6-7 May 2017.

UBC, Vancouver,

For enquiries, email Ben Williams at tbjw+58cascade@math.ubc.ca.

Schedule and Location

Map

All talks will take place in MATX 1100 in the UBC Mathematics Annex building on the main UBC Vancouver campus. A map is here. Coffee, when available, will be in MATX 1102 (in the same building).

Schedule

Saturday 6 May Sunday 7 May
9-10 Carbohydrates & Coffee 9:30-10:30 Mona Merling
10-11 Jonathan Beardsley 10:30-11 Coffee
11:30-12:30 Henry Adams 11-12 Thomas Koberda
Lunch
2:30-3:30 Akhil Mathew
4-5 Eugene Gorsky

Talks

Metric reconstruction via optimal transport

Given a metric space $X$ and a distance threshold $r > 0$, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of $X$ of diameter less than $r$. A theorem of Jean-Claude Hausmann states that if $X$ is a Riemannian manifold and $r$ is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Janko Latschev proves an analogous theorem for sufficiently dense samplings. Little is known about the behavior of Vietoris-Rips complexes for larger values of $r$, even though these complexes arise naturally when using persistent homology. We describe how as $r$ increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. These homotopy types are related to cyclic and centrally symmetric polytopes and orbitopes. We argue that infinite Vietoris-Rips complexes should be equipped with a different topology: an optimal transport or Wasserstein metric thickening the metric on $X$. With this new metric, we describe the first change in homotopy type (as $r$ increases) of Vietoris-Rips complexes of higher-dimensional spheres. Joint work with Michal Adamaszek and Florian Frick.

Jonathan Beardsley

University of Washington

Iterated Quotients of Ring Spectra and Spectral Torsors

I'll describe a method of taking iterated quotients of ring spectra (or any $E_n$-monoid in a stable, presentable infinity category) by actions of $E_n$-monoidal Kan complexes and show that in the presence of a fibration of $E_n$-monoidal Kan complexes one can take recognizable intermediate quotients. This will allow us to think of classical cobordism spectra as “spectral torsors.” If there is time I will describe a conjectural connection to Koszul duality.

Eugene Gorsky

University of California, Davis

Knot invariants and Hilbert schemes

I will discuss recent results and conjectures relating knot invariants (such as HOMFLY-PT polynomial and Khovanov-Rozansky homology) to algebraic geometry of Hilbert schemes of points on the plane. All notions will be introduced in the talk, no preliminary knowledge is assumed. This is a joint work with Andrei Negut and Jacob Rasmussen.

Thomas Koberda

University of Virginia

Square roots of Thompson's group $F$

I will discuss square roots of Thompson's group $F$, which are certain two-generator subgroups of the homeomorphism group of the interval, the squares of which generate a copy of Thompson's group $F$. We prove that these groups may contain nonabelian free groups, they can fail to be smoothable, and can fail to be finitely presented. This represents joint work with Y. Lodha.

Akhil Mathew

Harvard University

Polynomial functors and algebraic $K$-theory

The Grothendieck group $K_0$ of a commutative ring is well-known to be a λ-ring: although the exterior powers are non-additive, they induce maps on $K_0$ satisfying various universal identities. The λ-operations yield homomorphisms on higher $K$-groups. In joint work in progress with Glasman and Nikolaus, we give a general framework for such operations. Namely, we show that the $K$-theory space is naturally functorial for polynomial functors, and describe a universal property of the extended $K$-theory functor. This extends an earlier algebraic result of Dold for $K_0$. In this picture, the λ-operations come from the strict polynomial functors of Friedlander-Suslin.

Mona Merling

Johns Hopkins University

Equivariant $A$-theory

I will describe joint work with Cary Malkiewich on equivariant $A$-theory. I will describe an approach which is related the bivariant $A$-theory of Williams and another approach which we expect encodes information about equivariant pseudo-isotopies of $G$-manifolds.

Accommodation

Housing Options

Housing at UBC during the academic term is regrettably expensive, although the exchange rate between the US Dollar and the Canadian goes some way toward offsetting this.

• West Coast Suites are the on-campus hotel option.
• St John's College is a residential college on campus with a limited number of guest rooms. Please be sure to mention the Cascade Topology Seminar if attempting to book one of them.
• The Carey Centre is another on-campus option.
• St Andrew's College offers summer accommodation at reasonable rates. Be warned that there is a two-night minimum stay.
• Off campus options are also available. There is an excellent bus network in Vancouver, and UBC is 40 minutes by bus from downtown Vancouver