# Spring 2017

 Monday 1/9 Vincent Alberge (Fordham University): Extremal length geometry on Teichmüller spaces Abstract:  Given an oriented closed surface S of genus g ≥ 2, its Teichmüller space denoted by T(S) classifies conformal structures on S. It is endowed with a canonical metric (the Teichmüller metric) which is related to an important conformal invariant, the extremal length. This invariant allows us to define a compactification of Teichmüller space, called the Gardiner-Masur compactification. In this talk, we will deal with the horocyclic deformation, a conformal analogue of the Fenchel-Nielsen deformation, and we prove that in some cases these conformal deformations converge towards the Gardiner-Masur boundary. Time permitting, we will introduce the reduced Gardiner-Masur boundary and discuss its relationship with the so-called null-set lamination space. Monday 1/16 University holiday Monday 1/23 3:30 - 4:30 Please note unusual time Sheel Ganatra (USC): Floer theory and critical values of complex-valued functions Abstract: Given a (tame at infinity) holomorphic function on a complex variety with isolated (but possibly degenerate) singularities, what is the minimum number of critical values of any tame complex deformation (with isolated, but also possibly degenerate singularities)? We prove there is a symplectic Floer/Fukaya-theoretic lower bound for the answer, as conjectured by Seidel and previously studied in the non-degenerate singularity case. Using this bound, one can find examples of varieties and holomorphic functions having arbitrarily large numbers of critical values that cannot be holomorphically deformed away, even though they smoothly can. Monday 1/30 No seminar this week. Monday 2/6 No seminar this week. Monday 2/13 Inkang Kim (Korean Institute for Advanced Study): Gromov simplicial volume and bounded cohomology Abstract: Gromov introduced a topological volume both for compact and non-compact manifolds, called Gromov simplicial volume. The basic question is which manifolds admit non-zero simplicial volume. We address this issue for non-compact manifolds using bounded cohomology and barycenter methods. Monday 2/20 University holiday Monday 2/27 Sergio Fenley (Florida State University): Free Seifert fibered pieces of pseudo-Anosov flows Abstract: We prove a structure theorem for pseudo-Anosov flows restricted to Seifert fibered pieces of three manifolds. The piece is called periodic if there is a Seifert fibration so that a regular fiber is freely homotopic, up to powers, to a closed orbit of the flow. A non periodic Seifert fibered piece is called free. In this talk we consider free Seifert pieces. We show that, in a carefully defined neighborhood of the free piece, the pseudo-Anosov flow is orbitally equivalent to a hyperbolic blow up of a geodesic flow piece. A geodesic flow piece is a finite cover of the geodesic flow on a compact hyperbolic surface, usually with boundary. We introduce almost k-convergence groups, and an associated convergence theorem. We also introduce an alternative model for the geodesic flow of a hyperbolic surface that is suitable to prove these results, and we define what is a hyperbolic blow up. This is joint work with Thierry Barbot. Monday 3/6 Francis Bonahon (USC): Generalizing (X+Y)^n = X^n + Y^n Abstract: The convenient formula (X+Y)^n = X^n + Y^n is (unfortunately) frequently used by our calculus students. Our better students know that this relation does hold in some special cases, for instance in characteristic n or when YX=qXY with q a primitive n-root of unity.  I will discuss similar miraculous cancellations’’ in the context of the quantum group U_q(sl_2). Monday 3/13 4:00 - 6:00 PM Joint LATop seminar at UCLA, MS 6627 Mark Hughes (Brigham Young University): Neural networks and knot theory Abstract: In recent years neural networks have received a great deal of attention due to their remarkable ability to detect subtle and very complex patterns in large data sets. They have become an important machine learning tool and have been used extensively in many fields, including computer vision, fraud detection, artificial intelligence, and financial modeling. Knots in 3-space and their associated invariants provide a rich data set (with many unanswered questions) on which to apply these techniques. In this talk I will describe neural networks, and outline how they can be applied to the study of knots in 3-space. Indeed, these networks can be applied to answer a number of algebraic and geometric problems involving knots and their invariants. I will also outline how neural networks can be used together with techniques from reinforcement learning to construct explicit examples of slice and ribbon surfaces for certain knots. John Etnyre (Georgia Tech): Embeddings of contact manifolds Abstract: I will discuss recent results concerning embeddings and isotopies of one contact manifold into another. Such embeddings should be thought of as generalizations of transverse knots in 3-dimensional contact manifolds (where they have been instrumental in the development of our understanding of contact geometry). I will mainly focus on embeddings of contact 3-manifolds into contact 5-manifolds. In this talk I will discuss joint work with Ryo Furukawa aimed at using braiding techniques to study contact embeddings. Braided embeddings give an explicit way to represent some (maybe all) smooth embeddings and should be useful in computing various invariants. If time permits I will also discuss other methods for embedding and constructions one may perform on contact submanifolds. Monday 3/20 Ian Le (Perimeter Institute): Higher Laminations and Affine Buildings Abstract: Higher Teichmüller spaces are a component of the character variety for a topological surface S and groups like SL_n(R). These spaces have a parameterization by cluster coordinates, and these cluster coordinates have a natural tropicalization. This leads to the tropicalization of higher Teichmüller space, which can be considered as a generalization of laminations which we call higher laminations. I will show that higher laminations can be realized via actions on affine buildings. This generalizes the association of classical laminations to R-trees. I will emphasize how tropical geometry reflects the piecewise-linear metric geometry of the affine building. I will try to show that higher laminations are concrete and computable objects, and I will draw analogies between the cases of SL_n, n > 2, and the classical case where n=2. Monday 3/27 Priyam Patel (UC Santa Barbara): Algebraic and topological properties of big mapping class groups Abstract: The mapping class group of a surface is the group of