トポロジー火曜セミナー

過去の記録 ~08/07次回の予定今後の予定 08/08~

開催情報 火曜日 17:00~18:30 数理科学研究科棟(駒場) 056号室
担当者 河野 俊丈, 河澄 響矢, 北山 貴裕, 逆井卓也
セミナーURL http://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index.html
備考 Tea: 16:30 - 17:00 コモンルーム

過去の記録

2020年07月28日(火)

17:00-18:00   オンライン開催
参加を希望される場合は、下記URLから参加登録を行って下さい。
Anderson Vera 氏 (京都大学数理解析研究所)
A double filtration for the mapping class group and the Goeritz group of the sphere (ENGLISH)
[ 講演概要 ]
I will talk about a double-indexed filtration of the mapping class group and of the Goeritz group of the sphere, the latter is the group of isotopy classes of self-homeomorphisms of the 3-sphere which preserves the standard Heegaard splitting of $S^3$. In particular I will explain how this double filtration allows to write the Torelli group as a product of some subgroups of the mapping class group. A similar study could be done for the group of automorphisms of a free group. (work in progress with K. Habiro)
[ 参考URL ]
https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

2020年07月21日(火)

17:00-18:00   オンライン開催
参加を希望される場合は、下記URLから参加登録を行って下さい。
Sergei Burkin 氏 (東京大学大学院数理科学研究科)
Twisted arrow categories of operads and Segal conditions (ENGLISH)
[ 講演概要 ]
We generalize twisted arrow category construction from categories to operads, and show that several important categories, including the simplex category $\Delta$, Segal's category $\Gamma$ and Moerdijk--Weiss category $\Omega$ are twisted arrow categories of operads. Twisted arrow categories of operads are closely connected with Segal conditions, and the corresponding theory can be generalized from multi-object associative algebras (i.e. categories) to multi-object P-algebras for reasonably nice operads P.
[ 参考URL ]
https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

2020年07月21日(火)

18:00-19:00   オンライン開催
参加を希望される場合は、下記URLから参加登録を行って下さい。
Dexie Lin 氏 (東京大学大学院数理科学研究科)
Monopole Floer homology for codimension-3 Riemannian foliation (ENGLISH)
[ 講演概要 ]
In this paper, we give a systematic study of Seiberg-Witten theory on closed oriented manifold with codimension-3 oriented Riemannian foliation. Under a certain topological condition, we construct the basic monopole Floer homologies for a transverse spinc structure with a bundle-like metric, generic perturbation and a complete local system. We will show that these homologies are independent of the bundle-like metric and generic perturbation. The major difference between the basic monopole Floer homologies and the ones on manifolds is the necessity to use the complete local system to construct the monopole Floer homologies.
[ 参考URL ]
https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

2020年07月14日(火)

17:30-18:30   オンライン開催
Lie群論・表現論セミナーと合同。 参加を希望される場合は、セミナーのウェブページをご覧下さい。
奥田 隆幸 氏 (広島大学)
Kobayashi's properness criterion and totally geodesic submanifolds in locally symmetric spaces (JAPANESE)
[ 講演概要 ]
G をリー群とし,X を G-等質空間とする. X のいくつかの開集合を G 移動で貼り合わせて得られる多様体を(G,X)-多様体とよぶ. X の G 不変局所幾何構造(計量など)は(G,X)-多様体に移植可能であり, (G,X)-多様体はよい幾何構造を持った多様体の例を供給することが期待される. この意味で, (G,X)-多様体の構成は微分幾何学における重要な研究テーマの一つである.

G の離散部分群が X に固有不連続に作用するとき, その離散群を X の不連続群とよび, その作用による X の商多様体を Clifford--Klein 形と呼ぶ. Clifford--Klein 形は (G,X)-多様体である. これより G-等質空間 X 上の不連続群の構成や分類は重要な問題となる. G-等質空間 X のイソトロピーがコンパクトである場合には, Gの捻じれのない離散部分群はすべて不連続群である. しかし X のイソトロピーが非コンパクトであるような場合においては, G の捻じれのない離散群であっても, X の不連続群になるとは限らない.

以下, G が線型簡約リー群であり, G-等質空間 X として簡約型かつイソトロピーが非コンパクトであるような場合を考える (この設定では X は G 不変リーマンは許容しないが, G不変擬リーマン計量を許容する). 小林俊行氏は [Math.Ann.(1989)], [J. Lie Theory (1996)] において, 与えられた G の離散部分群が X の不連続群になるための判定条件を与えている. この判定法は与えられた離散部分群と X におけるイソトロピー部分群の ``固有値の分布'' の関係性に着目する画期的なものである.

