N+11th Southern California Topology Colloquium

During their distinguished careers, Francis Bonahon and Erica Flapan have been prominent figures in the Southern California topology circle. We wish to thank them for their service to the community in the past 35 years. All talks will be on this Zoom link (Meeting ID: 486 698 4535, Passcode: SCTC2021). All times displayed are Pacific Daylight Time (PDT).

There will be a virtually party following the conference. If you wish to say something at the party (either live or prerecorded), please contact Ko Honda.

Titles and Abstracts

David Gabai, Knotted 3-balls in the 4-sphere

Abstract: (Joint work with Ryan Budney) We give examples of codimension-1 knotting in the 4-sphere, i.e. there are 3-balls B_1 with boundary the standard 2-sphere, which are not isotopic rel boundary to the standard 3-ball B_0. In fact isotopy classes of such balls form a group which is infinitely generated. The existence of knotted balls implies that there exists inequivalent fiberings of the unknot in the 4-sphere, in contrast to the situation in dimension-3. Also, that there exists diffeomorphisms of S^1 x B^3 homotopic rel boundary to the identity, but not isotopic rel boundary to the identity.

Sheel Ganatra, Categorical non-properness in wrapped Floer theory

Abstract: In all known explicit computations on exact (Liouville) manifolds, the self-wrapped Floer homology of non-compact exact connected Lagrangian is always either infinite-dimensional or zero. We will explain why a global variant of this observed phenomenon necessarily holds: the wrapped Fukaya category of any Weinstein manifold must be either non-proper or zero. We will also describe a perspective on the proof in terms of certain "properness obstruction" invariants of the wrapped Fukaya category, which can be related to closed and open-string versions of Rabinowitz Floer theory (the latter by joint work in progress with Y. Gao and S. Venkatesh).

Ciprian Manolescu, Khovanov homology and the search for exotic 4-spheres

Abstract: A well-known strategy to disprove the smooth 4D Poincare conjecture is to find a knot that bounds a disk in a homotopy 4-ball but not in the standard 4-ball. Freedman, Gompf, Morrison and Walker suggested that Rasmussen’s invariant from Khovanov homology could be useful for this purpose. I will describe three recent results about this strategy: that it fails for Gluck twists (joint work with Marengon, Sarkar and Willis); that an analogue works for other 4-manifolds (joint work with Marengon and Piccirillo); and that 0-surgery homeomorphisms provide a large class of potential examples (joint work with Piccirillo). In particular, I will show 5 topologically slice knots such that if any of them were slice, then an exotic 4-sphere would exist.

Giuseppe Martone, Counting and Equidistribution for cusped Hitchin representations

Abstract: Hitchin representations provide a large class of discrete subgroups of the Lie group SL(d,R) which coincide with the holonomies of hyperbolic metrics on a closed surface when d=2. These representations can be studied via dynamical, representation theoretic, and complex geometric methods.

We develop a dynamical theory of Hitchin representations on surfaces with cusps. Our interest is in part motivated by classical results of Masur and McMullen on the Weil-Petersson metric for the Teichmuller space and its metric completion, and by Bridgeman, Canary, Labourie and Sambarino's introduction of a (pressure) metric for the space of Hitchin representations.

We show that cusped Hitchin representations are encoded by (well-behaved) potentials on a (well-behaved) countable Markov shift. We use Kessebohmer and Kombrink extension of Lalley's renewal theorem to prove equidistribution and counting results for this class of potentials and countable Markov shifts.

This talk is based on joint work with Harry Bray, Dick Canary, and Nyima Kao.

Danielle O’Donnol, Unknotting theta-curves and DNA replication

Abstract: Replication is when a single DNA molecule is reproduced to form two new identical DNA molecules. In the middle of replication a more complex structure is formed. When a circular piece of DNA is replicated the intermediate structure is that of a theta-graph. A theta-graph has two vertices and three edges between them. This talk will focus on unknotting numbers of embedded theta-graphs and understanding DNA replication.

Helen Wong, Topological descriptions of protein folding

Abstract: Knotting in proteins was once considered exceedingly rare. However, systematic analyses of solved protein structures over the last two decades have demonstrated the existence of many deeply knotted proteins, and researchers now hypothesize that the knotting presents some functional or evolutionary advantage for those proteins. Unfortunately, little is known about how proteins fold into knotted configurations. In this talk, we approach this problem from a theoretical point of view, using topological techniques. In particular, based on computational and experimental evidence, we propose a new theoretical pathway for proteins to form knots. We then use topological techniques to compare the configurations obtained from the theoretical pathways with known configurations of actual proteins. This is joint work with Erica Flapan and Adam He.

Organizers:

Ko Honda (UCLA)

Aaron Lauda (University of Southern California)

Yi Ni (California Institute of Technology)

Helen Wong (Claremont McKenna College)