Coffee and Refreshments 8:15 - 8:30 in Park 361
Talk 8:30 - 9:45 in Park 328
|Tuesday September 18, 2018||
|Using 2-torsion to obstruct topological isotopy||
Two knots in S^3 are ambiently isotopic if and only if there is an orientation preserving automorphism of S^3 carrying one knot to the other. In this talk, we will examine a family of smooth 4-manifolds in which the analogue of this fact does not hold, i.e. each manifold contains a pair of smoothly embedded, homotopic 2-spheres that are related by a diffeomorphism, but not smoothly isotopic. In particular, the presence of 2-torsion in the fundamental groups of these 4-manifolds can be used to obstruct even a topological isotopy between the 2-spheres; this shows that Gabai's recent ``4D Lightbulb Theorem" does not hold without the 2-torsion hypothesis.
September 25, 2018
|(No meeting due to PATCH on Friday September 28)|
Friday September 28, 2018
(University of Texas at Austin)
Mahler measure and the Vol-Det Conjecture
Characterizing slopes for hyperbolic and torus knots
A basic open problem is to understand how the hyperbolic volume of knots and links is related to diagrammatic knot invariants. The Vol-Det Conjecture relates the volume and determinant of alternating links. We prove the Vol-Det Conjecture for infinite families of alternating links using the dimer model, the Mahler measure of 2-variable polynomials, and the hyperbolic geometry of biperiodic alternating links. This is joint work with Abhijit Champanerkar and Matilde Lalin.
In the background talk (11:00 AM, in room 617), we will review some classical ways to get geometric invariants of alternating links, and then generalize these ideas to study the geometry of biperiodic alternating links.
Given a knot KK in S3S3, we say that p/qp/q is a characterizing slope if the oriented homeomorphism type of the p/qp/q-surgery on KK is sufficient to uniquely determine the knot KK. It is known that for a given torus knot all but finitely many non-integer slopes are characterizing and that for hyperbolic knots all but finitely many slopes with q>2q>2 are characterizing. I will discuss the proofs of both results, which have a surprising amount in common.
In the background talk, (at 9:30am), I will give an overview of Dehn surgery and some basic 3-manifold topology concepts that will appear in the main talk.
October 2, 2018
|Jason Cantarella (Univ. of Georgia)||The symplectic volume of polygon space, Fourier analysis, orthoschemes, and a puzzle.||Kapovich and Millson were among the first to discover that polygons with fixed edgelengths in Euclidean space carried a symplectic structure. In fact, these polygon spaces are (almost always) toric symplectic manifolds. Computing the symplectic volume of polygon space then became a natural problem in symplectic geometry, accomplished in various ways by Takakura, Khoi, and Mandini, among others. By the Duistermaat-Heckmann theorem, this reduces to the problem of computing the volume of the moment polytope, which can be done in various ways.
In this talk, we'll give two new ways to compute the volume of the moment polytope: an explicit simplicial decomposition using orthoschemes (due to Kyle Chapman) which reduces the problem to combinatorics, and an integral representation (due to the presenter and coauthors) inspired by Fourier analysis.
The last part of the story leads to a puzzle: the integral representation of the volume formula for closed polygons is clearly formally related to the integral representation for the volume formula for open polygons. This must somehow reflect the fact that closed polygons are a symplectic reduction of open polygons. But how?
This talk represents joint work with Clayton Shonkwiler (mathematics, Colorado State University), Bertrand Duplantier (physics, ITP-Saclay), Tetsuo Deguchi and Erica Uehara (physics, Ochanomizu University).
The PACT Seminar is funded by the Mellon Tri-College Faculty Forum Program and hosts research talks on a broad range of topics from contact/symplectic topology, low-dimensional topology, and algebraic topology. The seminar is jointly organized by Professor Thomas Hunter (Swarthmore), Professor Paul Melvin (Bryn Mawr), Professor Josh Sabloff (Haverford), and Professor Lisa Traynor (Bryn Mawr). We are a friendly and mathematically engaged group composed of members of mathematics departments around the Philadelphia area including Bryn Mawr, Haverford, Swarthmore, Penn, Temple, Eastern, and Widener.
Goals for the seminar are:
· To explore new developments in contact, symplectic, and low-dimensional topology;
· To encourage collaboration between participants, and to find entry points for such collaboration through the exchange of ideas and works in progress;
· To involve advanced undergraduate and graduate students in ongoing research in a seminar format; and
· To share, support, and critique work in progress.
The PACT seminar typically meets weekly (Tuesday mornings!) to enable detailed discussions of the participants’ own work or topics of particular interest to the participants. The seminar is usually conducted in a “mini-course” format, in which one member typically spends three to six sessions explaining research ideas in more depth than is feasible during a one-time seminar, and with more of an emphasis on underlying ideas than can easily be gleaned from a formal paper.
The PACT seminar meets on Bryn Mawr's beautiful campus, which is easy to reach from any of Philadelphia's campuses by car or train. The talk is from 8:30 - 9:45 a.m. in Park 328, and is preceded by coffee and breakfast refreshments at 8:15 in the math lounge, Park 361.