Topology at IU

Seminars During Academic Year 2016-17

Fall 2016

Wednesday September 7, 2016 in Rawles Hall 104: Carmen Rovi, Hirzebruch $\chi_y$-genera and the signature modulo $8$ of fibre bundles; In this talk we shall be concerned with the residues modulo $4$ and modulo $8$ of the signature $\sigma(M)$ in $\mathbb{Z}$ of an oriented 4k-dimensional geometric Poincare complex $M$. The precise relation between the signature modulo $8$ and the Brown-Kervaire invariant was worked out by Morita. We shall discuss how the relation between these invariants and the Arf invariant can be applied to the study of the signature modulo $8$ of a fibration. In particular it had been proved by Meyer in 1973 that a surface bundle has signature divisible by $4$. This was generalized to higher dimensions by Hambleton, Korzeniewski and Ranicki in 2007. I will explain two results from my thesis concerning the signature modulo $8$ of a fibration: firstly under what conditions can we guarantee divisibility of the signature by 8 and secondly what invariant detects non-divisibility by 8 in general. I will also talk about a joint project with S. Yokura where we investigate how multiplicativity properties of the signature modulo $8$ also hold for certain Hirzebruch $\chi_y$-genera.
Wednesday September 14, 2016 in Rawles Hall 104: Anthony Conway, Splitting numbers and signatures; The splitting number of a link is the minimal number of crossing changes between different components required to convert it into a split union of its components. Since 2012, this invariant has been studied using various tools such as Khovanov homology, covering link calculus, the Alexander polynomial and Heegaard-Floer homology. After briefly reviewing some of these methods, we will show how (multivariable) signatures give strong lower bounds on the splitting number. This is a joint work with David Cimasoni and Kleopatra Zacharova.
Wednesday September 21, 2016 in Rawles Hall 104: Henry Horton, A functorial symplectic instanton homology via traceless character varieties
Wednesday October 5, 2016 in Rawles Hall 104: Peter Lambert-Cole, Conway Mutation and knot Floer homology; Mutant knots are notoriously hard to distinguish. Many, but not all, knot invariants take the same value on mutant pairs. Khovanov homology with coefficients in Z/2Z is known to be mutation-invariant, while the bigraded knot Floer homology groups can distinguish mutants such as the famous Kinoshita-Terasaka and Conway pair. However, Baldwin and Levine conjectured that delta-graded knot Floer homology, a singly-graded reduction of the full invariant, is preserved by mutation. In this talk, I will give a new proof that Khovanov homology mod 2 is mutation-invariant. The same strategy can be applied to delta-graded knot Floer homology and proves theBaldwin-Levine conjecture for mutations on a large class of tangles.
Wednesday October 26, 2016 in Rawles Hall 104: Michael Mandell, The algebraic K-theory of the sphere spectrum; We've learned a lot about the algebraic K-theory of the sphere spectrum since I spoke on this topic 2 years ago. I'll say some things about this.
Wednesday November 2, 2016 in Rawles Hall 104: Gabe Angelini-Knoll, Red-Shift Phenomena in Iterated Algebraic K-theory of Finite Fields; In the early 2000âs, Ausoni and Rognes initiated a program to study the arithmetic of structured ring spectra. In particular, they conjectured that algebraic K-theory should take ring spectra of chromatic telescopic complexity n to ring spectra of chromatic telescopic complexity n + 1, generalizing the number theoretic Quillen-Lichtenbaum conjecture. To verify this conjecture, it is necessary to do computations. However, only a few algebraic K-theory computations of ring spectra which are not Eilenberg-Mac Lane, have been completed. The iterated algebraic K-theory of a finite field of order q, where q is a prime power generator of the p-adic units, is of special interest because, after p- completion algebraic K-theory of these finite fields is equivalent to the connective cover of the K(1)-local sphere, when p is odd. This computation therefore fits into an old program of Waldhausenâs, where he suggested computing algebraic K-theory of suitable connective covers of certain localizations of the sphere spectrum.In my talk, I will describe a method for computing topological Hochschild homology, a linear approximation to algebraic K-theory, using a multiplicative Whitehead tower of a connective commutative ring spectrum as a filtration. I will also show how certain height two classes in the homotopy groups of the Smith-Toda complex V(1) map nontrivially to the circle homotopy fixed points of mod (p,v_1) THH of p-complete algebraic K-theory of fields of order q. As a consequence, I can show that the Î²-family, a periodic family of height two in the homotopy groups of spheres, is detected in mod (p,v_1)-homotopy of the iterated algebraic K-theory a finite field of order q. I will also provide some evidence indicating that it may be infinitely generated as a P(v_2)-module, which would imply that iterated algebraic K-theory of finite fields does not have telescopic complexity two. This means that the red-shift conjecture may need to be reformulated for it to hold in this case, even though it certainly holds in spirit.
Thursday November 3, 2016 in Rawles Hall 104: Gabriel C. Drummond-Cole, Homotopy probability theory on a Riemannian manifold; Homotopy probability theory is a homological enrichment of algebraic probability theory, a toy model for the algebra of observables in a quantum field theory. I will introduce the basics of the theory and use it to describe a reformulation of fluid flow equations on a compact Riemannian manifold.
Monday November 7, 2016 in Swain West 218: Dror Bar-Natan, A Poly-Time Knot Polynomial Via Solvable Approximation; I will construct the first poly-time-computable knot polynomial since Alexander's (1928) by using some new commutator-calculus techniques and a Lie algebra $g_1$ which is at the same time solvable and an approximation of the simple Lie algebra $sl_2$
Monday November 7, 2016 in Swain West 218: Dror Bar-Natan, (cont) A Poly-Time Knot Polynomial Via Solvable Approximation; I will construct the first poly-time-computable knot polynomial since Alexander's (1928) by using some new commutator-calculus techniques and a Lie algebra $g_1$ which is at the same time solvable and an approximation of the simple Lie algebra $sl_2$
Wednesday November 30, 2016 in Rawles Hall 104: Wai-Kit Yeung, Homotopy braid closure and knot contact homology; In this talk, I will present a universal construction, called homotopy braid closure, that produces invariants of links in R^3 starting with a braid group action on objects of a (model) category C. Applying this construction to the natural action of the braid group B_n on the category of perverse sheaves on the two-dimensional disk with singularities at n marked points, we obtain a differential graded (DG) category that gives knot contact homology in the sense of L. Ng. As an application, we show that the category of finite-dimensional modules over the 0-th homology of this DG category is equivalent to the category of perverse sheaves on R^3 with singularities at most along the link. [This is joint work with Yu. Berest and A. Eshmatov.]
Wednesday December 7, 2016 in Swain West 220: Wenzhao Chen, Heegaard Floer inviants of cable knots; The Upsilon invariant is a knot concordance invariant introduced by Ozsvath, Stipsicz, and Szabo in 2014, using knot Floer homology. It can be viewed as powerful generalization of an earlier known concordance invariant, the Ozsvath-Szabo tau-invariant. In this talk, we will discuss how the Upsilon invariant behaves correspondingly under the cabling operation on knots. I will present a partial answer, which is an inequality relating the Upsilon invariant of a knot and that of its cable. This is a natural generalization of results of Hedden and Van Cott on the tau-invariant of cable knots. As an application, I will also show how one can use this inequality to easily see the existence of an infinite-rank summand of topologically slice knots.

