All talks are in Skiles 006.

* This is a colloquium talk

Lightning talks are 5 minute talks given by junior mathematicians to expose there work to a wide audience. We have two lightning talk sessions. One will is during lunch on Saturday and one is be between the two regular talks on Sunday.

Lightning talks session 1:

Slides for most of the lightning talks.

Lightning talks session 2:

Slides for most of the lightning talks.

Titles and Abstracts

Corrin Clarkson
Title: Heegaard Floer homology and 3-manifold mutation
Abstract: The mutation operation is easy to define: cut a 3-manifold along an embedded surfaces and gluing it back together. It is also familiar; special cases include Dehn surgery and the construction of mapping tori. In many cases, the effects of mutation are significant and easily detected, but some gluings are more subtle. One such subtle gluing map is the genus-2 hyperelliptic involution. This is the only higher genus mapping class whose mutations might preserve the total rank of Heegaard Floer homology. I will show that all other gluings can change the total rank of HF-hat and give and overview of our understanding of the exceptional case: mutating by the genus two hyper elliptic involution.

Neil Fullarton
Title: Cohomology of principal congruence subgroups of the mapping class group
Abstract: Recently, Church-Farb-Putman showed that the rational cohomology of the mapping class group is zero at its virtual cohomological dimension. We show that the level p principal congruence subgroup of the mapping class group of a genus g surface has non-zero rational cohomology at this dimension, for all primes p and g at least 2. To show this, we use Bieri-Eckmann duality, and a combinatorial description of the top dimensional homology group of the curve complex. This is joint work with Andrew Putman.

Jen Hom
Title: Symplectic four-manifolds and Heegaard Floer homology
Abstract: We give new constraints on the topology of symplectic four-manifolds using invariants from Heegaard Floer homology. In particular, we will prove that certain simply-connected four-manifolds with positive-definite intersection forms cannot admit symplectic structures. This is joint work with Tye Lidman.

Ko Honda
Title: Semi-global Kuranishi charts and the definition of contact homology
Abstract: In this talk I will explain how to define the full contact homology algebra for any contact manifold and show that it is an invariant of the contact manifold. Our approach uses a simplified version of Kuranishi perturbation theory, consisting of ``semi-global'' Kuranishi charts (for example, if a relevant moduli space of holomorphic maps is compact, then we only need one chart). This is joint work in progress with Erkao Bao.

Allison Moore
Title: Cosmetic surgery in L-spaces and nugatory crossings
Abstract: A classic problem in knot theory is the cosmetic crossing conjecture, which asserts that the only crossing changes which preserve the isotopy class of a knot are nugatory crossing changes. Previously, the knots known to satisfy this conjecture included two-bridge and fibered knots. We will show that knots with branched double covers that are L-spaces also satisfy the cosmetic crossing conjecture, provided that the first singular homology of the branched double cover decomposes into summands of square-free order. The proof relies on the surgery characterization of the unknot, a tool coming from Floer homology, along with the G-equivariant Dehn's Lemma. We'll also discuss some applications. This is joint work with Lidman.

Dylan Thurston
Title: Discrete measured foliations
Abstract: We introduce a notion of harmonic discrete measured foliation on a filling, elastic graph on a surface. This gives a discrete analogue of harmonic measured foliations and holomorphic quadratic differentials, and in particular a system of coordinates on measured foliations. Unlike other representations (e.g., train tracks), this gives a single uniform system of coordinates for all measured foliations on a closed surface. It is also useful for computationally approximating actual harmonic measured foliations.

Karen Vogtmann
Title: Moduli of graphs
Abstract: Finite metric graphs are used to describe many phenomena in mathematics and science, so we would like to understand the space of all such graphs, which is called the moduli space of graphs. This space is stratified by subspaces consisting of graphs with a fixed number of loops and leaves. These strata generally have complicated structure that is not at all well understood. For example, Euler characteristic calculations indicate a huge number of nontrivial homology classes, but only a very few have actually been found. I will discuss the structure of these moduli spaces, including recent progress on the hunt for homology based on joint work with Jim Conant, Allen Hatcher and Martin Kassabov.

Funda Gultepe
Title: Spheres, tori and outer automorphisms of the free group
Abstract: In the 3 dimensional 'model' for Out(F_n) we will discuss spheres, tori and how we can generate certain outer automorphisms of the free group using them.

