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:

Lightning talks session 2:

During lunch on Saturday there will be a lightning talk session: a series of 5 minute talks on the research of junior mathematicians, given by the junior mathematicians themselves or by their mentors.

Titles and Abstracts

Matt Clay
Title: L2-torsion of free-by-cyclic groups
Abstract: I will provide an upper bound on the L2-torsion of a free-by-cyclic group, in terms of a relative train-track representative for the monodromy. This result shares features with a theorem of Luck-Schick computing the L2-torsion of the fundamental group of a 3-manifold that fibers over the circle in that it shows that the L2-torsion is determined by the exponential dynamics of the monodromy. In light of the result of Luck-Schick, a special case of this bound is analogous to the bound on the volume of a 3-manifold that fibers over the circle with pseudo-Anosov monodromy by the normalized entropy recently demonstrated by Kojima-McShane.

David Futer
Title: Geometrically similar knots and 3-manifolds
Abstract: This talk is motivated by the following question: how well do geometric invariants (such as hyperbolic volume and the length of geodesics) distinguish knot complements, or more general 3-manifolds?

There are several known ways to produce hyperbolic 3-manifolds that are isospectral (i.e. have exactly the same spectrum of geodesic lengths) but not isometric. All known constructions of of this sort involve finite covers of the same base manifold, leading Reid to ask whether this is a necessary feature. That is, are isospectral manifolds necessarily commensurable? I will describe a way to build pairs of hyperbolic manifolds (including knot complements) that are incommensurable but have the same closed geodesics up to length L, where L is as large as one likes. This is joint work with Christian Millichap.

Eli Grigsby
Title: Annular Khovanov-Lee homology, Braids, and Cobordisms
Abstract: Khovanov homology associates to a knot K in the three-sphere a bigraded vector space arising as the homology groups of an abstract chain complex. Using a deformation of Khovanov's complex, due to Lee, Rasmussen defined an integer-valued knot invariant he called s(K) that gives a lower bound on the 4-ball genus of knots, sharp for knots that can be realized as quasipositive braid closures.

On the other hand, when K is a braid closure, its Khovanov complex can itself be realized in a natural way as a deformation of a triply-graded complex, defined by Asaeda-Przytycki-Sikora, further studied by L. Roberts, and now known as the (sutured) annular Khovanov complex.

In this talk, I will describe joint work with Tony Licata and Stephan Wehrli aimed at understanding an annular version of Lee's deformation of the Khovanov complex. In particular, we obtain a family of real-valued braid conjugacy class invariants generalizing Rasmussen's "s” invariant that give bounds on the Euler characteristic of smoothly-imbedded surfaces in the thickened solid torus as well as information about the associated mapping class of the punctured disk. The algebraic model for this construction is the recently-defined Upsilon invariant of Ozsvath-Stipsicz-Szabo.

Caitlin Leverson
Title: Legendrian contact homology and normal rulings
Abstract: Given a plane field $ker(dz-ydx)$ in $\mathbb{R}^3$, a Legendrian knot is a knot which is tangent to the plane at every point. One can similarly define a Legendrian knot in any contact $3$-manifold. In this talk, we will explore Legendrian knots in different contact manifolds. In particular, we will look at the relationship between two Legendrian knot invariants, normal rulings and augmentations of the Legendrian contact homology differential graded algebra.

Sara Maloni
Title: Polyhedra inscribed in quadrics and their geometry
Abstract: In 1832 Steiner asked for a characterization of polyhedra which can be inscribed in quadrics. In 1992 Rivin answered in the case of the sphere, using hyperbolic geometry. In this talk, I will describe the complete answer to Steiner's question, which involves the study of interesting analogues of hyperbolic geometry including anti de Sitter geometry. Time permitting, we will also discuss future directions in the study of convex hyperbolic and anti de Sitter manifolds. (This is joint work with J. Danciger and J.-M. Schlenker.)

Lisa Traynor
Title: A Quantitative Look at Lagrangian Cobordisms
Abstract: Lagrangian cobordisms between Legendrian submanifolds arise in Relative Symplectic Field Theory. In recent years, there has been much progress on answering qualitative questions such as: For a fixed pair of Legendrians, does there exist a Lagrangian cobordism? I will address two quantitative questions about Lagrangian cobordisms: For a fixed pair of Legendrians, what is the minimal "length" of a Lagrangian cobordism? What is the relative Gromov width of a Lagrangian cobordism? Regarding length, I will give examples of pairs of Legendrians where Lagrangian cobordisms are flexible in that the non-cylindrical region can be arbitrarily short; I will also give examples of other pairs of Legendrians where Lagrangian cobordisms are rigid in that there is a positive lower bound to their length. For the second quantitative measure, I will give some calculations and estimates of the relative Gromov width of particular Lagrangian cobordisms. This is joint work with Joshua M. Sabloff.

Daniel Wise
Title: The Cubical Route to Understanding Groups
Abstract: Cube complexes have come to play an increasingly central role within geometric group theory, as their connection to right-angled Artin groups provides a powerful combinatorial bridge between geometry and algebra. This talk will introduce nonpositively curved cube complexes, and then describe the developments that have recently culminated in the resolution of the virtual Haken conjecture for 3-manifolds, and simultaneously dramatically extended our understanding of many infinite groups.

Bahar Acu
Title: The Weinstein conjecture for iterated planar contact structures
Abstract: The Weinstein conjecture asserts that the Reeb vector field of every contact form carries at least one closed orbit. The conjecture was proven for all closed 3-dimensional manifolds by Taubes. Despite considerable progress, it is still open in higher dimensions. In this talk, we show that (2n+1)-dimensional "iterated planar” contact manifolds satisfy the Weinstein conjecture.