Spring 2017

Wednesday March 1, 2017 in Rawles Hall 104: Birgit Richter, Inhabitants of interesting subsets of the Bousfield lattice; Bousfield introduced an equivalence relation on the stable homotopy category that sorts spectra according to their acyclics. The set of Bousfield classes has some important subsets such as the distributive lattice, DL, of all classes which are smash idempotent and the complete Boolean algebra, cBA, cBA of closed classes. Hovey and Palmieri stated in 1999 that no explicit example of a class in DL \ cBA is known. In joint work with Brooke-Taylor and Loewe we provide uncountably many examples of spectra that are in DL but not in cBA; in particular, for every prime p, the Bousfield class of the Eilenberg-MacLane spectrum of Z/pZ is in DL but not in cBA.
Wednesday March 29, 2017: Matthias Nagel (UQEM)
Thursday March 30, 2017 in Rawles Hall 104: Inbar Klang, Factorization homology and topological Hochschild cohomology of Thom spectra; By a theorem of Lewis, the Thom spectrum of an n-fold loop map to BO is an E_n-ring spectrum. I will discuss a project studying the factorization homology and the higher topological Hochschild cohomology of such Thom spectra, and talk about some applications, such as computations, and a duality between higher topological Hochschild homology and cohomology of such Thom spectra. This talk will include an exposition to factorization homology and higher Hochschild cohomology.
Wednesday April 5, 2017 in RH 104: Kun Wang, The L-theoretic Farrell-Jones conjecture under group extensions and its applications; In this talk, I will introduce what the Farrell-Jones conjecture is and present a general result on its L-theoretic version under group extensions. I will then discuss its applications to the s-cobordism and h-cobordism Borel conjectures and to the Novikov conjecture.
Wednesday April 12, 2017 in RH 104: Peter Feller, "Concordance with a view toward embedding problems in affine algebraic geometry"; Abstract: We first discuss classical questions about polynomial embeddings of the complex line C into complex spaces such as C^m and affine algebraic groups. Next, we consider torus knots and discuss questions related to their non-sliceness motivated by singularity theory. Finally, we use a knot theory perspective to indicate proofs for the embedding questions discussed first.
Wednesday July 26, 2017 in RH 104: Matthias Nagel, Milnor's triple linking numbers and C-complexes; We relate fillability of two link exteriors, and the question when two links admit equivalent C-complexes to (a refinement) of Milnor's triple linking numbers. This extends a theorem of Davis-Roth to include also links with nonvanishing linking numbers. This is joint work with C. Davis, P. Orson and M. Powell.

IU/PU/IUPUI Joint Topology Seminar

Fall 2016: Saturday November 12, 2016: IU/PU/IUPUI Joint Topology Seminar: Mentor Stafa / Andrew Putman, Fall 2016 Meeting