Rutgers Geometry/Topology seminar: Fall 2018 - Spring 2019

Tuesdays 3:40-4:40 in Hill 525

Past seminars: 2017-2018

Fall 2018

Date Speaker Title (click for abstract)
Sep. 4th No Seminar
Sep. 11th Brian Klatt (Rutgers) The Inequalities of Hitchin-Thorpe and Thorpe
Sep. 18th Chenxi Wu (Rutgers) Normal generation and fibered cones
Sep. 25th Maggie Miller (Princeton) The Price twist and trisections
Oct. 2nd Nick Salter (Columbia) Continuous sections of families of complex algebraic varieties
Oct. 9th Ary Shaviv (Weizmann institute in Israel) Schwartz functions on sub-analytic manifolds
Oct. 16th Abhijit Champanerkar (CUNY) Geometry of biperiodic alternating links
Oct. 23rd Francesco Lin (Princeton) The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds
Oct. 30th Ilya Kofman (CUNY) Mahler measure and the Vol-Det Conjecture
Nov. 6th Boyu Zhang (Princeton)
Nov. 13th Tianqi Wu (NYU)
Nov. 20th No Seminar Happy Thanksgiving!
Nov. 27th Andrew Yarmola (Princeton)
Dec. 4th Zhengyi Zhou (IAS)
Dec. 11th

Spring 2019

Date Speaker Title (click for abstract)
Jan. 22nd No Seminar
Jan. 29th
Feb. 5th
Feb. 12th
Feb. 19th
Feb. 26th
March 5th
March 12th
March 19th No Seminar Spring break
March 26th
April 2nd
April 9th
April 16th
April 23rd
April 30rd

Abstracts

The Inequalities of Hitchin-Thorpe and Thorpe

J. A. Thorpe and N. Hitchin independently discovered that the Euler characteristic and signature of a compact, oriented, 4-dimensional Einstein manifold must satisfy a remarkable inequality. As a consequence there are infinitely many simply-connected topological manifolds that cannot support an Einstein metric. An underemphasized aspect of the story of this so-called Hitchin-Thorpe inequality is that it is only a special case of Thorpe's original and more general inequality. In this talk, we give some background on the classical Hitchin-Thorpe inequality as well as Thorpe's inequality before discussing our recent Generalized Thorpe Inequality, which identifies the classical inequalities as resulting from a pure result in Chern-Weil theory.

Normal generation and fibered cones

This is joint work with Harry Baik, Eiko Kin and Hyunshik Shin. We showed that for any 2d slice in the Thurston's fibered cone, all but finitely many primitive points correspond to pseudo anosov maps that normally generate the mapping class group.

The Price twist and trisections

Let S be an RP^2 embedded in a smooth 4-manifold X^4. With some mild conditions, the Price twist is a surgery operation on S that yields a 4-manifold homeomorphic (but not necessarily diffeomorphic) to X^4. In particular, for every RP^2 embedded in S^4, this operation yields a homotopy 4-sphere.

In this talk, we will understand the Price twist via the theory of trisections. In particular, I will show how to produce an explicit trisection diagram of the surgered 4-manifold.

This is joint work with Seungwon Kim.

Continuous sections of families of complex algebraic varieties

Families of algebraic varieties exhibit a wide range of fascinating topological phenomena. Even families of zero-dimensional varieties (configurations of points on the Riemann sphere) and one-dimensional varieties (Riemann surfaces) have a rich theory closely related to the theory of braid groups and mapping class groups. In this talk, I will survey some recent work aimed at understanding one aspect of the topology of such families: the problem of (non)existence of continuous sections of "universal" families. Informally, these results give answers to the following sorts of questions: is it possible to choose a distinguished point on every Riemann surface of genus g in a continuous way? What if some extra data (e.g. a level structure) is specified? Can one instead specify a collection of n distinct points for some larger n? Or, in a different direction, if one is given a collection of n distinct points on CP^1, is there a rule to continuously assign an additional m distinct points? In this last case there is a remarkable relationship between n and m. For instance, we will see that there is a rule which produces 6 new points given 4 distinct points on CP^1, but there is no rule that produces 5 or 7, and when n is at least 6, m must be divisible by n(n-1)(n-2). These results are joint work with Lei Chen.

Schwartz functions on sub-analytic manifolds

Schwartz functions are classically defined on R^n as smooth functions such that they, and all their (partial) derivatives, decay at infinity faster than the inverse of any polynomial. The space of Schwartz functions is a Frechet space, and its continuous dual space is called the space of tempered distributions. A third space that plays a key role in the Schwartz theory is the space of tempered functions – a function is said to be tempered if point-wise multiplication by it preserves the space of Schwartz functions. This theory was formulated on R^n by Laurent Schwartz, later on Nash manifolds (smooth semi-algebraic varieties) by Fokko du Cloux and by Avraham Aizenbud and Dmitry Gourevitch, and on singular algebraic varieties by Boaz Elazar and myself.

The goal of this talk is to present the recently developed Schwartz theory on sub-analytic manifolds. I will first explain how one can attach a Schwartz space to an arbitrary open subset of R^n. Then, I will define (globally) sub-analytic manifolds – loosely speaking these are manifolds that locally look like sub-analytic open sub-sets of R^n (I will explain what are these too) and have some ”finiteness” property. Model theorists may think of definable manifolds in R^{an}. I will prove that one can intrinsically define the space of Schwartz functions (as well as the spaces of tempered functions and of tempered distributions) on these manifolds, and prove that these spaces are ”well behaved” (in the sense that they form sheaves and co-sheaves on the Grothendieck sub-analytic topology). Along the way we will see where sub-analyticity is used, and why this theory is ill-defined in the category of smooth (not necessarily sub-analytic) manifolds. Mainly, some ”polynomially bounded behaviour” (that holds in the sub-analytic case thanks to Lojasiewicz’s inequality) is required. As time permits I will describe some possible applications.

Geometry of biperiodic alternating links

In this talk we will study the hyperbolic geometry of alternating link complements in the thickened torus. We will give conditions which imply the hyperbolicity of alternating link complements in the thickened torus. We show that these complements admit a positively oriented, unimodular geometric ideal triangulation, and determine sharp upper and lower volume bounds. For links which arise from semi-regular Euclidean tilings, called semi-regular links, we explicitly determine the complete hyperbolic structure on their complements and will discuss nice consequences like determination of exact volumes, arithmeticity and commensurability for this class of links. We will also discuss the Volume Density Conjecture and examples.

The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds

While both hyperbolic geometry and Floer homology have both been tremendously successful tools when studying three-dimensional topology, their relationship is still very mysterious. In this talk, we provide sufficient conditions for a hyperbolic rational homology sphere not to admit irreducible solutions to the Seiberg-Witten equations in terms of its volume and the length spectrum (i.e. the set of lengths of closed geodesics). We discus explicit examples in which this criterion can be applied. This is joint work with Michael Lipnowski.

Mahler measure and the Vol-Det Conjecture

A basic open problem is to understand how the hyperbolic volume of knots and links is related to diagrammatic knot invariants. The Vol-Det Conjecture relates the volume and determinant of alternating links. We prove the Vol-Det Conjecture for infinite families of alternating links using the dimer model, the Mahler measure of 2-variable polynomials, and the hyperbolic geometry of biperiodic alternating links. This is joint work with Abhijit Champanerkar and Matilde Lalin.

Organizers: Feng Luo, Xiaochun Rong, Hongbin Sun