Ian Banfield
Title: Khovanov homology and fibered strongly quasipositive braid closures
Abstract: I will discuss strongly quasipositive braid links, which are closures of positive words in the Birman-Ko-Lee generators of the braid group. Fibered links of this type are exactly the so-called tight links, which induce the tight contact structure on S^3. I will explain a simple criterion that ensures a strongly quasipositive braid is fibered; allude to its geometric interpretation and hint at exciting connections to Khovanov homology, L-Spaces, and an old conjecture of Murusugi-Przytycki.

Michael Dougherty
Title: The Milnor Fiber of the Braid Arrangement
Abstract: By a theorem of Milnor’s, the complement of any complex hyperplane arrangement is a fiber bundle over the circle. While we have known for some time that hyperplane complements have torsion-free homology, an analogous statement for their so-called Milnor fibers remained elusive. We show that the Milnor fiber for the braid arrangement - the most natural example of a hyperplane arrangement - has torsion in its homology. This is joint work with Jon McCammond.

Yen Duong
Title: Random groups and cubulations
Abstract: I’ll define random groups, show an example of Sageev’s wall construction for cubulation, and suggest how the two relate.

Neil Fullarton
Title: Top-dimensional cohomology in the mapping class group
Abstract: (Joint work with Andy Putman). A basic question about any group or space is: what are its (rational) cohomology groups? Putman and I found a vast amount of rational cohomology for principal congruence subgroups of the mapping class group, which I will discuss.

Funda Gultepe
Title: Calculating distance by twisting and projecting
Abstract: We will talk about ingredients of a partial distance formula for $Out(F_n)$ given by projecting arcs, curves and free factors and twisting along short conjugacy classes.

Kevin Kordek
Title: Picard groups of moduli spaces of Riemann surfaces with symmetry
Abstract: There is a well-known connection between moduli spaces of Riemann surfaces (geometry) and mapping class groups (topology). In this talk, I will show how to use symmetric mapping class groups to express properties of the Picard groups of moduli spaces of Riemann surfaces with an abelian group of automorphisms of a fixed topological type.

Ivan Levcovitz
Title: Divergence of CAT(0) Cube Complexes and Right-Angled Coxeter Groups
Abstract: The divergence function of a metric space, a quasi-isometry invariant, roughly measures the rate which pairs of geodesic rays stray apart. We will present new results regarding divergence functions of CAT(0) cube complexes. Right-angled Coxeter groups, in particular, exhibit a rich spectrum of possible divergence functions. The talk will give special focus to applications of our results to these groups. Applications to the theory of random right-angled Coxeter groups will also be briefly mentioned.

Mark Lowell
Title: When is a Knot Diagram Legendrian?
Abstract: We provide a method for determining if a knot diagram is combinatorially equivalent to the Lagrangian projection of a Legendrian knot, using techniques from linear programming.

Alex Moody
Title: Stein fillings of Legendrian surgeries with enough stabilizations
Abstract: After very briskly reviewing the background for the main problems in understanding Stein fillings of various contact 3-manifolds, we will state the solution to the geography problem for Stein fillings of Legendrian surgeries with enough stabilizations and ask the corresponding classification question.

Ziva Myer
Title: Algebraic Structures for Legendrian and Lagrangian Submanifolds with Generating Families
Abstract: In contact and symplectic geometry, there are different methods for constructing algebraic invariants for Legendrian and Lagrangian submanifolds. I will give an overview of some of my current and future research goals -- to build products and A-infinity algebras/categories using the Morse/Cerf theoretic tool of generating families to form moduli spaces of gradient flow trees.

Yu Pan
Title: Exact Lagrangian Fillings of Legendrian $(2,n)$ torus links
Abstract: For a Legendrian $(2,n)$ torus knot or link with maximal Thurston-Bennequin number, Ekholm, Honda, and K{\'a}lm{\'a}n constructed $C_n$ exact Lagrangian fillings, where $C_n$ is the $n$-th Catalan number. We show that these exact Lagrangian fillings are pairwise non-isotopic through exact Lagrangian isotopy. To do that, we compute the augmentations induced by the exact Lagrangian fillings $L$ to $\mathbb{Z}_2[H_1(L)]$ and distinguish the resulting augmentations.

Samantha Pezzimenti
Title: Immersed Lagrangian Fillings of Legendrian Knots
Abstract: The Poincaré polynomial associated to certain Legendrian knots with generating families provides an obstruction to the existence of embedded Lagrangian fillings. As a consequence of the Seidel Isomorphism, if the polynomial is not of the form 2-2g, the Legendrian cannot have an embedded Lagrangian filling. However, the existence of an immersed Lagrangian filling is not obstructed. In fact, any Legendrian knot with a generating family has an immersed Lagrangian filling. We investigate how this polynomial gives us information about the genus of the filling, the number of immersion points, and the indices of those immersion points.

Huygens Ravelomanana
Title: PSL_2(C) Character variety and Dehn surgeries
Abstract: Dehn surgery is a fundamental tool in 3-manifolds theory but it is not clear when two 3-manifolds obtained by Dehn surgeries with distinct slopes can be homeomorphic. I will discuss a result about the case of Seifert fibered surgeries on hyperbolic knots in Q-homology spheres which relates to PSL_2(C) Character variety of 3-manifolds.

Faramarz Vafaee
Title: The prism manifold realization problem
Abstract: A prism manifold is a quotient of the 3-sphere by the action of a central extension of a binary dihedral group. These form a two parameter family P(p,q) with p and q coprime integers and p positive. We determine which of the prism manifolds P(p,q) with q less than 0 are the result of integral Dehn surgery along knots in the 3-sphere. The methodology undertaken to obtain the classification is similar to that of Greene for lens spaces. This is joint work with W. Ballinger, C. Hsu, W. Mackey, Y. Ni, and T. Ochse.