David Krcatovich
Title: Optimal cobordisms between knots
Abstract: A smooth cobordism between knots is optimal if its genus is the difference between the slice genera of the two knots. These arise, for example, between knots which are intersections of the same algebraic curve in C^2 with different 3-spheres. We give an obstruction to the existence of optimal cobordisms using the Upsilon invariant from Heegaard Floer homology. This is joint work with Peter Feller.

Bo Hyun Kwon
Title: Nontrivial examples of bridge trisection of knotted surfaces in S^4
Abstract: Currently, I found interesting exams which are knotted surfaces obtained from (4,1)-bridge trisections. I hope that they are not unknotted surfaces. Also, there are interesting nontrivial examples which have (5,2,2,1)-bridge trisections.

Justin Lanier
Title: Generating mapping class groups with torsion elements
Abstract: Brendle and Farb showed that the mapping class group of a surface of genus at least three is generated by six involutions. Monden showed that the same mapping class groups are generated by three elements of order 3 and are also generated by four elements of order 4. We extend these results to elements of arbitrary order.

Ash Lightfoot
Title: Link maps in the 4-sphere
Abstract: The linking number decides if a two-component link in the 3-sphere is link homotopically trivial. It is an open problem whether Kirk's $\sigma$-invariant decides if a link map of two 2-spheres in the 4-sphere is link homotopically trivial. We describe past and recent work on the problem.

Ziva Myer
Title: An A-infinity structure for Legendrians from Generating Families
Abstract: The study of legendrian submanifolds is central to the field of contact topology, and invariants of these submanifolds have been obtained through a variety of techniques. There are remarkable parallels between legendrian invariants constructed through the two seemingly different techniques of generating families and pseudoholomorphic curves. I will give an overview of how I have been extending these parallels by defining an A-infinity product structure through Morse flow trees of generating families; this corresponds to the A-infinity product structure that has been defined through pseudoholomorphic curves by Etnyre, Sabloff, et al.

Huygens Ravelomanana
Title: Exceptional cosmetic surgeries on S^3
Abstract: Two distinct Dehn surgeries on the same Knot are called cosmetic if they produces homeomorphic 3-manifolds. The knot cosmetic surgery problem asks if cosmetic surgeries do exist. I will present a result on this problem for the case of hyperbolic knots in S^3.

Leona Sparaco
Title: Character Varieties of 2-Bridge Knot Complements
Abstract: The set of all homomorphisms from the fundamental group of a hyperbolic 3-manifold into SL_2(C) is an algebraic set. We can look at the characteristics of this algebraic set, such as the number of irreducible components and the geometric genus of each component, in order to better understand the manifold. In this talk we will look at the character varieties of a class of 2-bridge knot complements.

Balazs Strenner
Title: Penner’s conjecture
Abstract: In 1988, Penner conjectured that all pseudo-Anosov mapping classes arise up to finite power from a construction named after him. This conjecture was known to be true on some simple surfaces, including the torus, but has otherwise remained open. I will sketch the proof (joint work with Hyunshik Shin) that the conjecture is false for most surfaces.

Faramarz Vafaee
Title: Knots in S^1 x S^2 with L-space surgeries
Abstract: We briefly go over the properties of knots in S^1 x S^2 with L-space surgeries. More specifically, if a knot K in an L-space admits an S^1 x S^2 surgery, then K is rationally fibered, that is, the complement of K admits a fibration over the circle. Moreover, the induced contact structure on the ambient manifold is tight. This work is joint with Yi Ni.

Shida Wang
Title: Semigroups of L-space cable knots and the Upsilon function
Abstract: We will introduce the semigroup of algebraic knots and generalize it to L-space itrated torus knots. As an application, we will use the Upsilon function recently defined by Ozsvath, Stipsicz and Szabo to give genus bounds for cobordisms between such knots.

Becca Winarski
Title: The lifting mapping class group of a superelliptic cover
Abstract: The mapping class group of a surface S is the group of isotopy classes of homeomorphisms of S. In topology and algebraic geometry, branched covering spaces are widely used to understand the structure, subgroups, homology, and representations of the mapping class groups of surfaces. Let X -> S be a branched cover of surfaces. It is useful to study the subgroup of the mapping class group of S consisting of elements that have representatives that lift to homeomorphisms of X. We find a presentation for this lifting subgroup of the mapping class group of S for a family of superelliptic covers of the sphere.