本講演では正定値非コンパクトリーマン対称空間の全測地的部分多様体の族として実現されるような G-等質空間 X について, リーマン幾何学の言葉を用いて上記の小林氏の判定定理を翻訳したものを紹介する. この枠組みにおいては, 与えられた離散部分群の``固有値の分布''の代わりに, その群の定める局所対称空間の``測地ループの分布''に着目する.
[ 参考URL ]
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

2020年07月07日(火)

17:00-18:00   オンライン開催
参加を希望される場合は、下記URLから参加登録を行って下さい。
野崎 雄太 氏 (広島大学)
Abelian quotients of the Y-filtration on the homology cylinders via the LMO functor (JAPANESE)
[ 講演概要 ]
We construct a series of homomorphisms on the Y-filtration on the homology cylinders via the mod $\mathbb{Z}$ reduction of the LMO functor. The restriction of our homomorphism to the lower central series of the Torelli group does not factor through Morita's refinement of the Johnson homomorphism. We use it to show that the abelianization of the Johnson kernel of a closed surface has torsion elements. This is the joint work with Masatoshi Sato and Masaaki Suzuki.
[ 参考URL ]
https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

2020年06月30日(火)

17:00-18:00   オンライン開催
参加を希望される場合は、下記URLから参加登録を行って下さい。
Daniel Matei 氏 (IMAR Bucharest)
Homology of right-angled Artin kernels (ENGLISH)
[ 講演概要 ]
The right-angled Artin groups A(G) are the finitely presented groups associated to a finite simplicial graph G=(V,E), which are generated by the vertices V satisfying commutator relations vw=wv for every edge vw in E. An Artin kernel Nh(G) is defined by an epimorphism h of A(G) onto the integers. In this talk, we discuss the module structure over the Laurent polynomial ring of the homology groups of Nh(G).
[ 参考URL ]
https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

2020年06月23日(火)

17:00-18:00   オンライン開催
参加を希望される場合は、下記URLから参加登録を行って下さい。
今野 北斗 氏 (東京大学大学院数理科学研究科)
Gauge theory and the diffeomorphism and homeomorphism groups of 4-manifolds (JAPANESE)
[ 講演概要 ]
I will explain my recent collaboration with several groups that develops gauge theory for families
to extract difference between the diffeomorphism groups and the homeomorphism groups of 4-manifolds.
After Donaldson’s celebrated diagonalization theorem, gauge theory has given strong constraints on the topology of smooth 4-manifolds. Combining such constraints with Freedman’s theory, one may find many non-smoothable topological 4-manifolds.
Recently, a family version of this argument was started by T. Kato, N. Nakamura and myself, and soon later it was developed also by D. Baraglia and his collaborating work with myself. More precisely, considering gauge theory for smooth fiber bundles of 4-manifolds, they obtained some constraints on the topology of smooth 4-manifold bundles. Using such constraints, they detected non-smoothable topological fiber bundles of smooth 4-manifolds. The existence of such bundles implies that there is homotopical difference between the diffeomorphism and homeomorphism groups of the 4-manifolds given as the fibers.
If time permits, I will also mention my collaboration with Baraglia which shows that a K3 surface gives a counterexample to the Nielsen realization problem in dimension 4. This example reveals also that there is difference between the Nielsen realization problems asked in the smooth category and the topological category.
[ 参考URL ]
https://zoom.us/webinar/register/WN_oS594Z6BRyaKNCvlm3yCoQ

2020年01月28日(火)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
関野 希望 氏 (東京大学大学院数理科学研究科)
Existence problems for fibered links (JAPANESE)
[ 講演概要 ]
It is known that every connected orientable closed 3-manifold has a fibered knot. However, finding (and classifying) fibered links whose fiber surfaces are fixed homeomorphism type in a given 3-manifold is difficult in general. We give a criterion of a simple closed curve on a genus 2g Heegaard surface being a genus g fibered knot in terms of its Heegaard diagram. As an application, we can prove the non-existence of genus one fibered knots in some Seifert manifolds.
There is one generalization of fibered links, homologically fibered links. This requests that the complement of the "fiber surface" is a homologically product of a surface and an interval. We give a necessary and sufficient condition for a connected sums of lens spaces of having a homologically fibered link whose fiber surfaces are some fixed types as some algebraic equations.

