Topology at IU

Seminars During Academic Year 2015-16

Fall 2015

Wednesday September 2, 2015 in Rawles Hall 104: Agnes Beaudry, Gluing data in chromatic homotopy theory; Understanding the stable homotopy groups of spheres is one of the great challenges of algebraic topology. They form a ring which, despite its simple definition, carries an amazing amount of structure. A famous theorem of Hopkins and Ravenel states that it is filtered by simpler rings called the chromatic layers. This point of view organizes the homotopy groups into periodic families and reveals patterns. There are many structural conjectures about the chromatic filtration. I will talk about one of these conjectures, the chromatic splitting conjecture, which concerns the gluing data between the different layers of the chromatic filtration.
Wednesday September 30, 2015 in Rawles Hall 104: Gwenael Massuyeau, Poisson brackets in representation algebras generalizing the Atiyah-Bott-Goldman structures; Goldman showed how to use intersection of curves in order to compute the Poisson bracket induced by the Atiyah-Bott symplectic structure on the moduli space of representations of a surface group in a Lie group. We will present some algebraic generalizations of Goldman's formulas where the surface group is replaced by a cocommutative Hopf algebra and the Lie group is replaced by a group scheme. Next, we will explain how to apply this general framework to higher-dimensional manifolds using intersection of cycles in loop spaces.
Wednesday October 7, 2015 in Rawles Hall 104: Danielle O'Donnol, Minimal embedding of Legendrian graphs and the total Thurston-Bennequin number.; A Legendrian graph is a graph embedded in such a way that its edges are everywhere tangent to the contact structure. We extend the classical invariants Thurston-Bennequin number and rotation number to Legendrian graphs. A minimal embedding is one where all of the minimal length cycles are unknots with $tb=-1$. I will talk about some new results about a new invariant, the total Thurston-Bennequin number, and discuss the consequences of these results for embeddings of $K_4$ and $K_{3,3}$.
Wednesday October 14, 2015 in Rawles Hall 104: Emmanuel Dror Farjoun, A new look at Postnikov sections; (with Wojtek C., Ramon F., Jerome S.) It turns out that for a nilpotent space X, the Postnikov sections P_nX have a strong "cellular" relation to X itself: For example if X is acyclic with respect to any homology theory so is P_nX. This allows strong statements about the value of certain functors F(Y)—–>Y (e.g higher covers) on these sections. The main tool is a modified Bousfield Kan completion tower which is easier to understand in certain respects.
Wednesday October 21, 2015 in Rawles Hall 104: Sarah Yeakel, A chain rule for Goodwillie calculus; In the homotopy calculus of functors, Goodwillie defines a way of assigning a Taylor tower of polynomial functors to a homotopy functor and identifies the homogeneous pieces as being classified by certain spectra, called the derivatives of the functor. Michael Ching showed that the derivatives of the identity functor of spaces form an operad, and Arone and Ching developed a chain rule for composable functors. We will review these results and show that through a slight modification to the definition of derivative, we have found a more straight forward chain rule for endofunctors of spaces.
Tuesday October 27, 2015 in SE 140: Inna Zakharevich, Spectral sequences associated to the Grothendieck ring of varieties; The Grothendieck ring of varieties is defined to be the free abelian group generated by varieties, modulo the relation that for a closed embedding $Y to X$, [X] = [Y]+[XY]. This ring comes with a filtration by dimension of variety, but the associated graded of this filtration is very difficult to compute, as higher-dimensional varieties can induce relations between lower-dimensional ones. To get around this difficulty we use algebraic K-theory to construct a spectral version of this ring, which will also have a dimension filtration. We show how to compute the associated graded of this filtration and discuss the resulting spectral sequence.
Wednesday November 4, 2015 in Rawles Hall 104: Palanivel Manoharan, Geometry of manifolds modeled on Hilbert modules
Wednesday November 11, 2015 in Rawles Hall 104: Cary Malkiewich, Duality in topological Hochschild homology (THH); Topological Hochschild homology (THH) is a beautiful and computable invariant of rings and ring spectra. In this talk, I will focus on the ring spectrum DX, and discuss a few different aspects of THH(DX). For example, it splits when X is a suspension, and we can use this for computations in topological cyclic homology. I will also recall the "Atiyah duality" between THH(DX) and the free loop space LX, and prove that this duality preserves the genuine S^1-structure. This uses the new "norm" model of THH, and a surprising technical result about orthogonal G-spectra. If there is time, I will apply these tools once more and describe an enrichment of the character map from representation theory.
Wednesday November 18, 2015 in Rawles Hall 104: Ralph Kaufmann, Around Feynman categories; We recall the notion of Feynman categories and then go on to discuss examples and constructions. We then turn to newer developments like Hopf algebras arising out of them, which are known from physics and number theory as well as newer decorated version.

