Tuesday Seminar on Topology (April -- July, 2019)

[Japanese]   [Past Programs]
17:00 -- 18:30 Graduate School of Mathematical Sciences, The University of Tokyo
Tea: 16:30 -- 17:00 Common Room

Last updated September 20, 2019
Information :@
Toshitake Kohno
Nariya Kawazumi
Takahiro Kitayama
Takuya Sakasai

April 2 -- Room 056, 17:00 -- 18:30

Jongil Park (Seoul National University)

A topological interpretation of symplectic fillings of a normal surface singularity

Abstract: One of active research areas in symplectic 4-manifolds is to classify symplectic fillings of certain 3-manifolds equipped with a contact structure. Among them, people have long studied symplectic fillings of the link of a normal surface singularity. Note that the link of a normal surface singularity carries a canonical contact structure which is also known as the Milnor fillable contact structure.

In this talk, Ifd like to investigate a topological surgery description for minimal symplectic fillings of the link of quotient surface singularities and weighted homogeneous surface singularities with a canonical contact structure. Explicitly, Ifll show that every minimal symplectic filling of the link of quotient surface singularities and weighted homogeneous surface singularities satisfying certain conditions can be obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding surface singularity. This is joint work with Hakho Choi.

April 9 -- Room 056, 17:00 -- 18:30

Hiraku Nakajima (Kavli IPMU, The University of Tokyo)

Coulomb branches of 3d SUSY gauge theories

Abstract: I will give an introduction to a mathematical definition of Coulomb branches of 3-dimensional SUSY gauge theories, given by my joint work with Braverman and Finkelberg. I will emphasize on the role of hypothetical 3d TQFT associated with gauge theories.

April 16 -- Room 056, 17:00 -- 18:30

Kenfichi Ohshika (Gakushuin University)

Thurstonfs bounded image theorem

Abstract: The bounded image theorem by Thurston constitutes an important step in the proof of his unifomisation theorem for Haken manifolds. Thurstonfs original argument was never published and has been unknown up to now. It has turned out a weaker form of this theorem is enough for the proof, and books by Kappovich and by Otal use this weaker version. In this talk, I will show how to prove Thurstonfs original version making use of more recent technology. This is joint work with Cyril Lecuire.

April 23 -- Room 056, 17:00 -- 18:30

Christine Vespa (Université de Strasbourg)

Higher Hochschild homology as a functor

Abstract: Higher Hochschild homology generalizes classical Hochschild homology for rings. Recently, Turchin and Willwacher computed higher Hochschild homology of a finite wedge of circles with coefficients in the Loday functor associated to the ring of dual numbers over the rationals. In particular, they obtained linear representations of the groups Out(F_n) which do not factorize through GL(n,Z).

In this talk, I will begin by recalling what is Hochschild homology and higher Hochschild homology. Then I will explain how viewing higher Hochschild homology of a finite wedge of circles as a functor on the category of free groups provides a conceptual framework which allows powerful tools such as exponential functors and polynomial functors to be used. In particular, this allows the generalization of the results of Turchin and Willwacher; this gives rise to new linear representations of Out(F_n) which do not factorize through GL(n,Z).

(This is joint work with Geoffrey Powell.)

May 14 -- Room 056, 17:00 -- 18:30

J. Scott Carter (University of South Alabama, Osaka City University)

Diagrammatic Algebra

Abstract: Three main ideas will be explored. First, a higher dimensional category (a category that has arrows, double arrows, triple arrows, and quadruple arrows) that is based upon the axioms of a Frobenius algebra will be outlined. Then these structures will be promoted into one higher dimension so that braiding can be introduced. Second, relationships between braiding and multiplication will be studied from a homological perspective. Third, the next order relations will be used to formulate a system of abstract tensor equations that are analogous to the Yang-Baxter relation. In this way, a broad outline of the notion of diagrammatic algebra will be presented.

A handout is available.

May 21 -- Room 056, 17:00 -- 18:30

Maria de los Angeles Guevara (Osaka City University)

On the dealternating number and the alternation number

Abstract: Links can be divided into alternating and non-alternating depending on if they possess an alternating diagram or not. After the proof of the Tait flype conjecture on alternating links, it became an important question to ask how a non-alternating link is gclose toh alternating links. The dealternating and alternation numbers, which are invariants introduced by C. Adams et al. and A. Kawauchi, respectively, can deal with this question. By definitions, for any link, its alternation number is less than or equal to its dealternating number. It is known that in general the equality does not hold. However, in general, it is not easy to show a gap between these invariants. In this seminar, we will show some results regarding these invariants. In particular, for each pair of positive integers, we will construct infinitely many knots, which have dealternating and alternation numbers determined for these integers. Therefore, an arbitrary gap between the values of these invariants will be obtained.

May 28 -- Room 056, 17:00 -- 18:30

R. Inanc Baykur (University of Massachusetts)

Exotic four-manifolds via positive factorizations

Abstract: We will discuss several new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes, which yield novel constructions of various interesting four-manifolds, such as symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz pencils.

