2019年4月 -- 7月
[English]
[過去のプログラム]
17:00 -- 18:30 数理科学研究科棟(東京大学駒場キャンパス)
Tea: 16:30 -- 17:00 コモンルーム
Last updated September 20, 2019
世話係
河野 俊丈
河澄 響矢
北山 貴裕
逆井 卓也
4月2日 -- 056号室, 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’d 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’ll 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.
4月9日 -- 056号室, 17:00 -- 18:30
中島 啓 (Kavli IPMU, 東京大学大学院数理科学研究科)
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.
4月16日 -- 056号室, 17:00 -- 18:30
大鹿 健一 (学習院大学)
Thurston’s 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’s 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’s original version making use of more recent technology.
This is joint work with Cyril Lecuire.
4月23日 -- 056号室, 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.)
5月14日 -- 056号室, 17:00 -- 18:30
J. Scott Carter (University of South Alabama, 大阪市立大学)
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.
5月21日 -- 056号室, 17:00 -- 18:30
Maria de los Angeles Guevara (大阪市立大学)
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 “close to” 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.
5月28日 -- 056号室, 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.
6月4日 -- 056号室, 17:00 -- 18:30
福田 瑞季 (東京学芸大学)
Gluck twist on branched twist spins
Abstract: Branched twist spin とは４次元球面上の円作用の特異点集合として定義される埋め込まれた２次元球面であり，
スパン結び目やツイストスパン結び目などの２次元結び目の一般化となっている．
Gluck は４次元多様体内の２次元結び目に沿った向きを保つ手術は微分同相類を除いて２種類のみであることを示しており，
自明でない手術を Gluck twist と呼ぶ．一般に Gluck twist が全空間の微分同相を保つかどうかは知られていないが，
Pao によって branched twist spin に沿った Gluck twist は 再び４次元球面と微分同相になることが知られている．
本講演では，Pao の結果の別証明として円作用を用いて４次元球面の分解を与え，
各ピースが Gluck twist を通してどのように変化するかを説明する．また，２次元結び目に注目したとき，
Gluck twist によって branched twist spin は再び branched twist spin になることを証明する．
6月18日 -- 056号室, 17:00 -- 18:30
谷口 正樹 (東京大学大学院数理科学研究科)
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 ’90.
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.
6月25日 -- 056号室, 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.
7月2日 -- 056号室, 17:00 -- 18:30
若月 駿 (東京大学大学院数理科学研究科)
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.
7月9日 -- 056号室, 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.
7月16日 -- 056号室, 17:00 -- 18:30
茂手木 公彦 (日本大学)
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).