Spring 2016

Wednesday January 6, 2016 in Rawles Hall 104: Mark Bell
Wednesday January 20, 2016 in Rawles Hall 104: Peter Lambert-Cole, Classifying planar Legendrian graphs; A planar spatial graph in S^3 is a 1-complex embedded on a smooth sphere. The graph is Legendrian if it is everywhere tangent to the standard contact structure on the 3-sphere. Using convex surface theory, we solve several classification problems for planar Legendrian graphs, including Legendrian simplicity, the Legendrian mirror problem, and stabilization equivalence. This is joint with Danielle O'Donnol (Indiana).
Wednesday February 10, 2016 in Rawles Hall 104: Jim Davis, Topology of L^2-acyclic manifolds; A manifold is L^2-acyclic if its L^2-betti numbers vanish. An example is given by a manifold whose fundamental group is infinite cyclic where every homology class of its universal cover is annihilated by a nonzero Laurent polynomial in the deck transformation. After giving examples, we will discuss general fundamental groups and give analytic properties of such manifolds. This is likely to be the first in a series of talks.
Tuesday February 16, 2016 in Rawles Hall 104: Chuck Livingston, Survey of knot concordance
Wednesday March 9, 2016 in Rawles Hall 104: Birgit Richter, Ramified extensions of structured ring spectra; Well behaved multiplicative cohomology theories can be represented by ring spectra. Topological complex K-theory for instance is represented by a periodic ring spectrum $KU$ whose homotopy groups are a Laurent polynomial ring on a generator in degree two over the integers. Algebraic K-theory of ring spectra carries geometric information in some cases, but here the ring spectra involved are often connective, i. e., have homotopy groups concentrated in non-negative degrees, such as the sphere spectrum or connective complex K-theory. Even for Galois extensions of ring spectra, such as the complexification map from real to complex periodic K-theory, the associated map on connective covers behaves like a ramified extension of number rings. We discuss examples and present detection methods for ramification.
Wednesday March 23, 2016 in Rawles Hall 104: Nerses Aramian, The Integration Pairing and Extended Topological Field Theories; Given an hermitian line bundle with an hermitian connection on a manifold there is a straightforward way of producing 1-dimensional topological field theory over the manifold. One can generalize this procedure by replacing complex line bundles with $n$-gerbes bound by $U(1)$, and replacing connections with appropriate connective structures. The resulting topological field theories are $n$-dimensional and extended. In this talk, we provide an explicit construction of topological field theories using the higher categorical framework of Jacob Lurie, and show that the construction establishes equivalence between the smooth Deligne complex and the framed topological field theories.
Thursday March 24, 2016 in Rawles Hall 104: Qayum Khan, Stable existence of incompressible 3-manifolds in 4-manifolds; Given an injective amalgam at the level of fundamental groups and a specific 3-manifold, does there exist a corresponding geometric-topological decomposition of a given 4-manifold, in a stable sense? We find an algebraic-topological splitting criterion and express it via the associated orientation classes and universal covers. Along the way, we generalize the Lickorish–Wallace theorem to regular covers. This is joint work-in-progress with my PhD student, Gerrit Smith.