2020年01月28日(火)

18:00-19:00   数理科学研究科棟(駒場) 056号室
渡部 淳 氏 (東京大学大学院数理科学研究科)
Fibred cusp b-pseudodifferential operators and its applications (JAPANESE)
[ 講演概要 ]
Melrose's b-calculus and its variants are important tools to study index problems on manifolds with singularities. In this talk, we introduce a new variant "fibred cusp b-calculus", which is a generalization of fibred cusp calculus of Mazzeo-Melrose and b-calculus of Melrose. We discuss the basic property of this calculus and give a relative index formula. As its application, we prove the index theorem for a Z/k manifold with boundary, which is a generalization of the mod k index theorem of Freed-Melrose.

2020年01月14日(火)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
茅原 涼平 氏 (東京大学大学院数理科学研究科)
SO(3)-invariant G2-geometry (JAPANESE)
[ 講演概要 ]
Berger's classification of holonomy groups of Riemannian manifolds includes exceptional cases of the Lie groups G2 and Spin(7). Many authors have studied G2- and Spin(7)-manifolds with torus symmetry. In this talk, we generalize the celebrated examples due to Bryant and Salamon and study G2-manifolds with SO(3)-symmetry. Such torsion-free G2-structures are described as a dynamical system of SU(3)-structures on an SO(3)-fibration over a 3-manifold. As a main result, we reduce this system into a constrained Hamiltonian dynamical system on the cotangent bundle over the space of all Riemannian metrics on the 3-manifold. The Hamiltonian function is very similar to that of the Hamiltonian formulation of general relativity.

2020年01月14日(火)

18:00-19:00   数理科学研究科棟(駒場) 056号室
石橋 典 氏 (東京大学大学院数理科学研究科)
Algebraic entropy of sign-stable mutation loops (JAPANESE)
[ 講演概要 ]
Since its discovery, the cluster algebra has been developed with friutful connections with other branches of mathematics, unifying several combinatorial operations as well as their positivity notions. A mutation loop induces several dynamical systems via cluster transformations, and they form a group which can be seen as a combinatorial generalization of the mapping class groups of marked surfaces.
We introduce a new property of mutation loops called the sign stability, with a focus on an asymptotic behavior of the iteration of the tropicalized cluster X-transformation. A sign-stable mutation loop has a numerical invariant which we call the "cluster stretch factor", in analogy with the stretch factor of a pseudo-Anosov mapping class on a marked surface. We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor. This talk is based on a joint work with Shunsuke Kano.

2020年01月07日(火)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
浅尾 泰彦 氏 (東京大学大学院数理科学研究科)
Magnitude homology of crushable spaces (JAPANESE)
[ 講演概要 ]
The magnitude homology and the blurred magnitude homology are novel notions of homology theory for general metric spaces coined by Leinster et al. They are expected to be dealt with in the context of Topological Data Analysis since its original idea is based on a kind of "persistence of points clouds". However, little property of them has been revealed. In this talk, we see that the blurred magnitude homology is trivial when a metric space is contractible by a distance decreasing homotopy. We use techniques from singular homology theory.

2020年01月07日(火)

18:00-19:00   数理科学研究科棟(駒場) 056号室
浅野 知紘 氏 (東京大学大学院数理科学研究科)
Intersection number estimate of rational Lagrangian immersions in cotangent bundles via microlocal sheaf theory (JAPANESE)
[ 講演概要 ]
Guillermou associated sheaves to exact Lagrangian submanifolds in cotangent bundles and proved topological properties of the Lagrangian submanifolds. In this talk, I will give an estimate on the displacement energy of rational Lagrangian immersions in cotangent bundles with intersection number estimates via microlocal sheaf theory. This result overlaps with results by Chekanov, Liu, and Akaho via Floer theory. This is joint work with Yuichi Ike.

2019年12月17日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
入江 慶 氏 (東京大学大学院数理科学研究科)
Symplectic homology of fiberwise convex sets and homology of loop spaces (JAPANESE)
[ 講演概要 ]
シンプレクティック・ベクトル空間の(コンパクト)部分集合に対して、シンプレクティック・ホモロジー(Floer ホモロジーの一種)を用いてそのシンプレクティック容量(capacity)を定義することができる。一般に、Floerホモロジーの定義には非線形偏微分方程式(いわゆるFloer方程式)の解の数え上げが関わるため、容量を定義から直接計算したり評価したりするのは難しい。この講演では(シンプレクティック・ベクトル空間をEuclid空間の余接空間とみなしたとき)fiberwiseに凸な集合のシンプレクティック・ホモロジーおよび容量をループ空間のホモロジーから計算する公式を示し、その応用を二つ与える。

