All talks will be in Skiles 006.

* This will be a colloquium talk.

Danny Calegari
Title: Roots, Schottky semigroups, and a proof of Bandt's Conjecture
Abstract: In 1985, Barnsley and Harrington defined a "Mandelbrot Set" M for pairs of similarities — this is the set of complex numbers z with norm less than 1 for which the limit set of the semigroup generated by the similarities x -> zx and x -> z(x-1)+1 is connected. Equivalently, M is the closure of the set of roots of polynomials with coefficients in {-1,0,1}. Barnsley and Harrington already noted the (numerically apparent) existence of infinitely many small "holes" in M, and conjectured that these holes were genuine. These holes are very interesting, since they are "exotic" components of the space of (2 generator) Schottky semigroups. The existence of at least one hole was rigorously confirmed by Bandt in 2002, but his methods were not strong enough to show the existence of infinitely many holes; one difficulty with his approach was that he was not able to understand the interior points of M, and on the basis of numerical evidence he conjectured that the interior points are dense away from the real axis. We introduce the technique of *traps* to construct and certify interior points of M, and use them to prove Bandt's Conjecture. Furthermore, our techniques let us certify the existence of infinitely many holes in M. This is joint work with Sarah Koch and Alden Walker.

James Conway
Title: Transverse Surgery in Contact 3-Manifolds
Abstract: Much ink has been spilled on surgery on Legendrian knots; much less well studied is surgery on transverse knots. We will investigate transverse surgery, how it relates to surgery on Legendrian knots, and see how it interacts with open book decompositions, the Heegaard Floer contact invariant, and tightness. This investigation leads to the result that surgery on the connected binding of a genus g open book supporting a tight contact structure preserves tightness if the surgery coefficient is greater than 2g-1. In a complementary direction, we partially generally a result of Lisca and Stipsicz to give criteria for when positive contact surgery on Legendrian knots will result in an overtwisted manifold.

Laura DeMarco
Title: Complex dynamics, elliptic curves, and topology
Abstract: In the classical theory of elliptic curves, circa 1960, it was shown that an elliptic curve defined over a function field has only finitely many rational torsion points. This statement was recently reproved and generalized by Matt Baker to the setting of complex dynamical systems; specifically, to the study of rational maps on P^1. In this talk, I'll present a topological perspective on Baker's theorem and the Mordell-Weil theorem for function fields, and I will explain how these statements play a central role in today's study of rational maps.

Nathan Dunfield
Title: Random knots: their properties and algorithmic challenges
Abstract: I will discuss various models of random knots in S3, surveying what is known about them theoretically and what is conjectured about them experimentally. In particular, I will discuss experiments that probe the practical/average case complexity of questions like computing the genus of a knot.

Chris Leininger
Title: Free-by-cyclic groups and dynamics
Abstract: A 3-manifold that fibers over the circle often does so in many ways. The combined work of Thurston, Fried, and McMullen provides a framework for organizing all the monodromies and analyzing them simultaneously. With Dowdall and Kapovich, we provide a similar organizational tool for the "algebraic monodromies" of certain free-by-cyclic groups and relate this to the BNS invariant (which was already known to have close ties to Thurston's work). I'll describe this, explain some contrasts with the 3-manifold setting, and give an application to dynamics of free group automorphisms.

Kathryn Mann
Title: Left orderable groups that don't act on the line
Abstract: The study of left-orderable groups has deep connections to geometry, topology and dynamics. A well known and very useful tool is the fact that all countable L.O. groups embed in the group of homeomorphisms of the line. Uncountable L.O. groups act on ordered spaces too, but whether they act on the line is a complicated question.

In this talk, I'll present a natural (though perhaps surprising!) example of a L.O. group that does not act on the line. The proof is entirely self-contained, using one sophisticated fact -- on the group cohomology of Homeo(S^1) -- but otherwise a very hands-on approach.

Laura Starkston
Title: Relations in mapping class groups and embeddings of pseudoholomorphic curves
Abstract: For certain contact 3-manifolds supported by a planar open book decomposition, there are two ways of constructing and classifying symplectic fillings whose boundary is that contact 3-manifold. One way involves understanding factorizations of the monodromy of the open book into positive Dehn twists. The other way is to look at embeddings of concave neighborhoods of a collection of surfaces into a well understood (rational) symplectic 4-manifold. Depending on the bounding contact 3-manifold, each of these methods has different strengths and data from one method can provide information about the other. Therefore it is very useful to understand how to translate the information coming from pseudoholomorphic curve embeddings to mapping class group relations or vice versa. The goal of this talk will be to discuss some correspondences between these two methods in large classes of examples, where the boundary 3-manifold is Seifert fibered over the 2-sphere.