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Geometry and Topology Seminar

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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.

Monday 8/29
Peter Samuelson (University of Edinburgh): The Kauffman bracket and double affine Hecke algebras
Abstract: The Kauffman bracket skein module of a 3-manifold M is a q-deformation of the ring of functions on the SL_2 character variety of M. It turns out that the skein module of T^2 x [0,1] is the double affine Hecke algebra, and this algebra has a second deformation parameter t. We present a conjecture relating this second parameter to skein modules of knot complements, and also relate this conjecture to a conjecture of Brumfiel and Hilden. This is joint work with Yuri Berest.
Monday 9/5
University holiday
Monday 9/12
Chris Tuffley (Massey University, New Zealand): Intrinsic linking in higher dimensions, and with linking numbers divisible by q
Abstract: In 1983 Conway and Gordon proved that every embedding of the complete graph K_6 in R^3 contains a pair of disjoint cycles that form a non-separable link -- a fact that is expressed by saying K_6 is intrinsically linked. Since then, a number of authors have shown that embeddings of larger complete graphs necessarily exhibit more complicated linking behaviour, such as links with many components and/or large pairwise linking numbers.
With some adaptions to the proofs, similar results can be established for embeddings of large n-complexes in R^{2n+1}. We will look at some of the adaptions required, in the context of proving the existence of two component links with linking number a nonzero multiple of a given integer q. In the course of this we will obtain an improved bound for n>0 on the number of vertices needed to force a two component link
with linking number at least q in absolute value.
Monday 9/19
No seminar this week, but see Colloquium on Wednesday
Monday 9/26
Matt Hogancamp (USC): Stability phenomena in link homology
Abstract: In topology it often happens that cohomology of a stable limit is simpler than the cohomology of its finite approximations.  A classic example is cohomology of Grassmannian of k dimensional planes in C^n, in the limit as n goes to infinity.  In this talk I will report on similar phenomena in link homology.  In 2015 I showed that the triply graded homology of torus links stabilizes, and that the stable limit is surprisingly simple: it is a super polynomial ring.  More recently, Michael Abel and I showed that that there is a *second* stable limit, which we computed explicitly.  In both cases, our results verify stable versions of conjectures of Gosky-Negut-Rasmussen, which relate the homology of torus knots with flag Hilbert schemes.   I hope also to address the category theoretic reason that underlies the formal similarities between various such stablity phenomena.
Monday 10/3
Ramanujan Santharoubane (University of Virginia): Quotient of surface groups and homology of finite covers via Topological Quantum Field Theories
Abstract: I will show how we can produce interesting representations of surface groups from the Witten-Reshetikhin-Turaev TQFT. The key fact is the following : these "quantum representations" of surface groups have infinite images but every simple loop acts with finite order. Using this key fact and integral TQFT, we will build regular finite covers of surfaces where the integral homology is not generated by pullbacks of simple closed curves on the base. This is joint work with Thomas Koberda.
Monday 10/10
Bahar Acu (USC): On finite energy foliations of contact manifolds by planar J-holomorphic curves
Abstract: In this talk, we will explain how to foliate a special class of higher dimensional contact manifolds (odd dimensional manifolds equipped with a “maximally non-integrable hyperplane field”) with planar J-holomorphic curves and show that, by using this kind of technology, one can prove the long-standing Weinstein conjecture for that class.
Monday 10/17
Caltech, Sloan 151
Note time and location
Joint LA Topology seminar at Caltech, Sloan 151
4:00. Hongbin Sun (UC Berkeley): NonLERFness of arithmetic hyperbolic manifold groups
Abstract: We will show that, for "almost" all arithmetic hyperbolic manifolds with dimension >3, their fundamental groups are not LERF. The main ingredient in the proof is a study of certain graph of groups with hyperbolic 3-manifold groups being the vertex groups. We will also show that a compact irreducible 3-manifold with empty or tori boundary does not support a geometric structure if and only if its fundamental group is not LERF.
5:00. Sucharit Sarkar (UCLA): Equivariant Floer homology
Abstract: Given a Lie group G acting on a symplectic manifold preserving a pair of Lagrangians setwise, I will describe a construction of G-equivariant Lagrangian Floer homology. This does not require G-equivariant transversality, which allows the construction to be flexible. Time permitting, I will talk about applying this for the O(2)-action on Seidel-Smith's symplectic Khovanov homology. This is joint with Kristen Hendricks and Robert Lipshitz.
 