2019年12月10日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
小木曽 岳義 氏 (城西大学)
q-Deformation of a continued fraction and its applications (JAPANESE)
[ 講演概要 ]
Morier-Genoud と Ovsienko によって連分数のある種の q-変形が導入された。このq-変形の最大の応用はそれを用いて向きづけられた有理絡み目の Jones 多項式がそれから直接求めることができることである。またこの連分数のq-変形は結び目理論への応用以外にも、2次無理数論、組み合わせ論への応用もあり、それについても紹介する。

一方、Lee-Schiffler の snake graph を用いる方法や Kogiso-Wakui による Conway-Coxeter frieze を持ちいる方法で Jones 多項式を計算するレシピが与えられている。そのことから、Morier-Genoud and Ovsienko の結果のそれらの観点からの別証明が考えられるが、それについて紹介し、さらに, Kogiso-Wakui の研究で用いた Ancestoral triangles の観点から連分数のq-変形をさらに一般化でき、連分数の cluster-variable 変形が出来ることを紹介する。

2019年12月03日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Anton Zeitlin 氏 (Louisiana State University)
Homotopy Gerstenhaber algebras, Courant algebroids, and Field Equations (ENGLISH)
[ 講演概要 ]
I will talk about the underlying homotopical structures within field equations, which emerge in string theory as conformal invariance conditions for sigma models. I will show how these, often hidden, structures emerge from the homotopy Gerstenhaber algebra associated to vertex and Courant algebroids, thus making all such equations the natural objects within vertex algebra theory.

2019年11月26日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Marco De Renzi 氏 (早稲田大学)
$2+1$-TQFTs from non-semisimple modular categories (ENGLISH)
[ 講演概要 ]
Non-semisimple constructions have substantially generalized the standard approach of Witten, Reshetikhin, and Turaev to quantum topology, producing powerful invariants and TQFTs with unprecedented properties. We will explain how to use the theory of modified traces to renormalize Lyubashenko’s closed 3-manifold invariants coming from finite twist non-degenerate unimodular ribbon categories. Under the additional assumption of factorizability, our renormalized invariants extend to $2+1$-TQFTs, unlike Lyubashenko’s original ones. This general framework encompasses important examples of non-semisimple modular categories which were left out of previous non-semisimple TQFT constructions.

Based on a joint work with Azat Gainutdinov, Nathan Geer, Bertrand Patureau, and Ingo Runkel.

2019年11月19日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Ramón Barral Lijó 氏 (立命館大学)
The smooth Gromov space and the realization problem (ENGLISH)
[ 講演概要 ]
The n-dimensional smooth Gromov space consists of the pointed isometry classes of complete Riemannian n-manifolds. In this talk we will present the definition and basic properties of this space as well as two different applications: The first addresses the following classical problem in foliation theory.

Realization problem: what kind of manifolds can be leaves of compact foliations?

Our joint work with Álvarez López has produced the following solution in the context of foliated spaces.

Theorem. Every Riemannian manifold of bounded geometry is a leaf in a compact foliated space X endowed with a metric tensor. Moreover, we can assume that X has trivial holonomy and is transversely Cantor.

The second application is the recent research by Abert and Biringer on the subject of unimodular random Riemannian manifolds.

2019年11月05日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
五味 清紀 氏 (東京工業大学)
Magnitude homology of geodesic space (JAPANESE)
[ 講演概要 ]
Magnitude is an invariant which counts `effective number of points' on a metric space. Its categorification is magnitude homology. This notion is first formulated for metric spaces associated to simple graphs by Hepworth and Willerton, and then for any metric spaces by Leinster and Shulman. The definition of the magnitude homology is easy, but its calculation is rather difficult. For example, the magnitude homology of the circle with geodesic metric was known partially. In my talk, I will explain my result that fully determines the magnitude homology of any geodesic metric space subject to a certain non-branching assumption. In this result, the magnitude homology is described in terms of geodesics. Complete and connected Riemannian manifolds are examples of the geodesic metric spaces satisfying the assumption.

2019年10月29日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Chung-Jun Tsai 氏 (National Taiwan University)
Strong stability of minimal submanifolds (ENGLISH)
[ 講演概要 ]
It is well known that the distance function to a totally geodesic submanifold of a negatively curved ambient manifold is a convex function. One can identify a strong stability condition on minimal submanifolds that generalizes the above scenario. Besides a strong local uniqueness property, a strongly stable minimal submanifold is also Lipschitz stable under the mean curvature flow. We will also discuss some famous local (complete, non-compact) models. This is based on a joint work with Mu-Tao Wang.

