Program

Friday, November 11

  • 14:45-15:45 == Jenna Rajchgot (McMaster)
  • 15:45-16:15 == Coffee Break
  • 16:15-17:15 == Egor Shelukhin (Université de Montréal)

Saturday, November 12

  • 9:00-10:00 == Duncan McCoy (UQAM)
  • 10:00-10:30 == Coffee Break
  • 10:30-11:30 == Sébastien Picard (UBC)
  • 13:30-14:30 == Jean-Philippe Burelle (Sherbrooke)
  • 14:45-15:45 == Catherine Pfaff (Queen’s)
  • 15:45-16:15 == Coffee Break
  • 16:15-17:15 == Alexander Kupers (Toronto)

Sunday, November 13

  • 9:00-10:00 == Martin Frankland (Regina)
  • 10:00-10:30 == Coffee Break
  • 10:30-11:30 == Keegan Boyle (UBC)

Talks

Keegan Boyle

Title: Involutions on the 4-sphere and SNAcKs

I will start with a brief overview of what is known about involutions on the 4-sphere. For example, it is unknown whether every involution with a fixed-point set of S0 or S1 is conjugate to an isometry. I will then discuss joint work with Wenzhao Chen in which we develop an equivariant version of the Conway polynomial for certain symmetric knots (strongly negative amphi-chiral knots, or SNAcKs). One of our results leads to the construction of involutions on the 4-sphere with a fixed-point set S1, which we believe are not conjugate to an isometry.

Jean-Philippe Burelle

Title: Symplectic Schottky groups and crooked surfaces

The simplest examples of convex-cocompact subgroups in Isom(Hn)=PO(n,1) are obtained by a Ping-pong construction and called Schottky groups. Their explicit nature makes it easy to compute their limit set and domain of discontinuity in the boundary of Hn, and to describe the quotient manifold obtained by taking the quotient of this domain. In higher rank, the dynamics of a discrete group playing Ping-pong are more complicated, and the “right” generalization of convex-cocompact subgroups is the notion of Anosov subgroups. In joint work with Fanny Kassel, we describe how to construct projective Anosov Schottky subgroups in real symplectic groups which have domains of discontinuity in the Lagrangian Grassmannian. For these groups, we also construct fundamental domains bounded by finitely many hypersurfaces and count the number of connected components of their moduli spaces.

Martin Frankland

Title: Multiparameter persistence modules in the large scale

A persistence module with m discrete parameters is a diagram of vector spaces indexed by the poset Nm. If we are only interested in the large scale behavior of such a diagram, then we can consider two diagrams equivalent if they agree outside of a “negligeable” region. In the 2-dimensional case, we classify the indecomposable diagrams up to finitely supported diagrams. In higher dimension, we partially classify the indecomposable diagrams up to suitably finite diagrams.

Along the way, we classify the tensor closed Serre subcategories of the category of finitely generated m-parameter persistence modules: they are in bijection with the simplicial complexes on m vertices. This is joint work with Don Stanley.

Alexander Kupers

Title: Embedding calculus and moduli spaces of manifolds

Embedding calculus assigns to a manifold M the data of the collection of configuration spaces of open balls in M and natural maps between these. The Goodwillie-Klein-Weiss convergence results say that you can recover embeddings from their action on this data, as long as the codimension is at least 3. I will explain how this may be used to obtain information about moduli spaces of manifolds, and present some novel convergence results in low dimensions. This includes joint work with Manual Krannich and Mauricio Bustamante.

Duncan McCoy

Title: The search for alternating surgeries

Dehn surgery is an operation where one constructs a 3-manifold by taking a knot in the 3-sphere, cutting out a tubular neighbourhood and then gluing in another solid torus. We say that a Dehn surgery is an “alternating surgery” if it produces a manifold which arises as the double branched cover of an alternating link. I will try to justify why alternating surgeries are interesting and explain some of what is known about them. In particular, I will discuss the existence of an algorithm to calculate all possible alternating surgeries on a given knot and describe the results of implementing such an algorithm. This is joint work with Ken Baker and Marc Kegel.

Catherine Pfaff

Title: Typical Trees: An Out(Fr) Excursion

Answering questions posed by Bestvina, Handel, and Mosher, we prove which properties are held by random outer automorphisms of free groups and points in the boundary of Culler-Vogtmann outer space. This is joint work with Ilya Kapovich (Hunter College), Joseph Maher (CUNY), and Samuel J. Taylor (Temple University).

Sébastien Picard

Title: Non-Kahler Transitions of Calabi-Yau Threefolds

It was proposed in the works of Clemens, Reid and Friedman to connect Calabi-Yau threefolds with different topologies by a process which degenerates 2-cycles and introduces new 3-cycles. This operation may produce a non-Kahler complex manifold with trivial canonical bundle. In this talk, we will discuss the geometrization of this process by special non-Kahler metrics. This is joint work with T.C. Collins and S.-T. Yau.

Jenna Rajchgot

Title: Symmetric quivers and symmetric varieties

Since the 1980s, mathematicians have found connections between orbit closures in type A quiver representation varieties and Schubert varieties in type A flag varieties. For example, singularity types appearing in type A quiver orbit closures coincide with those appearing in Schubert varieties in type A flag varieties; combinatorics of type A quiver orbit closure containment is governed by Bruhat order on the symmetric group; and formulas for classes of type A quiver orbit closures in torus equivariant cohomology and K-theory can be expressed in terms of Schubert polynomials, Grothendieck polynomials, and other objects from Schubert calculus.

After recalling some of this story, I will discuss the related setting of H. Derksen and J. Weyman’s symmetric quivers and their representation varieties. I will show how one can adapt results from the ordinary type A setting to unify aspects of the equivariant geometry of type A symmetric quiver representation varieties with Borel orbit closures in corresponding symmetric varieties G/K (G = general linear group, K = orthogonal or symplectic group). This is ongoing joint work with Ryan Kinser and Martina Lanini.

Egor Shelukhin

Title: Persistent homology of eigenfunctions

We present recent joint work with Buhovsky, Payette, Polterovich, Polterovich, and Stojisavljevic which proves and applies new properties of persistence modules in order to obtain new results about the eigenfunctions of the Laplacian and their linear combinations. In particular we prove coarse versions of the Courant theorem for all Betti numbers and of Bézout’s theorem in this context.