshtikhin-Turaev (WRT) invariants for closed 3-manifolds. The homological block is a q-series with integer powers and integer coefficients, which are necessary properties for the categorification. In physics, such q-series can be interpreted as certain BPS invariants. I will also discuss the WRT invariants from the Chern-Simons matrix model. Then, I will talk about the calculation of homological blocks for Seifert manifolds with gauge group $G=SU(N)$.
28 May 2019 16:00--17:00 at 1424
Atiyah class and Todd class of dg manifolds
Prof. Dr. Mathieu Stiénon (Pennsylvania State University in State College PA, USA)
Exponential maps arise naturally in the contexts of Lie theory and of smooth manifolds. The infinite jets of these exponential maps are related to the Poincare--Birkhoff--Witt isomorphism and the complete symbols of differential operators. We will discuss how these exponential maps can be extend to the context of dg manifolds. As an application, we will describe a natural L-infinity structure associated with the Atiyah class of a dg manifold.
22 May 2019 11:00--13:00 at 1423
Group study: Operad and convolution Lie algebra Notes
Dr. Hsuan-Yi Liao (KIAS)
We will discuss about the lecture notes by Dolgushev and Rogers: https://arxiv.org/abs/1202.2937.
24 May 2019 14:00--15:00 at 8309
Exotic Lagrangian tori and versal deformations
Mr. Joe Brendel (Université de Neuchâtel in Neuchâtel, Switzerland)
Monotone Lagrangians are an important class of submanifolds of a symplectic manifold. In this talk we will concentrate on monotone Lagrangian tori. In particular we will discuss the construction of exotic examples of these objects. Furthermore, we will insist on the method of versal deformations for creating suitable invariants for distinguishing them up to symplectomorphism or up to Hamiltonian isotopy. If time permits, we will touch on recent work which tries to apply these invariants to monotone Lagrangian tori in toric symplectic four-manifolds. This talk is based on work by Y. Chekanov and F. Schlenk.
17 May 2019 16:00--17:00 at 1423
On definite intersection forms of 4-manifolds with boundary
Dr. Kyungbae Park (Seoul National University in Seoul, Korea)
The intersection pairing of the second homology group of a compact oriented 4-manifold is a symmetric bilinear form on an integer lattice, called the intersection form. In this talk, we discuss which definite forms can be realized as the intersection forms of 4-manifolds bounded by a given 3-manifold. We also introduce our results to this question by using the conditions on such forms induced from the Donaldson’s diagonalization theorem and Heegaard Floer theory. This is joint work with Dong Heon Choe.
14 May 2019 16:00--17:00 at 8101
An explicit solution of a mean field equation on hyperelliptic curves
Prof. Dr. Jia-Ming Liou (National Cheng Kung University in Tainan, Taiwan)
We will give an explicit construction of a solution to the mean field equation on hyperelliptic curves with respect to the canonical metric. In the first part of the talk, I will briefly review the notion of the canonical metric and motivate the problems we studied. In the second part of the talk, I will give an explicit solution of a solution to the MFE using an algebraic construction under certain algebraic conditions. In the end of the talk, I will present the future work in this direction.
15 March 2019 13:00--14:00 at 1424
Isotopies of surfaces in 4-manifolds via banded unlink diagrams
Dr. Seungwon Kim (National Institute for Mathematical Sciences in Daejeon, Korea)
In this talk, we consider surfaces embedded in 4-manifolds. We give a complete set of moves relating banded unlink diagrams of isotopic surfaces in an arbitrary 4-manifold. This extends work of Swenton and Kearton-Kurlin in $S^4$. As an application, we show that bridge trisections of isotopic surfaces in a trisected 4-manifold are related by a sequence of perturbations and deperturbations, affirmatively proving a conjecture of Meier and Zupan. We also exhibit several isotopies of unit surfaces in $\mathbb{CP}^2$ (i.e. spheres in the generating homology class), proving that many explicit unit surfaces are isotopic to the standard $\mathbb{CP}^1$. This strengthens some previously known results about the Gluck twist in $S^4$, related to Kirby problem 4.23.
04 March 2019 10:00--12:00 at 1423
Symplectic coordinates on 3-Hitchin components
Mr. Hongtaek Jung (KAIST in Daejeon, Korea)
Goldman parametrizes the 3-Hitchin component of a closed oriented hyperbolic surface of genus g by 16g-16 parameters. Among them, 10g-10 coordinates are canonical. In this paper, we prove that the 3-Hitchin component equipped with the Goldman symplectic form admits a global Darboux coordinate system such that the half of its coordinates are canonical Goldman coordinates. To this end, we prove a version of action-angle principle and a Zocca type decomposition formula for the symplectic form of H. Kim and Guruprasad-Huebschmann-Jeffrey-Weinstein given to each symplectic leaf of the Hitchin component of a compact surface.
13 February 2019 13:30--15:30 at 8101
On the smoothness of generic torus orbit closures in Schubert varieties
Dr. Eunjeong Lee (IBS Center for Geometry and Physics in Pohang, Korea)
The standard action of a complex torus $(\mathbb{C}^*)^n$ on the complex vector space $\mathbb{C}^n$ induces an action of $(\mathbb{C}^*)^n$ on the full flag variety $\mathcal{F}l(\mathbb{C}^n)$. It has been studied that the closure of a generic torus orbit in the full flag variety is a smooth projective toric variety whose fan is associated to the Weyl chamber. In this talk, we define a generic torus orbit in the Schubert variety $X_w$ and its closure $Y_w$ for a permutation $w\in \mathfrak{S}_n$. Moreover we associate a graph $\Gamma_w(u)$ to each $u\leq w$, and prove that $Y_w$ is smooth at the fixed point $uB$ if and only if the graph $\Gamma_w(u)$ is acyclic. This is joint work with Mikiya Masuda.
07 January 2019 16:00--17:30 at 1424
Cheeger-Gromov L^2 rho-invariants of 3-manifolds
Mr. Geunho Lim (Indiana University in Bloomington IN, USA)
I'll talk about Cheeger-Gromov L^2 rho-invariants of 3-manifolds. Cheeger and Gromov analytically defined L^2 rho-invariants to Riemannian manifolds and showed L^2 rho-invariants has a universal bound by using deep analytic arguments. Chang and Weinberger extended the definition to topological manifolds. Cha proved the existence of universal bound for L^2 rho-invariants of topological manifolds and found an explicit bound in terms of complexity of given 3-manifold. To be specific, L^2 rho-invariants of 3-manifolds can be linearly bounded by the number of 2-handles of a 4-manifold which has a boundary of given 3-manifold. We will discuss the topological definition and proof about Cheeger-Gromov L^2 rho-invariants of 3-manifolds.
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Address: 85 Hoegiro Dongdaemungu, Seoul 02455, Republic of Korea
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