Speakers & Abstracts
Ronan Conlon (The University of Texas at Dallas)
Title: Shrinking Kahler-Ricci solitons on complex surfaces
Abstract: Shrinking Kahler-Ricci solitons model finite-time singularities of the Kahler-Ricci flow, hence the need for their classification. I will talk about the classification of such solitons in four real dimensions. This is joint work with Deruelle-Sun, Cifarelli-Deruelle, and Bamler-Cifarelli-Deruelle.
Renato Bettiol (Lehman College, CUNY)
Title: Curvature operators and rational cobordism
Abstract: In this talk, I will discuss generalizations of the well-known fact (due to Lichnerowicz) that closed spin manifolds with positive scalar curvature have vanishing A-hat genus. More precisely, we determine linear inequalities on the eigenvalues of the curvature operator that imply vanishing of the index of Dirac operators twisted with prescribed tensor bundles. The vanishing of such indices has topological implications, e.g., in terms of Pontryagin classes, rational cobordism type, signature, elliptic genus, and Witten genus of the manifold. The key algebraic estimate in our proof is a Lie-theoretic generalization of recent results by Petersen and Wink on the Bochner technique, which is also of interest outside the realm of spin geometry. (This is based on joint work with Jackson Goodman.)
Saman Esfahani (Duke University)
Title: Gauge theory in higher dimensions, non-compactness problems, and the Fueter equation
Abstract: Gauge-theoretic invariants, and in particular Yang-Mills instantons and Seiberg-Witten invariants, revolutionized the study of 3- and 4-dimensional manifolds. Donaldson and Thomas proposed a program to generalize these gauge theories to higher-dimensional manifolds with special holonomy groups and define similar invariants.
The main difficulty in following this program comes from the non-compactness of the moduli spaces of solutions of these gauge theories. The non-compactness of these spaces is closely related to certain minimal (calibrated) submanifolds, and the Fueter equation defined on them. We prove partial results in this direction, examining different sources of non-compactness, and proving some of them, in fact, do not occur.
The study of the Fueter equation motivates defining new invariants of 3- and 4-dimensional manifolds. We will discuss these invariants and their relations to some other known invariants of 3-manifolds, including the Casson invariant and the Rozansky-Witten invariant.
Daniel Ketover (Rutgers University)
Title: Stabilizations of Heegaard surfaces and minimal surfaces
Abstract: In the 1930s, Reidemeister and Singer showed that any two Heegaard surfaces in a three-manifold become isotopic after adding sufficiently many trivial handles. I will show how this topological result gives rise to minimal surfaces of Morse index 2 in many ambient geometries. In particular, applied to most lens spaces we produce genus 2 minimal surfaces. I’ll show using this that the number of genus g minimal surfaces in the round sphere tends to infinity as g does (previously the lower bound for all large genera was two).