Tuesday 10/18
12-1PM, KAP 427
Note unusual day and location
Brian Collier (University of Maryland): Connected components of SO(n,n+1) surface group representations
Abstract: Since Higgs bundles were introduced, they have been an effective tool for parameterizing connected components the character variety of representations of  a surface group into a Lie group G. Generalizing Hitchin's results for SO(1,2)=PSL(2,R) and his description of the Hitchin component, we will parameterize many new connected components of the SO(n,n+1) character variety. Once these new components have been described we will discuss the geometry of the representations. In particular, we will give good evidence that they consist entirely of Anosov representations. Time permitting, we will see what aspects generalize to SO(n,m).
Monday 10/24
Brice Loustau (Rutgers University): Generalized Weil-Petersson metrics on character varieties
Abstract: I will present the construction of a (hyper-)Kähler metric on the character variety X(pi, G) associated to a closed surface group pi = pi_1(S) and a semisimple algebraic group G. This metric generalizes both the Weil-Petersson metric on Teichmüller space T(S) and the Hitchin metric on the moduli space of G-Higgs bundles. The idea of this construction is due to Andrew Sanders (in a forthcoming paper) and I have been studying the properties of this metric with him.
Monday 10/31
Joint LA Topology seminar at USC
Kasra Rafi (University of Toronto and MSRI):
Shape of the moduli space
Abstract: The thick part of moduli space is compact and its rich complex geometry if is often ignored while studying the coarse
geometry of Teichmüller space. We examine the systole function on moduli space which is known to be a morse function and
we will show that is has many local maxima. We will also review some older results and discuss some open questions. 
Tian Yang (Stanford University): Volume conjectures for Reshetikhin-Turaev and Turaev-Viro invariants
Abstract: In a joint work with Qingtao Chen we conjecture that, at the root of unity exp(2πi/r) instead of the root exp(πi/r) usually considered, the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic 3-manifold grow exponentially, with growth rates respectively connected to the hyperbolic and complex volume of the manifold. This reveals an asymptotic behavior of the relevant quantum invariants that is different from that of Witten's invariants (which grow polynomially by the Asymptotic Expansion Conjecture), and may indicate a geometric interpretation of the Reshetikhin-Turaev invariants that is different the SU(2) Chern-Simons gauge theory. Recent progress toward these conjectures will be summarized, including joint work with Renaud Detcherry and Effie Kalfagianni. 
Monday 11/7
LATop seminar at UCLA
Matt Hogancamp (USC):
Burak Ozbagci (Koc University):

Tuesday 11/8, 12:00 - 12:50 in KAP XXX.
Note unusual day and location
Valentina Disarlo (Indiana University and MSRI): On the geometry of the flip graph
Abstract: The flip graph of an orientable punctured surface is the graph whose vertices are the ideal triangulations of the surface (up to isotopy) and whose edges correspond to flips. Its combinatorics is crucial in works of Thurston and Penner’s decorated Teichmuller theory. In this talk we will explore some geometric properties of this graph, in particular we will see that it provides a coarse model of the mapping class group in which the mapping class groups of some subsurfaces are convex. We will also establish upper and lower bounds on the growth of the diameter of the flip graph modulo the mapping class group, extending a result of Sleator-Tarjan-Thurston. This is a joint work with Hugo Parlier.
Monday 11/14
Jeremy Toulisse (USC):The geometry of maximal representations in SO(2,3)
Abstract: Given a closed oriented surface S of genus at least 2, the notion of maximal representations in SO(2,3) naturally generalizes the Teichmueller space of S. In this talk, I will explain how to construct a unique maximal surface into the pseudo-hyperbolic space H^{2,2} which is preserved by the action of a maximal representation. It generalizes in particular a recent result of Labourie. If time permits, I will explain how to extend the construction to maximal representations in SO(2,n). This is a joint work with Brian Collier and Nicolas Tholozan.
Monday 11/21

Monday 11/28

Monday 12/5



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