Monday, April 25, 2016 - 14:00 , Location: Skiles 006 , Nicholas J. Kuhn , University of Virginia , Organizer: Kirsten Wickelgren
In a 1958 paper, Milnor observed that then new work by Bott allowed him to show that the n sphere admits a vector bundle with non-trivial top Stiefel-Whitney class precisely when n=1,2,4, 8. This can be interpreted as a calculation of the mod 2 Hurewicz map for the classifying space BO, which has the structure of an infinite loopspace. I have been studying Hurewicz maps for infinite loopspaces by showing how a filtration of the homotopy groups coming from stable homotopy theory (the Adams filtration) interacts with a filtration of the homology groups coming from infinite loopspace theory. There are some clean and tidy consequences that lead to a new proof of Milnor's theorem, and other applications.
Tuesday, April 19, 2016 - 13:00 , Location: Skiles 006 , David Gay , University of Georgia , Organizer: John Etnyre
Please note different day and time for the seminar
In honor of John Stallings' great paper, "How not to prove the Poincare conjecture", I will show how to reduce the smooth 4-dimensional Poincare conjecture to a (presumably incredibly difficult) statement in group theory. This is joint work with Aaron Abrams and Rob Kirby. We use trisections where Stallings used Heegaard splittings.
Monday, April 11, 2016 - 14:00 , Location: Skiles 006 , Jonathan Beardsley , Johns Hopkins University , Organizer: Kirsten Wickelgren
Given an action by a loop space on a structured ring spectrum we describe how to produce its associated quotient ring spectrum. We then describe how this structure may be leveraged to produce intermediate Hopf-Galois extensions of ring spectra, analogous to the way one produces intermediate Galois extensions from normal subgroups of a Galois group. We will give many examples of this structure in classical cobordism spectra and in particular describe an entirely new construction of the complex cobordism spectrum which bears a striking resemblance to Lazard's original construction of the Lazard ring by iterated extensions.
Friday, April 1, 2016 - 14:05 , Location: Skiles 006 , William Menasco , U Buffalo , Organizer: Dan Margalit
In Watchareepan Atiponrat's thesis the properties of decomposable exact Lagrangian codordisms betweenLegendrian links in R^3 with the standard contact structure were studied. In particular, for any decomposableexact Lagrangian filling L of a Legendrian link K, one may obtain a normal ruling of K associated with L.Atiponrat's main result is that the associated normal rulings must have an even number of clasps. As a result, there exists a Legendrian (4,-(2n +5))-torus knot, for each n >= 0, which does not have a decomposable exact Lagrangian filling because it has only 1 normal ruling and this normal rolling has odd number of clasps.
Wednesday, March 30, 2016 - 17:05 , Location: Skiles 006 , Henry Segerman , University of Oklahoma , firstname.lastname@example.org , Organizer: Stavros Garoufalidis
When visualising topological objects via 3D printing, we need athree-dimensional geometric representation of the object. There areapproximately three broad strategies for doing this: "Manual" - usingwhatever design software is available to build the object by hand;"Parametric/Implicit" - generating the desired geometry using aparametrisation or implicit description of the object; and "Iterative" -numerically solving an optimisation problem.The manual strategy is unlikely to produce good results unless the subjectis very simple. In general, if there is a reasonably canonical geometricstructure on the topological object, then we hope to be able to produce aparametrisation of it. However, in many cases this seems to be impossibleand some form of iterative method is the best we can do. Within theparametric setting, there are still better and worse ways to proceed. Forexample, a geometric representation should demonstrate as many of thesymmetries of the object as possible. There are similar issues in makingthree-dimensional representations of higher dimensional objects. I willdiscuss these matters with many examples, including visualisation offour-dimensional polytopes (using orthogonal versus stereographicprojection) and Seifert surfaces (comparing my work with Saul Schleimerwith Jack van Wijk's iterative techniques).I will also describe some computational problems that have come up in my 3D printed work, including the design of 3D printed mobiles (joint work withMarco Mahler), "Triple gear" and a visualisation of the Klein Quartic(joint work with Saul Schleimer), and hinged surfaces with negativecurvature (joint work with Geoffrey Irving).
