Distinguished Lectures in Topology and Geometry – Distinguished Lectures in Topology and Geometry

The lectures are held several times per semester, usually on Mondays at 4 pm in Smith hall, room 204 (Rutgers-Newark campus), preceded by tea at 3:30 pm.

Upcoming lectures:

November 18th, Marina Ville from the University of Tours, France

Title: Minimal surfaces in R⁴

Abstract. A minimal surface increases its area under a small local deformation. Embedded minimal surfaces in R³ have been intensively studied, with many examples, on the one hand, and many non-existence theorems on the other hand. By contrast, very little is known about embedded minimal surfaces in R⁴ and most of the known examples are given by the algebraic curves of C². After an overview of the R³ case, I will ask some questions about the R⁴ case, indicate some tools and give a couple of examples. No prior knowledge of minimal surfaces is required

Link to research papers: https://www.researchgate.net/profile/Marina-Ville

November 4th, Professor Kate Petersen from University of Minnesota-Duluth

Website: https://sites.google.com/d.umn.edu/katepetersen

Title: Triangulations, Trace Fields and Lacunary Polynomials

Abstract: The trace field of a hyperbolic 3-manifold M is a number field which captures the arithmetic complexity of M. We study how the degrees of trace fields vary under Dehn filling of a 1-cusped 3-manifold M. By studying Lacunary (sparse) polynomials, we show that for Dehn fillings, M(p/q), this growth is bounded below (and above) by a constant (depending only on M) times max{|p|, |q|} for “most” p/q. Further, given any epsilon>0 the degree bounded below by a constant times max{|p|, |q|}^{1-epsilon} for all p/q. Previous work by Garoufalidis and Jeon showed a similar bound (without the epsilon) conditional on Lehmer’s conjecture. As an application of our result, we show that there are fillings M(p_n/q_n) of M such that the number of tetrahedra needed to triangulate M(p_n/q_n) grows linearly n, but the degrees of the trace fields grow exponentially in n. This is joint work with Paul Fili and Neil Hoffman.

Past lectures:

Speaker: Professor Jennifer Schultens from the University of California, Davis

Website: https://www.math.ucdavis.edu/~jcs/
Wikipedia page: https://en.wikipedia.org/wiki/Jennifer_Schultens

Title: Surfaces in Seifert fibered spaces

Abstract: Curve complexes and surface complexes come in several varieties. They have been studied by geometric group theorists and employed by low-dimensional topologists. The unique structure of Seifert fibered spaces allows us to describe certain surface complexes in terms of certain curve complexes.

Speaker: Professor Efstratia Kalfagianni from Michigan State University

Website: https://users.math.msu.edu/users/kalfagia/
Wikipedia page: https://en.wikipedia.org/wiki/Efstratia_Kalfagianni

Title: Knot crossing numbers and Jones polynomials

Abstract: The crossing number of a knot is the smallest number of “double points” (crossings) over all planar projections of the knot. Crossing numbers are hard to compute and their behavior under basic topological operations is poorly understood. In this talk I will discuss how the knot Jones polynomial and its relative the colored Jones polynomial can be used to determine the crossing numbers for large families of knots. The talk is partly based on joint work with Christine Lee.

Speaker: Professor Shelly Harvey from Rice University

Website: https://math.rice.edu/~shelly/
Wikipedia page: https://en.wikipedia.org/wiki/Shelly_Harvey

Title: Linking in 4-dimensions
Abstract: Knots and links play an essential role in classifying 3- and 4-dimensional manifolds (in both the smooth and topological category). Where knots/links up to isotopy is the correct equivalence relation on knots/links to understand 3-manifolds. In this talk, we will be concerned with knots/links up to concordance – the correct equivalence relation to understand 4-manifolds. After a review of concordance, we will discuss new work (with C. Leidy, C. Davis, and J.H. Park) towards an understanding of links up to what could be called algebraic concordance. While the classification of knots up to algebraic concordance has been well understood since the 60’s, we still know little about links. This talk will be accessible to a wide audience. In particular, I will not assume any knowledge of knot theory or low-dimensional topology.

Organizers: Kyle Hayden, John Loftin and Anastasiia Tsvietkova. Funded by NSF CAREER DMS-2142487 grant.