Events

Sasha Zotine, Queen's University

The Kawaguchi-Silverman Conjecture (KSC) is a recent conjecture equating two invariants of a dominant rational map between projective varieties: the first dynamical degree and arithmetic degree. The first dynamical degree measures the topological mixing of the map, and the arithmetic degree measures how complicated rational points become after iteration. Recently, the conjecture was established for several classes of varieties, including projectivized bundles over any non-elliptic curve. We will discuss my recent work with Brett Nasserden to resolve the elliptic case, hence proving KSC for all projectivized bundles over curves. 

QNC 2501

Lukas Mueller, Perimeter Institute for Theoretical Physics

Frederik Benirschke, University of Chicago

Classical results by Royden, Earle, and Kra imply that the biholomorphism group of Teichmüller space, the isometry group of the Teichmüller metric, and the mapping class group of the underlying surface are all isomorphic. In other words, every isometry of Teichmüller space is induced by a homeomorphism of the underlying surface.

In this talk, we present a generalization, obtained in joint work with Carlos Serván, where we relax isometries to isometric embeddings. The main result is that isometric embeddings of Teichmüller spaces are coverings constructions, except for some low-dimensional special cases. In other words: Isometric embeddings are induced by branched coverings of the underlying surfaces.

Time permitting, we explain how our techniques can be used to rule out the existence of certain totally geodesic submanifolds.

QNC 2501

Junsheng Zhang, University of California Berkeley

We proved a "no semistability at infinity" result for complete Calabi-Yau metrics asymptotic to cones, by eliminating the possible appearance of an intermediate K-semistable cone in the 2-step degeneration theory developed by Donaldson-Sun. As a consequence, a classification result for complete Calabi-Yau manifolds with Euclidean volume growth and quadratic curvature decay is given. Moreover a byproduct of the proof is a polynomial convergence rate  to the asymptotic cone for such manifolds. Joint work with Song Sun.

QNC 2501

Fried Tong, Harvard University

The Monge-Ampere equation is one of the most important PDE in geometry, and an important tool in the study of Monge-Ampere equations is the Monge-Ampere functional. In this talk, I will discuss a new interesting family of Monge-Ampere type functionals and their relations to some geometric problems. This is based on joint work with S.-T. Yau.

QNC 2501

Lawrence Mouillé, Syracuse University

The Grove-Searle Maximal Symmetry Rank Theorem (MSRT) is a foundational result in the study of manifolds with positive sectional curvature and large isometry groups. It provides a classification of closed, positively curved manifolds that admit isometric actions by tori of large rank. In this talk, I will present progress towards extending the MSRT to positive intermediate Ricci curvature, a condition that interpolates between positive sectional curvature and positive Ricci curvature. Grove and Searle were able to employ concavity of distance functions to establish their MSRT, but this feature is not available for positive intermediate Ricci curvature. I will discuss how we can overcome this barrier using a strengthening of Wilking's Connectedness Lemma. A portion of this talk is from joint work with Lee Kennard.

MC 5417

Changho Han, Department of Pure Mathematics, University of Waterloo

It is well-known that the Torelli map, that turns a smooth curve of genus g into its Jacobian (a principally polarized abelian variety of dimension g), extends to a map from the Deligne—Mumford moduli of stable curves to the moduli of semi-abelic varieties by Alexeev. Moreover, it is also known that the Torelli map does not extend over the alternative compactifications of the moduli of curves as described by the Hassett—Keel program, including the moduli of pseudostable curves (can have nodes and cusps but not elliptic tails). But it is not yet known whether the Torelli map extends over alternative compactifications of the moduli of curves described by Smyth; what about the moduli of curves of genus g with rational m-fold singularities, where m is a positive integer bounded above? As a joint work in progress with Jesse Kass and Matthew Satriano, I will describe moduli spaces of curves with m-fold singularities (with topological constraints) and describe how far the Torelli map extends over such spaces into the Alexeev compactifications.

MC 5417

Panagiotis Dimakis, Université du Québec à Montréal, CIRGET

Since their introduction in 2006, the Kapustin-Witten (KW) equations have become the subject of a number of conjectures. Given a knot $K$ embedded in a closed $3$-manifold $Y$, the most prominent conjecture predicts that the number of solutions to the KW equations on $Y\times\mathbb{R}_+$ with boundary conditions determined by the embedding and with fixed topological charge, is a topological invariant of the knot. A major obstacle with this conjecture is the difficulty of constructing solutions satisfying these boundary conditions. In this talk we assume $Y\cong \Sigma\times\mathbb{R}_+$ and study solutions to the dimensionally reduced KW equations with the required boundary conditions. We prove that the moduli spaces are diffeomorphic to certain holomorphic lagrangian sub-manifolds inside the moduli of Higgs bundles associated to $\Sigma$. Time permitting, we explain how one could use this result to construct knot invariants.

MC 5417

Michael Francis, Western University

The b-tangent bundle (terminology due to Melrose) is defined so that its sections are smooth vector fields on the base manifold tangent along a given hypersurface. Complex b-manifolds, studied by Mendoza, are defined just like ordinary complex manifolds, replacing the usual tangent bundle by the b-tangent bundle. Recently, a Newlander-Nirenberg theorem for b-manifolds was obtained by Francis-Barron, building on Mendoza's work. This talk will discuss the extension of the latter result to the setting of b^k-geometry for k>1. The original approach to b^k-geometry is due to Scott. A slightly different approach that allows for global holonomy phenomena not present in Scott's framework was introduced by Francis and, independently, by Bischoff-del Pino-Witte.

This seminar will be held both online and in person:

• Room: MC 5417
• Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09

Wanchun Rosie Shen, Harvard University

We survey some extensions of the classical notions of Du Bois and rational singularities, known as the k-Du Bois and k-rational singularities. By now, these notions are well-understood for local complete intersections (lci). We explain the difficulties beyond the lci case, and propose new definitions in general to make further progress in the theory. This is joint work (in progress) with Matthew Satriano, Sridhar Venkatesh and Anh Duc Vo.

MC 5417