Thursday March 31, 2016 in RH104: Alexandra Kjuchukova, Branched covers of four-manifolds and related questions in knot theory; Let $f:Y\to X$ be a branched cover of closed oriented four-manifolds with an oriented, topologically flatly embedded branching set $B$. By an old formula of Viro, the signature of $Y$ can be computed in terms of data about $X$, $B$ and $[B]\in H_2(X; Z)$. I give a generalization of this formula to the case where $B$ is embedded in $X$ with an isolated cone singularity of type $K$ (here, $K$ is a knot in $S^3$) and $f$ is an irregular dihedral cover. The defect to signature of $Y$ resulting from the presence of a singularity can be expressed in terms of classical invariants of $K$. I discuss several natural questions that follow: which knot types can arise as singularities? what is the range of signatures of dihedral covers of $S^4$? can all possible signatures over a given base be realized using two-bridge knots? I'll report on some on-going work on these questions.
Wednesday April 6, 2016 in SE 140: Chris Herald, Traceless SU(2) character varieties of tangles and their symplectic properties; In this talk, we discuss the character variety of traceless fundamental group representations to SU(2) of a tangle complement. Here, traceless means meridians of all n strands are sent to traceless SU(2) elements. There is a restriction map from the traceless character variety of an n-strand tangle to the traceless character variety of the 2n-punctured sphere. Weâll describe the structure of these traceless character varieties and their symplectic properties, and outline a program to define a Lagrangian Floer homology counterpart to Kronheimer Mrowka reduced singular instanton homology, which uses these traceless character varieties. This is work by various combinations of Paul Kirk, Matt Hedden, and Matt Hogancamp and myself
Tuesday April 12, 2016 in Rawles Hall 104: Allan Edmonds, Geography of Aspherical 4-Manifolds; I will survey what is know about the pairs (signature, Euler characteristic) of closed, oriented, aspherical 4-manifolds and sketch a construction of new examples.
Thursday April 28, 2016 in Rawls Hall 104: Arunima Ray, Shake slice and shake concordant knots.; If K is a knot in the 3-sphere, then the 4-manifold W_K obtained by adding a single 2-handle to B^4 along K with zero framing has H_2 isomorphic to Z. If a generator of H_2(W_K) can be represented by an embedded sphere, K is said to be shake-slice. Any slice knot is shake-slice, but the converse is unknown. We define a relative version of this concept, known as shake concordance, and construct infinite families of knots that are pairwise shake-concordant but not concordant. We show that the concordance invariants tau, s, and slice genus are not invariants of shake-concordance. We also give a characterization of shake-concordant and shake-slice knots in terms of concordance. (This is joint work with Tim Cochran.)
Tuesday June 14, 2016 in Rawles 104: Brandy Doleshal, The search for distinct primitive/Seifert knots that admit the same Seifert fibered space surgery; Given a knot in a 3-manifold, one can obtain a new 3-manifold by removing a neighborhood of the knot and refilling the empty space created. This process, called Dehn surgery, is a great way of understanding 3-manifolds, which is a major goal of low dimensional topology. Primitive/Seifert knots are a class of knots that lie on the genus two surface in a way that guarantees that they admit a Seifert fibered space Dehn surgery along a particular slope. We would like to answer the following question: is it possible for two distinct primitive/Seifert knots to have the same Seifert fibered space surgery after surgery at the surface slope? In this talk, we will discuss some necessary definitions, the origin of this problem and the things that went wrong but turned out alright. This is joint work with Jeff Meier.

IU/PU/IUPUI Joint Topology Seminar

Fall 2015: Saturday December 5, 2015: IU/PU/IUPUI Joint Topology Seminar: Charmaine Sia / Marco Varisco, Fall Seminar
Spring 2016: Saturday April 16, 2016: IU/PU/IUPUI Joint Topology Seminar: Jeremy Miller / Cary Malkievich, Spring 2016 Seminar