2019年10月15日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Gwénaël Massuyeau 氏 (Université de Bourgogne)
Generalized Dehn twists on surfaces and surgeries in 3-manifolds (ENGLISH)
[ 講演概要 ]
(Joint work with Yusuke Kuno.) Given an oriented surface S and a simple closed curve C in S, the "Dehn twist" along C is the homeomorphism of S defined by "twisting" S around C by a full twist. If the curve C is not simple, this transformation of S does not make sense anymore, but one can consider two possible generalizations: one possibility is to use the homotopy intersection form of S to "simulate" the action of a Dehn twist on the (Malcev completion of) the fundamental group of S; another possibility is to view C as a curve on the top boundary of the cylinder S×[0,1], to push it arbitrarily into the interior so as to obtain, by surgery along the resulting knot, a new 3-manifold. In this talk, we will relate two those possible generalizations of a Dehn twist and we will give explicit formulas using a "symplectic expansion" of the fundamental group of S.

2019年10月08日(火)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 17:00-17:30
塚本 真輝 氏 (九州大学)
いかにして双曲的力学系を群作用に拡張するか? (JAPANESE)
[ 講演概要 ]
双曲性は通常の力学系(1パラメータ群作用の研究)において最も基本的な重要性を持つ概念です.それは,十分な豊かさ(拡大性や正エントロピー)を持ちながらも,同時に制御可能(安定性や適切な意味での良い測度の一意性)な力学系の例を与えます.ではこれを群作用に拡張できるでしょうか?

ナイーブには困難です.例えば $Z^2$ の作用を考えましょう(つまり可換な 2 パラメータ作用)・簡単にわかるのは,有限次元のコンパクト多様体に $Z^2$ が可微分に作用するとき,その $Z^2$ 作用としてのエントロピーはゼロになります.つまり,通常の有限次元の状況には,豊かな $Z^2$ 作用は存在しません.言い換えると,十分に豊かな群作用を得るためには無限次元の世界に行かざるを得ません.しかし,無限次元の世界でどのような構造を見出せばよいのでしょうか?

この講演では,このような方向性にアプローチする際に,平均次元と呼ばれる量が大きな役割を果たす可能性を説明します.特に,次のような原理についてお話します:

$Z^k$(可換な $k$ パラメータ群)が空間 $X$ に何らかの「双曲性」を持って作用するとき,$Z^k$ のランク $k-1$ の部分群 $G$ の部分作用に対する平均次元が制御できる.

この講演はTom Meyerovitch,篠田万穂との共同研究に基づきます.

2019年10月01日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
村上 順 氏 (早稲田大学)
Quantized SL(2) representations of knot groups (JAPANESE)
[ 講演概要 ]
Let K be a knot and G be a group. The representation space of K for the group G means the space of homomorphisms from the knot group to G and is defined by using the group ring C[G], where C[G] is the ring of functions on G and has a commutative Hopf algebra structure. This construction can be generalized to any commutative Hopf algebras.

In this talk, we extend this construction to any braided Hopf algebras with braided commutativity. A typical example is BSL(2), which is the braided SL(2) introduced by S. Majid. Applying the above construction to BSL(2), we get the space of BSL(2) representations, which provides a quantization of SL(2) representations of a knot. This is joint work with Roloand van der Veen.

2019年07月16日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
茂手木 公彦 氏 (日本大学)
Seifert vs. slice genera of knots in twist families and a characterization of braid axes (JAPANESE)
[ 講演概要 ]
Twisting a knot $K$ in $S^3$ along a disjoint unknot $c$ produces a twist family of knots $\{K_n\}$ indexed by the integers. Comparing the behaviors of the Seifert genus $g(K_n)$ and the slice genus $g_4(K_n)$ under twistings, we prove that if $g(K_n) - g_4(K_n) < C$ for some constant $C$ for infinitely many integers $n > 0$ or $g(K_n) / g_4(K_n)$ limits to $1$, then the winding number of $K$ about $c$ equals either zero or the wrapping number. As a key application, if $\{K_n\}$ or the mirror twist family $\{\overline{K_n}\}$ 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 $\{K_n\}$ and $\{\overline{K_n}\}$ each contain infinitely many tight fibered knots. We also give a necessary and sufficient condition for $\{K_n\}$ 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).

2019年07月09日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Florent Schaffhauser 氏 (Université de Strasbourg)
Mod 2 cohomology of moduli stacks of real vector bundles (ENGLISH)
[ 講演概要 ]
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.

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