Fudan Topology Seminar

Spring 2025 Schedule

The regular meeting time for our seminar this semester will be Friday 4:30pm to 5:30pm. The seminar is joint with SIMIS.

Fall 2024 Schedule

The regular meeting time for our seminar this semester has changed to Friday 4pm to 5pm. Starting from this semester, the seminar is joint with SIMIS.

Time: 10:00 - 11:00

Zoom Meeting No.: 910 3139 9342 Passcode: 957825

Title: Scaled homology and topological entropy

Abstract:In this talk, I will introduce a scaled homology theory, lc-homology, for metric spaces such that every metric space can be visually regarded as “locally contractible” with this newly-built homology as well as its connection to classic singular homology theory. In addition, after briefly introducing topological entropy, I will discuss how to generalize one of the existing results of entropy conjecture, relaxing the smooth manifold restrictions on the compact metric spaces, by using lc-homology groups. This is joint work with Bingzhe Hou and Kiyoshi Igusa.

Time: 13:30 - 15:00

Place: SCMS 102

Title: Ring operads and symmetric bimonoidal categories

Abstract: We generalize the classical operad pair theory to a newmodel for E_\infty ring spaces, which we call ring operad theory, and stablishaconnection with the classical operad pair theory, allowingtheclassical multiplicative infinite loop machine to be appliedtoalgebras over any E_\infty ring operad. As an application, we showthat classifying spaces of symmetric bimonoidal categories are directlyhomeomorphic to certain E_\infty ring spaces in the ring operadsense. Consequently, this provides a simpler construction of the algebraicK-theory as an E_\infty ring spectra.

Time: 16:00 - 17:00

Place: SCMS 102

Title: Relative Bounded Cohomology on Groups with Contracting Elements

Abstract: Let $G$ be a countable group acting properly on a metric spacewith contracting elements and ${H_i:1\le i\le n}$ be a finite collection of Morse subgroups in $G$. We prove that each $H_i$ has infinite index in $G$ if and only if the relative second boundedcohomology $H^{2}b(G, {H_i}{i=1}^n; \mathbb{R})$is infinite-dimensional. In addition, we also prove that for any contracting element $g$, there exists $k>0$ such that $H^{2}_b(G, \llangle g^k\rrangle; \mathbb{R})$ is infinite-dimensional. Our results generalize a theorem of Pagliantini-Rolli for finite-rank free groups and produce some new results on the (relative) second bounded cohomology of groups. Under the same conditions, we also prove a Gap Theorem stating that any $C$-contracting element $g$ in $G$ either has a power which is conjugate to its inverse, or else the stable commutator length of $g$ is at least equal to some constant $\tau=\tau(C)>0$. This generalizes the Gap Theorem obtained by Calegari-Fujiwara for hyperbolic groups and mapping class groups. Joint work with Renxing Wan.

Time: 16:00 - 17:00

Place: SCMS 102

Title: This talk presents a study of the well-known marked length spectrum rigidity problem in the coarse-geometric setting.

For any two (possibly non-proper) group actions $G\curvearrowright X_1$ and $G\curvearrowright X_2$ with contracting property, we prove that if the two actions have the same marked length spectrum, then the orbit map $Go_1\to Go_2$ must be a rough isometry. In addition, we prove a finer marked length spectrum rigidity from confined subgroups and further, geometrically dense subgroups. Our proof is based on the Extension Lemma and uses purely elementary metric geometry. This is joint work with Xiaoyu Xu and Wenyuan Yang.

Title: Length bounds for hyperbolic Heegaard splittings

Abstract: In theory, by Mostow rigidity, the geometry of a hyperbolic 3-manifold is a function of its topology, but how to predict explicitly geometric features from a combinatorial presentation of the 3-manifold? By groundbreaking work of Minsky and Brock, Canary, and Minsky such a formula exists for the class of hyperbolic 3-manifolds fibering over the circle. Their model allows to understand explicitly quantities such as volume, length spectrum, and Laplace spectrum directly from topological data. It is desirable to develop a similar picture for Heegaard splittings as it would apply to all 3-manifolds. In this talk, I will present some joint work with Alessandro SIsto and Peter Feller where we develop an explicit purely topological formula for the length of a (short) geodesic in a hyperbolic Heegaard splitting.

Time: 16:00 - 17:00

Place: SIMIS 1610

Tilte: Minimal Surface Entropy of Hyperbolic Manifolds

Abstract: I will discuss the definition of minimal surface entropy and review the results for both closed hyperbolic 3-manifolds and the finite volume case. On one hand, for metrics with sectional curvature no greater than -1, the entropy achieves its minimum value if and only if it is hyperbolic. On the other hand, among all metrics with scalar curvature bounded from below by -6, the entropy reaches its maximum at the hyperbolic metric.

Spring 2024 Schedule

The regular meeting time for our seminar this semester has changed to Friday 4pm to 5pm.

Fall 2023 Schedule

The regular meeting time for our seminar this semester has changed to Friday 4pm to 5:30pm.

Poster

Spring 2023 Schedule

The regular meeting time for our seminar this semester has changed to Tuesday 3pm to 5pm.

Poster

Poster

Poster

Poster

Fall 2022 Schedule

Spring 2022 Schedule

Poster

Fall 2021 Schedule