During the Fall 2026 quarter, all of our meetings will be held in person in Skye Hall 268.
Talks will run from 11 am to 11:50 am PT, but you are welcome to arrive early to socialize with the speaker.
FALL 2026 SCHEDULE
September 25, 2026
Mario Rodriguez Sanchez de Toca, U. de Santiago de Compostela (and UCR)
Title: Harmonic manifolds: an overview.
Abstract: Given a Riemannian manifold, we say that it is locally harmonic if its volume density function is radial at any point. It was conjectured by Lichnerowicz that all such manifolds are symmetric spaces of rank one. While this is true for compact manifolds, in the non-compact case the Damek-Ricci spaces arise as a counterexample. In this talk we will discuss the basic aspects of locally harmonic manifolds and some of the open problems about them.
October 2, 2026
Shane Rankin and David Weed, Univ. of Calf., Riverside
Title: Curvature of Left-Invariant Magnetic Systems
Abstract: A magnetic system is a choice of Riemannian metric paired with a closed two-form, referred to as the magnetic form. Recently, Assenza introduced a notion of magnetic sectional curvature that generalizes the classical Riemannian notion to account for the influence of the induced Lorentz force on particle motion at various energy levels. In this talk, we will discuss recent joint work with Ivo Terek describing this curvature in the setting of left-invariant systems on Lie groups, extending in part classical work of Milnor on left-invariant metrics.
October 9, 2026
Tian Yang Texas A&M University
Title: Volume Conjecture and some recent progress
Abstract: In this talk, I will first give an overview of the Volume Conjecture, which connects two seeming different approaches to 3-dimensional topology – the hyperbolic geometry and the quantum invariants. Then I will talk about a recent joint work with Tianyue Liu, Shuang Ming, Xin Sun and Baojun Wu, where we define a family of quantum invariants for hyperbolic 3-manifolds with cusp ends and with totally geodesic boundary from the representation theory of the a quantum group, called the modular double of Uqsl(2; R), and prove that asymptotically these new invariants behave exactly in the way that the Volume Conjecture predicted.
October 16, 2026
Ivo Terek, Univ. of Calf., Riverside
Title: Symplectic connections with parallel Weyl tensor
Abstract: Unlike in Riemannian geometry, where the Levi-Civita connection is uniquely determined by the metric, the Darboux theorem implies that symplectic manifolds admit infinitely many torsionfree connections compatible with the symplectic form. Therefore, the connection must be explicitly included in the initial geometric data. We present an overview of symplectic connections, a natural decomposition of the space of symplectic curvature tensors, and show that connections with parallel Weyl tensor must be locally symmetric or Weyl-flat. This is joint work with Shane Rankin.
October 23, 2026
Tommy Murphy, CSU, Fullerton
Title: k-symmetry and nilmanifold
Abstract: Riemannian k-symmetric spaces arise as a generalization of
the classical Riemannian symmetric spaces. I will report on recent
progress on their classification, new examples, and investigate a
circle of conditions connecting this conditions to negative curvature
and homogeneous Einstein metrics. Based on joint projects with P.A.
Nagy (I.C.G., Korea) and T. Tatsuno (Oklahoma).
October 30
Shane Rankin, Univ. of Calf., Riverside
Title: Algebroid Desingularizable Poisson Structures
Abstract: A Poisson manifold can loosely be thought of as a collection of symplectic manifolds of varying dimension stitched together, and the Poisson structure degenerates where these dimension jumps occur. Sometimes, this degeneration can be resolved by modifying the tangent bundle in an appropriate sense. In this talk, we'll discuss an obstruction to the existence of any such modification, and show that this obstruction is sharp in the linear case (the duals of Lie algebras). The talk will be example-driven, and no background in Poisson geometry will be assumed.
November 6
Title: Higgs bundles and pseudo Kahler metric structures
Abstract: Let S be a closed surface of genus at least 2. For each choice of complex structure on S, the nonabelian Hodge correspondence defines a real analytic isomorphism between the moduli space of flat connections and a moduli space of holomorphic objects called polysyable Higgs bundles on the associated Riemann surface. The bridge between these two spaces is given by an equivariant harmonic map from the universal cover of S to the symmetric space SL(n,C)/SU(n), more generally G/K for a semisimple Lie group G. After reviewing how this correspondence works, I will describe a mapping class group invariant pseudo-Kahler structure on the moduli space of equivariant minimal surfaces whose symplectic form is given by the Atiyah-Bott-Goldman form. Most of the time will be devoted to the more explicit example of SL(3,R) Hitchin representations. This is based on joint work with Jérémy Toulisse and Richard Wentworth.
November 13
Title: A complex hyperKahler structure on the space of minimal surfaces inside PSL(2,C)
Abstract: The complex Lie group PSL(2,C), together with its Killing form, is a complex Riemannian manifold of constant sectional curvature -1 that contains totally geodesic copies of both hyperbolic 3-space and anti-de Sitter 3-space. Consequently, the study of surface immersions inside PSL(2,C) (initiated by F. Bonsante and C. El Emam) provides a unifying framework in which one can transition between hyperbolic and anti-de Sitter geometry.
In joint work with C. El Emam, N. Rungi, A. Seppi, and A. Tamburelli, we leverage this phenomenon to introduce a complex deformation space of equivariant minimal immersions in PSL(2,C), which naturally contains the space of almost-Fuchsian manifolds AF(S) and the space of globally hyperbolic anti-de Sitter (2+1)-spacetimes GH(S) as totally real, half-dimensional submanifolds. By a process of analytic continuation, we prove the existence of a complex hyperKähler structure that simultaneously extends Donaldson’s hyperKähler structure on AF(S), the para-hyperKähler structure on GH(S) introduced by Mazzoli–Seppi–Tamburelli, and the Loustau–Sanders bi-Kähler structure on the space of quasi-Fuchsian manifolds.
November 20
Alex Tao, , Univ. of Calf., Riverside
Title: TBA
Abstract: TBA
December 4
Title: TBA
Abstract: TBA
January 8
Johnathan Delgado (UC Irvine)
Title: TBA
Abstract: TBA
March 12
Nathaniel Sagman (UNC Chapel-Hill)
Title: TBA
Abstract: TBA