Monday, March 14, 2016 - 14:05 , Location: Skiles 006 , Jing Tao , U Oklahoma , Organizer: Dan Margalit
The commutator length of an element g in the commutator subgroup [G,G] of agroup G is the smallest k such that g is the product of k commutators. WhenG is the fundamental group of a topological space, then the commutatorlength of g is the smallest genus of a surface bounding a homologicallytrivial loop that represents g. Commutator lengths are notoriouslydifficult to compute in practice. Therefore, one can ask for asymptotics.This leads to the notion of stable commutator length (scl) which is thespeed of growth of the commutator length of powers of g. It is known thatfor n > 2, SL(n,Z) is uniformly perfect; that is, every element is aproduct of a bounded number of commutators, and hence scl is 0 on allelements. In contrast, most elements in SL(2,Z) have positive scl. This isrelated to the fact that SL(2,Z) acts naturally on a tree (its Bass-Serretree) and hence has lots of nontrivial quasimorphisms.In this talk, I will discuss a result on the stable commutator lengths inright-angled Artin groups. This is a broad family of groups that includesfree and free abelian groups. These groups are appealing to work withbecause of their geometry; in particular, each right-angled Artin groupadmits a natural action on a CAT(0) cube complex. Our main result is anexplicit uniform lower bound for scl of any nontrivial element in anyright-angled Artin group. This work is joint with Talia Fernos and MaxForester.
Friday, March 4, 2016 - 14:05 , Location: Skiles 006 , Roland van der Veen , University of Leiden , email@example.com , Organizer: Stavros Garoufalidis
I will give an elementary introduction to Majid's theory of braided groups and how this may lead to a more geometric, less quantum, interpretation of knot invariants such as the Jones polynomial. The basic idea is set up a geometry where the coordinate functions commute according to a chosen representation of the braid group. The corresponding knot invariants now come out naturally if one attempts to impose such geometry on the knot complement.
Friday, February 5, 2016 - 14:05 , Location: Skiles 006 , Matthias Goerner , Pixar , firstname.lastname@example.org , Organizer: Stavros Garoufalidis
We call a 3-manifold Platonic if it can be decomposed into isometric Platonic solids. Many key examples in 3-manifold topology are Platonic manifolds, e.g., the Poincar\'e homology sphere, the Seifert-Weber dodecahedral space and the complements of the figure eight knot, the Whitehead link, and the minimally twisted 5-component chain link. They have a strong connection to regular tessellations and illustrate many phenomena such as hidden symmetries.I will talk about recent work on a census of hyperbolic Platonic manifolds and some new techniques we developed for its creation, e.g., verified canonical cell decompositions and the isometry signature which is a complete invariant of a cusped hyperbolic manifold.
Thursday, December 3, 2015 - 14:00 , Location: Skiles 006 , Jeff Meier , University of Indiana , Organizer: John Etnyre
Please not non-standard day for seminar.
A trisection is a decomposition of a four-manifold into three trivial pieces and serves as a four-dimensional analogue to a Heegaard decomposition of a three-manifold. In this talk, I will discuss an adaptation of the theory of trisections to the relative setting of knotted surfaces in the four-sphere that serves as a four-dimensional analogue to bridge splittings of classical knots and links. I'll show that every such surface admits a decomposition into three standard pieces called a bridge trisection. I'll also describe how every such decomposition can be represented diagrammatically as a triple of trivial tangles and give a calculus of moves for passing between diagrams of a fixed surface. This is joint work with Alexander Zupan.
Monday, November 16, 2015 - 14:00 , Location: Skiles 006 , Nick Castro , University of Georgia , Organizer: John Etnyre
A trisection of a smooth, oriented, compact 4-manifold X is a decomposition into three diffeomorphic 4-dimensional 1-handlebodies with certain nice intersections properties. This is a very natural 4-dimensional analog of Heegaard splittings of 3-manifolds. In this talk I will define trisections of closed 4-manifolds, but will quickly move to the case of 4-manifolds with connected boundary. I will discuss how these "relative trisections" interact with open book decompositions on the bounding 3-manifold. Finally, I will discuss a gluing theorem which allows us to glue together relative trisections to induce a trisection on a closed 4-manifold.