We now have a website dedicated to the UCR Geometry and Topology Seminar. Please visit
https://sites.google.com/ucr.edu/ucrgeomtop/home
for the most up-to-date seminar information and the seminar archive, including the former Differential Geometry Seminar.
Below, I keep the Spring 2023 Schedule, the last quarter I organized before the dedicated website.
Spring 2023
April 14th: Chao-Ming Li, University of California, Irvine
On the convexity of general inverse $\sigma_k$ equations and some applications
Abstract:
In this talk, I will show my recent work on general inverse $\sigma_k$ equations and the deformed Hermitian—Yang—Mills equation (hereinafter the dHYM equation). First, I will show my recent result. This result states that if a level set of a general inverse $\sigma_k$ equation (after translation if needed) is contained in the positive orthant, then this level set is convex. As an application, this result justifies the convexity of the Monge—Ampère equation, the J-equation, the dHYM equation, the special Lagrangian equation, etc. Second, I will introduce some semialgebraic sets and a special class of univariate polynomials and give a Positivstellensatz type result. These give a numerical criterion to verify whether the level set will be contained in the positive orthant. Last, as an application, I will prove one of the conjectures by Collins—Jacob—Yau when the dimension equals four. This conjecture states that under the supercritical phase assumption, if there exists a C-subsolution to the dHYM equation, then the dHYM equation is solvable.
April 21th: Mohandas Pillai, University of California, Sen Diago
Title: Global Non-scattering solutions to the energy critical wave maps equation
Abstract:
We consider the 1-equivariant energy critical wave maps problem with two-sphere target. Using a method based on matched asymptotic expansions, we construct infinite time relaxation, blow-up, and intermediate types of solutions that have topological degree one. More precisely, for a symbol class of admissible, time-dependent length scales, we construct solutions which can be decomposed as a ground state harmonic map (soliton) re-scaled by an admissible length scale, plus radiation, and small corrections which vanish (in a suitable sense) as time approaches infinity. Our class of admissible length scales includes positive and negative powers of t, with exponents sufficiently small in absolute value. In addition, we obtain solutions with soliton length scale undergoing damped or undamped oscillations in a bounded set, or undergoing unbounded oscillations, for all sufficiently large t.
April 28th: Ming Zhang, University of California, Sen Diago
Orbifold quantum K-theory
Abstract:
Givental and Lee introduced quantum K-theory, a K-theoretic generalization of Gromov--Witten theory. It studies holomorphic Euler characteristics of coherent sheaves on moduli spaces of stable maps to given target spaces. In this talk, I will introduce the quantum K-theory for orbifold target spaces which generalizes the work of Tonita-Tseng. In genus zero, I will define a quantum K-ring which specializes to the full orbifold K-ring introduced by Jarvis-Kaufmann-Kimura. As an application, I will give a detailed description of the quantum K-ring of weighted projective spaces, which generalizes a result by Goldin-Harada-Holm-Kimura. This talk is based on joint work with Yang Zhou.
May 5th: Li-Sheng Tseng, University of California, Irvine
Abstract: A cone story for smooth manifolds
Abstract:
For manifolds such as special holonomy and symplectic manifolds that are equipped with a geometrical structure specified by a distinguished closed form, we will motivate the usefulness of considering pairs of differential forms that are linked together by a map of the distinguished form. We will show how this lead to new notions of Morse theory and flat connections, and also novel Yang-Mills type functionals. This talk is based on joint works with David Clausen, Xiang Tang, and Jiawei Zhou.
May 12th: Xiangwen Zhang, University of California, Irvine
Geometric flows and Type IIA equation
Abstract:
Geometric flows have been proven to be powerful tools in the study of many important problems arising from both geometry and theoretical physics. Aiming to study the equations from the flux compactifications of Type IIA superstrings, we introduce the so-called Type IIA flow, which is a flow of closed and primitive 3-forms on a symplectic Calabi-Yau 6-manifold. Remarkably, the Type IIA flow can also be viewed as a flow as a coupling of the Ricci flow with a scalar field. In this talk, we will discuss the progress on this flow.
May 19th: Jonathan Campbell
Some Applications of Bicategorical Thinking
Abstract:
Many seemingly ad hoc constructions in algebra become simpler and much more natural through the lens of bicategories. In this talk I'll describe a series of papers with Kate Ponto touching on Euler characteristics, 2 dimensional field theories, and topological Hochschild homology, which never could have been written without thinking bicategorically. Particular focus will be put on iterated traces (relating to 2d field theories) and the structure of topological Hochschild homology.
May 26th: Konstantinos Varvarezos, University of California, Los Angeles
Heegaard Floer homology and chirally cosmetic surgeries
Abstract:
Dehn surgery on knots is an important method for obtaining 3-manifolds. A pair of Dehn surgeries on a knot are said to be cosmetic if the resulting 3-manifolds are homeomorphic. Specifically, they are purely cosmetic if the resulting manifolds have the same orientation and chirally cosmetic otherwise. Applying the immersed curve formulation of Bordered Heegaard Floer homology developed by Hanselman, Rasmussen, and Watson,we find new obstructions to the existence of chirally cosmetic surgeries. As an application, we completely classify cosmetic surgeries on odd alternating pretzel knots.
June 2th: Yu-Shen Lin, Boston University
On the moduli spaces of ALH*-gravitational instantons
Abstract:
Gravitational instantons are defined as non-compact hyperKahler 4-manifolds with L^2 curvature decay. They are all bubbling limits of K3 surfaces and thus serve as stepping stones for understanding the K3 metrics. In this talk, we will focus on a special kind of them called ALH*-gravitational instantons. We will explain the Torelli theorem, describe their moduli spaces and some partial compactifications of the moduli spaces. This talk is based on joint works with T. Collins, A. Jacob, R. Takahashi, X. Zhu and S. Soundararajan.
June 9th: Hasan el-Hasan, University of California, Riverside
Random 3-Manifolds Have No Totally Geodesic Submanifolds
Abstract:
We show that generic closed Riemannian 3-manifolds have no non-trivial totally geodesic submanifolds. In 2018, Wilhelm and Murphy proved this for manifolds of dimension greater than or equal to 4. This question was originally posed by Spivak. We will also discuss some consequences of this result for the isometry group of a generic Riemannian metric.