Events
Geometry and Topology Seminar
- Events
- Geometry and Topology Seminar
Allison Miller, UT Austin, Winding number of satellite operators and concordance
SAS 4201Historically, the study of the collection of concordance classes of knots has focused on understanding its group structure while devoting relatively little attention to the natural metric induced by the 4-genus. Cochran and Harvey investigated the metric properties of the maps on concordance induced by satellite operators, asking when two patterns P and Q are of bounded distance in their…
Jennifer Hom, Georgia Tech, Heegaard Floer and homology cobordism
SAS 4201We study applications of Heegaard Floer homology to homology cobordism. In particular, to a homology sphere Y, we associate a module HF_conn(Y), called the connected Heegaard Floer homology of Y, and show that this module is invariant under homology cobordism and isomorphic to a summand of HF_red(Y). The definition of this invariant relies on involutive…
Elmas Irmak, University of Michigan, Simplicial Maps of Complexes of Curves and Mapping Class Groups of Surfaces
SAS 4201I will talk about recent developments on simplicial maps of complexes of curves on both orientable and nonorientable surfaces. I will also talk about joint work with Prof. Luis Paris. We prove that on a compact, connected, nonorientable surface of genus at least 5, any superinjective simplicial map from the two-sided curve complex to itself is induced…
Jeff Meier, University of Georgia Athens, Filling transverse links with trisected surfaces
SAS 4201We will describe an adaptation of the theory of trisections to the setting of properly embedded, smooth, compact surfaces in smooth, compact, orientable four-manifolds with boundary. The main result is that any such surface can be isotoped to lie in bridge trisected position with respect to a given trisection on the ambient four-manifold. The trisection…
Robin Koytcheff, University of Louisiana at Lafayette, Graph complexes, formality, and configuration space integrals for braids
SAS 4201In joint work with Rafal Komendarczyk and Ismar Volic, we study the space of braids, that is, the loopspace of the configuration space of points in a Euclidean space. We relate two different integration-based approaches to its cohomology, both encoded by complexes of graphs. On the one hand, we can restrict configuration space integrals for…
Andrew Cooper, NC State, Simplicial Configuration Spaces and Chromatic Homology
SAS 4201The configuration space of some $n$ points in a space $X$ is a well-studied object in topology, geometry, and combinatorics. We present a generalization, the simplicial configuration space $M(S,X)$, which takes as its data a simplicial complex $S$ on $n$ points. In this talk, we will describe how $M(S,X)$ gives rise to polynomial and homological invariants of…
Katherine Raoux, Michigan State University, τ-invariants for knots in rational homology spheres
SAS 4201Using the knot filtration on the Heegaard Floer chain complex, Ozsváth and Szabó defined an invariant of knots in the three sphere called τ(K) and showed that it is a lower bound for the 4-ball genus. Generalizing their construction, I will show that for a (not necessarily null-homologous) knot, K, in a rational homology sphere, Y, we obtain…
Aliakbar Daemi, Simons Center for Geometry and Physics, Chern-Simons functional and the Homology Cobordism Group
SAS 4201The set of 3-manifolds with the same homology as the 3-dimensional sphere, modulo an equivalence relation called homology cobordance, forms a group. The additive structure of this group is given by taking connected sum. This group is called the homology cobordism group and plays a special role in low dimensional topology and knot theory. In this talk, I…
Tye Lidman, NC State, Spineless 4-manifolds
We construct smooth compact 4-manifolds homotopy equivalent to S^2 which do not contain nicely embedded spheres realizing the homotopy equivalence.
Curtis Porter, NC State, Spinning Black Holes and CR 3-Folds
Some physically significant solutions to Einstein's field equations are spacetimes which are foliated by a family of curves called a shear-free null geodesic congruence (SFNGC). Examples include models of gravitational waves that were recently detected, and rotating black holes. The properties of a SFNGC induce a CR structure on the 3-dimensional leaf space of the…
Alex Chandler, NC State, Thin Posets and Homology Theories
Inspired by Bar-Natan's description of Khovanov homology, we discuss thin posets and their capacity to support homology and cohomology theories which categorify rank-statistic generating functions. Additionally, we present two main applications. The first, a categorification of certain generalized Vandermonde determinants gotten from the Bruhat order on the symmetric group by applying a special TQFT to…
Guangbo Xu, Simons Center for Geometry and Physics, Mirror Symmetry and Gauged Linear Sigma Model
In this overview talk, I will first explain several mathematical approaches of mirror symmetry. Then I will focus on the mathematical theory of Witten's gauged linear sigma model based on the recent joint work with Gang Tian, which settles the mathematical foundation of the approach of Hori and Vafa.
Alex Zupan, University of Nebraska, A special case of the Smooth 4-dimensional Poincare Conjecture
The smooth version of the 4-dimensional Poincare Conjecture (S4PC) states that every homotopy 4-sphere is diffeomorphic to the standard 4-sphere. One way to attack the S4PC is to examine a restricted class of 4-manifolds. For example, Gabai's proof of Property R implies that every homotopy 4-sphere built with one 2-handle and one 3-handle is standard. …
Juanita Pinzon Caicedo, NC State, Four–manifolds and knot concordance
SAS 4201The main goal of geometric topology is the classification of manifolds within a certain framework (topological, piecewise linear, smooth, simply-connected, symplectic, etc.). Dimension four is special, as it is the only dimension in which a manifold can admit infinitely many non-equivalent smooth structures, and the only dimension in which there exist manifolds homeomorphic but not…
Yakov Berchenko-Kogan, Washington University in St. Louis, Variational numerical methods in geometric PDE
Variational methods can be used to create numerical methods that respect conservation laws. I will discuss applications to electromagnetism, the Yang-Mills equations, and mean curvature flow. I will also discuss some new ideas about finite element spaces of differential forms.