During the Fall 2025 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.
Current organizers: Brian Collier, Filippo Mazzoli
Fall 2025 schedule
October 3, 2025
Shane Rankin (UCR)
Title: Shadows of Symplectic Geometry in some Poisson Manifolds
Abstract: In this talk, we’ll discuss a broad class of Poisson manifolds that are "almost" symplectic in some sense. We’ll first discuss how one can recover a remnant of "symplectic symmetry" inside the tangent algebra via a natural subalgebra that carries a rich algebraic structure. We'll then discuss a long exact sequence linking Poisson cohomology with Lie algebroid cohomology, which computes Poisson Cohomology and clarifies some discrepancies such as those appearing in E-Poisson cohomology. If time permits, we will also describe ongoing work toward assigning harmonic representatives to certain Poisson cohomology classes.
October 10, 2025
Ivo Terek (UCR)
Title: Compact plane waves with parallel Weyl curvature
Abstract: We present some background and discuss recent developments in the study of essentially conformally symmetric (ECS) manifolds - that is, those with parallel Weyl curvature which are not locally symmetric or conformally flat - including the topological structure in the rank-one case and the existence of compact examples.
Joint work with Andrzej Derdzinski. If you are interested in learning more about this topic, you can read about it in this recently published book of proceedings: https://link.springer.com/book/10.1007/978-3-031-99212-4
October 17, 2025
Leslie Mavrakis (University of Utah)
Title: Combinatorial Characterizations and Branched Manifolds
Abstract: A family F of compact n-manifolds is locally combinatorially defined (LCD) if there is a finite number of triangulated n-balls such that every manifold in F has a triangulation that locally looks like one of these n-balls. In joint work with Daryl Cooper and Priyam Patel, we show that LCD is equivalent to the existence of a compact branched n-manifold W, such that F is precisely those manifolds that immerse into W. In this way, W can be thought of as a universal branched manifold for F. In current and future work, we use this equivalence to show that, for each of the eight Thurston geometries, the family of closed 3-manifolds admitting that geometry is LCD. In this talk, I will present the main ideas of the proof of the equivalence and then construct branched 3-manifolds for a few of the geometries.
October 24, 2025
Filippo Mazzoli (UCR)
Title: Geometric transitions between the deformation spaces of quasi-Fuchsian representations in hyperbolic and anti-de Sitter space
Abstract: Let S be a closed oriented surface of genus g larger than or equal to 2. By the work of Donaldson, the deformation space of almost-Fuchsian hyperbolic metrics on the 3-manifold SxR admits a mapping class group-invariant hyperKahler structure. More recently, Seppi, Tamburelli and I showed that a similar phenomenon occurs for the deformation space of quasi-Fuchsian anti-de Sitter structures on SxR, which turns out to carry a natural para-hyperKahler structure. Hyperbolic and anti-de Sitter geometries are both subgeometries of the 3-dimensional projective space. Since the work of Danciger, geometric transitions between hyperbolic and anti-de Sitter structures via half-pipe geometry have been studied. In a joint work in progress with El Emam, Seppi, and Tamburelli, we show that a similar phenomenon of geometric transition occurs at the level of the defomation spaces between the aforementioned hyperKahler and para-hyperKahler structures, passing through so-called quasi-Fuchsian half-pipe 3-manifolds and their deformation spaces.
October 31, 2025
Kai-Wei Zhao (UC Irvine)
Title: Classification of Finite-Entropy Shortening Curves
Abstract: Curve shortening flow is a one-dimensional variant of mean curvature flow. In the compact case, it can be considered as the gradient flow of the arc-length functional. The classification of ancient solutions of CSF under some geometric conditions is a parabolic version of Liouville-type theorem. The previous results by Daskalopoulos–Hamilton–Sesum, X.-J. Wang, and Bourni–Langford–Tinaglia technically rely on the assumption of convexity of the curves. In the joint work with Kyeongsu Choi, Donghwi Seo, and Weibo Su, we replace the convexity condition by finite entropy, a mild measure of geometric complexity. In this talk, we will show that an ancient, embedded, smooth solution of CSF with finite entropy must be a static line, a shrinking circle, a paper clip, a translating grim reaper, or one in the large class of trombones constructed by Angenent–You.
November 7, 2025
Brian Collier (UCR)
Title: Energy and a generically non degenerate closed 2-form
Abstract: Let S be a closed topological surface of genus at least 2 and G be a semisimple complex Lie group, such as the group of determinant 1 matrices. Associated to data there is an interesting symplectic manifold X known as the G-character variety of S, whose points correspond to isomorphism classes of group homomorphisms from the fundamental group of S into G. If one a complex structure j on S, then there is an interesting complex manifold associated to (S,j,G) known as the moduli space of G-Higgs bundles. Remarkably, these very different spaces are diffeomorphic and the explicit diffeomorphism is known as the nonabelian Hodge correspondence. For a fixed point of the character variety, it is very hard to understand how various properties of the associated Higgs bundle depends on or varies with the choice of Riemann surface. This talk will be about trying to understand such questions using a differential geometric construction of a moduli space of Higgs
November 14, 2025
Julien Paupert (Arizona State University)
Title: Complex hyperbolic deformations of cusped hyperbolic lattices
Abstract: After reviewing some classical results about rigidity and flexibility of lattices in semisimple Lie groups and known results in dimension 2, we will discuss the existence and potential discreteness/faithfulness of deformations of cusped lattices of SO(3,1) into SU(3,1). We will focus on the cases of small Bianchi groups (joint with M. Thistlethwaite) and the figure-eight knot group (joint with S. Ballas and P. Will), as well as the general behavior of bending deformations in all dimensions.
November 21, 2025
Fred Wilhelm (UCR)
Title: Gradient and Hessian are metric notions
Abstract: We will revisit the notions of gradient and hessian for smooth functions, through a lens that emphasizes geometry. We will then discuss these notions in the context of some of my joint work with Luis Guijarro, which includes the following results:
Theorem 1. A submanifold of the n-sphere is positively curved provided if its focal radius is \geq \pi/4 and its dimension is \geq 2.
Theorem 2. A submanifold of any hyperbolic manifold is non-positively curved if it has infnite focal radius and its dimension is \geq 2.
Note that these statements do not require that the submanifolds be complete, and they hold regardless of the codimension, as long as the notion of curvature makes sense. The Clifford torus in S^3 shows that the first result is optimal. I have made a special effort to make the talk accessible to graduate students.
November 28, 2025
NO SEMINAR (Thanksgiving break)