UCR Geometry & Topology Seminar

During the Fall 2024 quarter, all of our meetings will be held in-person in Skye Hall 268.

Talks will run from 11a-12p PT, but feel free to show up early for socializing with the speaker.

Fall 2024 schedule

September 27 2024:

Nicholas Rungi (Université Grenoble-Alpes)
- Bi-complex hyperbolic space and SL(3,C)-quasi-Fuchsian representations
  - The space of quasi-Fuchsian surface group representations into PSL(2,C) has been largely studied in recent years, also due to the direct relation with hyperbolic geometry. In this framework, Bers' simultaneous uniformization theorem shows that it is parameterized by two copies of Teichmüller space of the closed surface. In this seminar, we are interested in surface group representations into SL(3,C) acting naturally on a homogeneous space called the bi-complex hyperbolic space. After briefly describing its main features, we will define minimal complex Lagrangian surface immersions and their structural equations. If they are equivariant under the action of a representation into SL(3,C), we will see that their embedding data provide a parameterization of the space of SL(3,C)-quasi-Fuchsian representations by an open subset in two copies of the bundle of holomorphic cubic differentials over Teichmüller space. This is a joint work with Andrea Tamburelli.

October 4 2024:

Curtis Pro (CSU Stanislaus)
- A brief survey of Riemannian Geometry with a lower curvature bound
  - In the mid 19th century, one of Riemann's first observations after his introduction of his Curvature Tensor wasthat from a conformal change of the standard Euclidean metric, one could construct all three constant curvature geometries. By the mid$20^{\mathrm{th}}$ century, after abstract manifolds were better understood, Riemannian Geometry began uncovering how constraints oncurvature relative to the three constant curvature geometries constrained the topology of the underlying manifold. In this talk, we'll survey the main results and tools developed in Riemannian Geometry when curvature (primarily Sectional Curvature) is bounded from below.

October 11 2024:

Edgar Bering (San José State University)
- Two-generator subgroups of free-by-cyclic groups
  - In general, the classification of finitely generated subgroups of a given group is intractable. Restricting to two-generator subgroups in a geometric setting is an exception. For example, a two-generator subgroup of a right-angled Artin group is either free or free abelian. Jaco and Shalen proved that a two-generator subgroup of the fundamental group of an orientable atoroidal irreducible 3-manifold is either free, free-abelian, or finite-index. In this talk I will present recent work proving a similar classification theorem for two generator mapping-torus groups of free group endomorphisms: every two generator subgroup is either free or conjugate to a sub-mapping-torus group. As an application we obtain an analog of the Jaco-Shalen result for free-by-cyclic groups with fully irreducible atoroidal monodromy. This is joint work with Naomi Andrew, Ilya Kapovich, and Stefano Vidussi.

October 18 2024:

Becky Eastham (UC Riverside)
- Separable homology of graphs and the Whitehead complex
  - We introduce a 1-complex associated with a finite regular cover of the rose and show that it is connected if and only if the fundamental group of the associated cover is generated by lifts of elements in a proper free factor of the free group. When the associated cover represents a characteristic subgroup of the free group, the complex admits an action of Out(F_n) by isometries. We then explore the coarse geometry of the 1-complex, showing that every component has infinite diameter, and that the 1-complex associated with the rose is nonhyperbolic. As corollaries, we obtain that the Cayley graph of the free group with the infinite generating set consisting of all primitive elements is nonhyperbolic.

October 25 2024:

Xiaolong Li (Wichita State University)
- New sphere theorems under curvature operator of the second kind
  - The curvature operator of the second kind has received recent attention following the resolution of Nishikawa's conjecture by Cao-Gursky-Tran, Nienhaus-Petersen-Wink, and myself. These works have shown that a closed Riemannian manifold with three-nonnegative curvature operator of the second kind is either flat or diffeomorphic to a spherical space form. In this talk, I will introduce the curvature operator of the second kind and talk about the proof of Nishikawa's conjecture. Then I will talk about my recent work on new sphere theorems under negative lower bounds of the curvature operator of the second kind.

November 1 2024:

Zihao Liu (Rice University)
- Scaled homology and topological entropy
  - 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.

November 8 2024:

Kejia Zhu (UC Riverside)
- Nonpositive curvature in complex curve complements and families
  - Motivated by the question of whether braid groups are CAT(0), we investigate CAT(0) behavior of fundamental groups of plane curve complements and certain universal families of curves. If the monodromy of the complement of a plane curve $C$ is finite, we show that $\PP^2\setminus C$ admits a complete nonpositively curved metric, and give an example with infinite monodromy where the fundamental group of complement is not CAT(0). When $C$ is the branch locus of a smooth, complete intersection, we also show that $\pi_1(\PP^2\setminus C)$ is CAT(0). In the another direction, we prove that the fundamental group of the universal family associated with the singularities of type $E_6$, $E_7$, and $E_8$ is not CAT(0). This is joint work with Corey Bregman and Anatoly Libgober.

November 15 2024:

Jeff Meyer (UC Riverside)
- Systoles of Arithmetic Manifolds
  - Pick your favorite compact space. How short is the shortest closed loop on it? Now look at your favorite cover of this space. Did that loop unwrap to a longer loop? These are systole questions. The systole of a manifold is the minimal length of a non-contractible closed loop. Systoles in arithmetic manifolds have many fascinating relationships with deep problems in number theory, such as Lehmer’s Mahler measure problem. In recent years, there have been numerous papers studying systoles, their bounds, and their growth up covers as you vary the underlying manifolds. In this talk, I will discuss interesting systole problems, survey known results, and present recent work with collaborators Sara Lapan and Benjamin Linowitz.

November 22 2024:

Nic Brody (UC Santa Cruz)
- Rational Fuchsian Groups
  - We will survey the class of linear groups, and identify three methods of constructing linear groups: algebraic, geometric, and arithmetic. We will propose that every subgroup comes from one of these types of constructions. There are many interesting consequences if this is indeed the case. Focusing on the case of 2-by-2 matrix groups, the algebraic and arithmetic subgroups are very well-understood. We will fill in some understanding of geometric subgroups in this setting, by showing that for any prime p, there is a closed surface group in PSL_2(Z[1/p]). These surface groups will have genus proportional to p, and we speculate that the construction is optimal.

December 6, 2024