homeomorphisms of the surface up to isotopy (a natural equivalence). Mapping class groups of finite type surfaces have been extensively studied and are, for the most part, well-understood. There has been a recent surge in studying surfaces of infinite type and in this talk, we shift our focus to their mapping class groups, often called big mapping class groups. The groups arise naturally when studying group actions on surfaces (dynamics) and foliations of 3-manifolds. In contrast to the finite type case, there are many open questions regarding the basic algebraic and topological properties of big mapping class groups. Until now, for instance, it was unknown whether or not these groups are residually finite. We will discuss the answer to this and several other open questions after providing the necessary background on surfaces of infinite type. This work is joint with Nicholas G. Vlamis. Monday 4/3 Catherine Pfaff (UC Santa Barbara): Outer automorphism of free group invariants and the behavior of geodesics in outer space Abstract: Outer automorphism groups of free groups are studied via their action on Culler-Vogtmann Outer Space. We study in particular the interplay between outer automorphism singularity invariants and the behavior of geodesics in Outer space. This is joint work with Yael Algom-Kfir and Ilya Kapovich. Wednesday 4/5 Please note unusual day Yu Pan (Duke University): Augmentations and Exact Lagrangian cobordisms. Abstract: To a Legendrian knot, one can associate an $A_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between  two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of  objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials. Monday 4/10 Joint LATop seminar at USC Julien Paupert (Arizona State U.): Rank 1 deformations of non-cocompact hyperbolic lattices Abstract: Let X be a negatively curved symmetric space and Gamma a noncocompact lattice in Isom(X).  We show that small, parabolic-preserving deformations of Gamma into the isometry group of any negatively curved symmetric space containing X remain discrete and faithful (the cocompact case is due to Guichard).  This applies in particular to a version of Johnson-Millson bending deformations,  providing for all n infnitely many noncocompact lattices in SO(n,1) which admit discrete and faithful deformations into SU(n,1). We also produce deformations of the figure-8 knot group into SU(3,1), not of bending type, to which the result applies.This is joint work with Sam Ballas and Pierre Will. Oleg Lazarev (Stanford):  Contact manifolds with flexible fillings Abstract: In this talk, I will show that all flexible Weinstein fillings of a given contact manifold have isomorphic integral cohomology. As an application, in dimension at least 5 any almost contact class that has an almost Weinstein filling has infinitely many exotic contact structures. Using similar methods, I will also construct the first known infinite family of almost symplectomorphic Weinstein domains whose contact boundaries are not contactomorphic. These results are proven by studying Reeb chords of loose Legendrians and positive symplectic homology. Monday 4/17 4-6PM, Sloan 151 Note location Joint LATop seminar at Caltech. Duncan McCoy (U. Texas): Characterizing slopes for torus knots Abstract: We say that p/q is a characterizing slope for a knot K in the 3-sphere if the oriented homeomorphism type of p/q-surgery is sufficient to determine the knot K uniquely. I will discuss the problem of determining which slopes are characterizing for torus knots, paying particular attention to non-integer slopes. This problem is related to the question of which knots in the 3-sphere have Seifert fibered surgeries. Steven Frankel (Yale University): Calegari's conjecture for quasigeodesic flows Abstract: We will discuss two kinds of flows on 3-manifolds: quasigeodesic and pseudo-Anosov. Quasigeodesic flows are defined by a tangent condition, that each flowline is coarsely comparable to a geodesic. In contrast, pseudo-Anosov flows are defined by a transverse condition, where the flow contracts and expands the manifold in different directions. When the ambient manifold is hyperbolic, there is a surprising relationship between these apparently disparate classes of flows. We will show that a quasigeodesic flow on a closed hyperbolic 3-manifold has a coarsely contracting-expanding transverse structure, a generalization of the strict transverse contraction-expansion of a pseudo-Anosov flow. This behavior can be seen at infinity, in terms of a pair of laminar decompositions of a circle, which we use to proof Calegari's conjecture: every quasigeodesic flow on a closed hyperbolic 3-manifold can be deformed into a pseudo-Anosov flow. Wednesday 4/19 11:00 AM, KAP 245 Please note unusual time and day Neslihan Gügümcü (National Technical University of Athens): Knotoids and some of their invariants Abstract: The theory of knotoids was introduced by V.Turaev in 2012, and extends the theory of classical knots.  A knotoid is defined as an open-ended knot diagram, that is, as the image of the unit interval in an oriented surface under an immersion with finitely many transversal double points endowed with under/overcrossing data and with two distinct endpoints; for knotoids we allow the endpoints to be in different regions of the diagram. In this talk, we will first go through the basic notions of knotoids in S^2 and R^2 and review basic notions from virtual knot theory. We will introduce the virtual closure map and show that it is a non-injective and non-surjective map. We then introduce some invariants of knotoids including the height of a knotoid, the affine index polynomial and the arrow polynomial. We show both the affine index polynomial and the arrow polynomial are used to determine the type of a knotoid. There is a corresponding theory so called the theory of braidoids. Braidoids are braid-like objects whose closures are knotoids. This is joint work with Lou Kauffman. Monday 4/24 Qionling Li (Caltech):  On the uniqueness of the coupled vortex equation Abstract: We study the uniqueness of a coupled vortex equation involving a holomorphic k-differential on the complex plane. As geometric applications, we show that there is a unique non-branching harmonic map from the complex plane into the hyperbolic 2-space with prescribed polynomial Hopf differential; there is a unique affine spherical immersion from the complex plane into the Euclidean 3-space with prescribed polynomial Pick differential. We also show that the uniqueness always fails for non-polynomial k-differentials with finite zeros.

# Fall 2016

This semester, the seminar usually meets on Mondays, from 4:30 to 5:30 in KAP 245.