June 4 -- Room 056, 17:00 -- 18:30

Mizuki Fukuda (Tokyo Gakugei University)

Gluck twist on branched twist spins

Abstract: A branched twist spin is an embedded two sphere in the four sphere and it is defined as the set of singular points of a circle action on the four sphere. Gluck showed that the set of isotopy classes of diffeomorphisms on $S^1 _times S^2$ is isomorphic to $Z_2$, and an operation of removing a neighborhood of 2-knot from the four sphere and regluing it by the generator of $Z_2$ is called a Gluck twist. It is known by Pao that the Gluck twist along a branched twist spin does not change the four sphere. In this talk, we give an another proof of Paofs result by using a decomposition of $S^4$ associated with the circle action, and we show that the set of branched twist spins does not change by the Gluck twist.

June 18 -- Room 056, 17:00 -- 18:30

Masaki Taniguchi (The University of Tokyo)

Filtered instanton homology and the homology cobordism group

Abstract: We give a new family of real-valued invariants {r_s} of oriented homology 3-spheres. The invariants are defined by using some filtered version of instanton Floer homology. The invariants are closely related to the existence of solutions to ASD equations on Y~R for a given homology sphere Y. We show some properties of {r_s} containing a connected sum formula and a negative definite inequality. As applications of such properties of {r_s}, we obtain several new results on the homology cobordism group and the knot concordance group. As one of such results, we show that if the 1-surgery of a knot has the Froyshov invariant negative, then all positive 1/n-surgeries of the knot are linearly independent in the homology cobordism group. This theorem gives a generalization of the theorem shown by Furuta and Fintushel-Stern in f90. Moreover, we estimate the values of {r_s} for a hyperbolic manifold Y with an error of at most 10^{-50}. It seems the values are irrational. If the values are irrational, we can conclude that the homology cobordism group is not generated by Seifert homology spheres. This is joint work with Yuta Nozaki and Kouki Sato.

June 25 -- Room 056, 17:00 -- 18:30

Tian-Jun Li (University of Minnesota)

Geometry of symplectic log Calabi-Yau surfaces

Abstract: This is a survey on the geometry of symplectic log Calabi-Yau surfaces, which are the symplectic analogues of Looijenga pairs. We address the classification up to symplectic deformation, the relations between symplectic circular sequences and anti-canonical sequences, contact trichotomy, and symplectic fillings. This is a joint work with Cheuk Yu Mak.

July 2 -- Room 056, 17:00 -- 18:30

Shun Wakatsuki (The University of Tokyo)

Brane coproducts and their applications

Abstract: The loop coproduct is a coproduct on the homology of the free loop space of a Poincaré duality space (or more generally a Gorenstein space). In this talk, I will introduce two kinds of brane coproducts which are generalizations of the loop coproduct to the homology of a sphere space (i.e. the mapping space from a sphere). Their constructions are based on the finiteness of the dimensions of mapping spaces in some sense. As an application, I will show the vanishing of some cup products on sphere spaces by comparing these two brane coproducts. This gives a generalization of a result of Menichi for the case of free loop spaces.

July 9 -- Room 056, 17:00 -- 18:30

Florent Schaffhauser (Université de Strasbourg)

Mod 2 cohomology of moduli stacks of real vector bundles

Abstract: The rational cohomology ring of the moduli stack of holomorphic vector bundles of fixed rank and degree over a compact Riemann surface was studied by Atiyah and Bott using tools of differential geometry and algebraic topology: they found generators of that ring and computed its Poincaré series. In joint work with Chiu-Chu Melissa Liu, we study in a similar way the mod 2 cohomology ring of the moduli stack of real vector bundles of fixed topological type over a compact Riemann surface with real structure. The goal of the talk is to explain the principle of that computation, emphasizing the analogies and differences between the real and complex cases, and discuss applications of the method. In particular, we provide explicit generators of mod 2 cohomology rings of moduli stacks of vector bundles over a real algebraic curve.

July 16 -- Room 056, 17:00 -- 18:30

Kimihiko Motegi (Nihon University)

Seifert vs. slice genera of knots in twist families and a characterization of braid axes

Abstract: Twisting a knot K in S3 along a disjoint unknot c produces a twist family of knots {Kn} indexed by the integers. Comparing the behaviors of the Seifert genus g(Kn) and the slice genus g4(Kn) under twistings, we prove that if g(Kn) - g4(Kn) < C for some constant C for infinitely many integers n > 0 or g(Kn) / g4(Kn) limits to 1, then the winding number of K about c equals either zero or the wrapping number. As a key application, if {Kn} or the mirror twist family {Kn} contains infinitely many tight fibered knots, then the latter must occur. This leads to the characterization that c is a braid axis of K if and only if both {Kn} and {Kn} each contain infinitely many tight fibered knots. We also give a necessary and sufficient condition for {Kn} to contain infinitely many L-space knots, and apply the characterization to prove that satellite L-space knots have braided patterns, which answers a question of both Baker-Moore and Hom in the positive. This result also implies an absence of essential Conway spheres for satellite L-space knots, which gives a partial answer to a conjecture of Lidman-Moore. This is joint work with Kenneth Baker (University of Miami).