I will begin the talk by reviewing the definition of commutative K-theory, a generalized cohomology theory introduced by Adem and Gomez. It is a refinement of topological K-theory, where the transition functions of a vector bundle satisfy a commutativity condition. The theory is represented by an infinite loop space which is called a “classifying space for commutativity”. I will describe the homotopy type of this infinite loop space. Then I will discuss the graded ring structure on its homotopy groups, which corresponds to the tensor product of vector bundles.

# Past Topology Seminar

Configuration spaces of points in a manifold are well studied. Giving the points thickness has obvious physical meaning: the configuration space of non-overlapping particles is equivalent to the phase space, or energy landscape, of a hard spheres gas. But despite their intrinsic appeal, relatively little is known so far about the topology of such spaces. I will overview some recent work in this area, including a theorem with Yuliy Baryshnikov and Peter Bubenik that related the topology of these spaces to mechanically balanced, or jammed, configurations. I will also discuss work in progress with Robert MacPherson on hard disks in an infinite strip, where we understand the asymptotics of the Betti numbers as the number of disks tends to infinity. In the end, we see a kind of topological analogue of a liquid-gas phase transition.

(Joint work with Marco Golla and József Bodnár)

We will give a general overview of the plethora of concordance invariants which can be extracted from Ozsváth-Szabó-Rasmussen's knot Floer homology.

We will then focus on the $\nu^+$ invariant and prove some of its useful properties.

Furthermore we will show how it can be used to obstruct the existence of cobordisms between algebraic knots.

I will report on joint work with Michael Weiss (https://arxiv.org/pdf/1503.00498.pdf):

Let K be a subset of a smooth manifold M. In some cases, functor calculus methods lead to a homotopical formula for M \ K in terms of the spaces M \ S, where S runs through the finite subsets of K. This is for example the case when K is a smooth compact sub manifold of co-dimension greater or equal to three.

A cornerstone of the study of mapping class groups is the

Nielsen--Thurston classification theorem. I will outline a

polynomial-time algorithm that determines the Nielsen--Thurston type and

the canonical curve system of a mapping class. Time permitting, I shall

describe a polynomial-time algorithm to compute the quotient orbifold of

a periodic mapping class, and I shall discuss the conjugacy problem for

the mapping class group. This is joint work with Mark Bell.

Decorate knot cobordisms functorially induce maps on knot Floer homology.

We compute these maps for elementary cobordisms, and hence give a formula for

the Alexander and Maslov grading shifts. We also show a non-vanishing result in the case of

concordances and present some applications to invertible concordances.

This is joint work with Marco Marengon.

We characterize cutting arcs on ber surfaces that produce new ber surfaces,

and the changes in monodromy resulting from such cuts. As a corollary, we

characterize band surgeries between bered links and introduce an operation called

generalized Hopf banding. We further characterize generalized crossing changes between

bered links, and the resulting changes in monodromy.

This is joint work with Matt Rathbun, Kai Ishihara and Koya Shimokawa

In this introduction for topologists, we explain the role that extensions of L-infinity algebras by taking homotopy fibers plays in physics. This first appeared with the work of physicists D'Auria and Fre in 1982, but is beautifully captured by the "brane bouquet" of Fiorenza, Sati and Schreiber which shows how physical objects such as "strings", "D-branes" and "M-branes" can be classified by taking successive homotopy fibers of an especially simple L-infinity algebra called the "supertranslation algebra". We then conclude by describing our joint work with Schreiber where we build the brane bouquet out of the homotopy theory of an even simpler L-infinity algebra called the superpoint.

The dimension of a finite-dimensional vector space V can be computed as the trace of the identity endomorphism id_V. This dimension is also the value F_V(S^1) of the circle in the 1-dimensional field theory F_V associated to the vector space. The trace of any endomorphism f:V-->V can be interpreted as the value of that field theory on a circle with a defect point labeled by the endomorphism f. This last invariant makes sense even when the vector space is infinite-dimensional, and gives the trace of a trace-class operator on Hilbert space. We introduce a 2-dimensional analog of this invariant, the `2-trace'. The 2-dimension of a finite-dimensional separable k-algebra A is the dimension of the center of the algebra. This 2-dimension is also the value F_A(S^1 x S^1) of the torus in the 2-dimensional field theory F_A associated to the algebra. Given a 2-endomorphism p of the algebra (that is an element of the center), the 2-trace of p is the value of the field theory on a torus with a defect point labeled by p. Generalizations of this invariant to other defect configurations make sense even when the algebra is not finite-dimensional or separable, and this leads to a general notion of 2-trace class and 2-trace in any 2-category. This is joint work with Andre Henriques.