2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021
Current contact: Dave Futer or Matthew Stover
The Seminar usually takes place on Wednesdays at 2:45 PM in Room 617 on the sixth floor of Wachman Hall.
Ivan Izmestiev, FU Berlin, Variational properties of the discrete Hilbert-Einstein functional
The discrete Hilbert-Einstein functional (also known as Regge action) for a 3-manifold glued from euclidean simplices is the sum of edge lengths multiplied with angular defects at the edges. There is an analog for hyperbolic cone-manifolds; a discrete total mean curvature term appears if the manifold has a non-empty boundary. Variational properties of this functional are similar to those of its smooth counterpart. In particular, critical points correspond to vanishing angular defects, i.e. to metrics of constant curvature. We give a survey on isometric embeddings and rigidity results that can be obtained by studying the second derivative of the discrete Hilbert-Einstein and speak about possible future developments.
Jason Behrstock, CUNY, Curve complexes for cube complexes
For \(CAT(0)\) cubical groups we develop analogues of tools which have played a key role in the study of the mapping class group, namely, the theory of curve complexes and subsurface projections. We will describe these parallel structures and also some new results that can be proven as a result of this new approach. This is joint work with Mark Hagen and Alessandro Sisto.
PATCH seminar, joint with Bryn Mawr, Haverford, and Penn
Robert Young, Courant Institute, Filling multiples of embedded curves
Filling a curve with an oriented surface can sometimes be "cheaper by the dozen". For example, L. C. Young constructed a smooth curve drawn on a projective plane in \(\mathbb{R}^n\) which is only about 1.5 times as hard to fill twice as it is to fill once and asked whether this ratio can be bounded below. We will use methods from geometric measure theory to answer this question and pose some open questions about systolic inequalities for surfaces embedded in \(\mathbb{R}^n\).
PATCH seminar, joint with Bryn Mawr, Haverford, and Penn
Elisenda Grigsby, Boston College, (Sutured) Khovanov homology and representation theory
Khovanov homology associates to a link \(L\) in the three-sphere a bigraded vector space arising as the homology groups of an abstract chain complex defined combinatorially from a link diagram. It detects the unknot (Kronheimer-Mrowka) and gives a sharp lower bound (Rasmussen, using a deformation of E.S. Lee) on the 4-ball genus of torus knots.
When \(L\) is realized as the closure of a braid (or more generally, of a "balanced" tangle), one can use a variant of Khovanov's construction due to Asaeda-Przytycki-Sikora and L. Roberts to define its sutured Khovanov homology, an invariant of the tangle closure in the solid torus. Sutured Khovanov homology distinguishes braids from other tangles (joint with Ni) and detects the trivial braid conjugacy class (joint with Baldwin).
In this talk, I will describe some of the representation theory of the sutured Khovanov homology of a tangle closure. It (perhaps unsurprisingly) carries an action of the Lie algebra \(sl(2)\). More surprisingly, this action extends to the action of a slightly larger Lie superalgebra whose structure hints at a unification with the Lee deformation. This is joint work with Tony Licata and Stephan Wehrli.
Genevieve Walsh, Tufts University, Boundaries of Kleinian groups
A hyperbolic group is endowed with a topological space, its boundary, which is well-defined up to homeomorphism. We will discuss hyperbolic groups that have boundaries homeomorphic to the boundaries of different types of Kleinian groups. In particular, we will discuss the boundaries of a type of group which is built up from surface groups, graph-Kleinian groups. This is joint, preliminary work with Peter Haissinsky and Luisa Paoluzzi.
Joseph Maher, CUNY, Random walks on weakly hyperbolic groups
Let \(G\) be a group acting by isometries on a Gromov hyperbolic space, which need not be proper. If \(G\) contains two hyperbolic elements with disjoint fixed points, then we show that a random walk on \(G\) converges to the boundary almost surely. This gives a unified approach to convergence for the mapping class groups of surfaces, \(Out(F_n)\) and acylindrical groups. This is joint work with Giulio Tiozzo.
Brian Rushton, Temple University, Detecting large-scale invariants of infinite groups
Finitely presented groups can be studied geometrically by means of the Cayley graph. The geometry of the Cayley graph has a direct influence on the algebraic properties of the group; for instance, the growth rate of the graph determines if the group is nilpotent. However, it can be difficult to determine the geometric properties of the group. We show how subdivision rules and cube complexes can be used to calculate geometric invariants of infinite groups.
Natasa Sesum, Rutgers University, Ancient solutions in geometric flows
I will discuss ancient solutions in the context of the mean curvature flow, the Ricci flow and the Yamabe flow. I will discuss the classification result in the Ricci flow, construction result of infinitely many ancient solutions in the Yamabe flow. In the last part of the talk I will mention the most recent result about the unique asymptotics of non-collapsed ancient solutions to the mean curvature flow which is a joint work with Daskalopoulos and Angenent.
Joel Fish, Institute for Advanced Study, Symplectic topology, Hamiltonian flows, and invariant subsets: not just going in going in circles anymore
I will discuss some current joint work with Helmut Hofer in which we make use of symplectic topology and pseudoholmorphic curves to study properties of Hamiltonian flows on compact regular hypersurfaces of symplectic manifolds. In particular, I will show how pseudoholomorphic curve techniques can be used to prove that every non-empty, compact, regular energy surface in \(R^4\) has a trajectory which is not dense in the energy level.
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Christian Millichap, Temple University, Mutations and geometric invariants of hyperbolic 3-manifolds (thesis defense)
In this talk, we will examine how a topological cut and paste operation known as mutation can be used to create geometrically similar hyperbolic manifolds: they are non-isometric yet they have a number of geometric invariants in common. Ruberman has shown that this mutation process preserves the volume of a hyperbolic 3-manifold. Building off of his work, we show that mutations also preserve sufficiently short geodesic lengths. As a result, we are able to construct large classes of hyperbolic knot complements that have the same volume, the same shortest geodesic lengths, but are pairwise incommensurable, i.e., do not share a common finite sheeted cover.
Patricia Cahn, University of Pennsylvania, Knots transverse to a vector field
We study knots transverse to a fixed vector field \(V\) on a 3-manifold \(M\) up to the corresponding isotopy relation. We show this classification is particularly simple when \(V\) is the co-orienting vector field of a tight contact structure, or when \(M\) is irreducible and atoroidal. We also apply our results to study loose Legendrian knots in overtwisted contact manifolds, and generalize results of Dymara and Ding-Geiges. This work is joint with Vladimir Chernov.
Helen Wong, Carleton College, Representations of the Kauffman skein algebra
The Kauffman skein algebra of a surface was originally defined to be a straightforward generalization of the Kauffman bracket polynomial for knots. Only later was it realized as a quantization of the \(PSL(2,\mathbb{C})\) character variety of the surface. The Kauffman skein algebra thus emerged as an important connector between quantum topology and hyperbolic geometry. In this talk, we'll describe how to construct representations of the Kauffman skein algebra and how to construct invariants to help tell them apart. This is joint work with F. Bonahon.
Ailsa Keating, Columbia University, Lagrangian tori in four-dimensional Milnor fibres
The Milnor fibre of any isolated hypersurface singularity contains exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. I will explain how to construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four. Time allowing, I will use these to give examples of fibres whose Fukaya categories are not generated by vanishing cycles, and explain applications to mirror symmetry for those fibres.
Giulio Tiozzo, Yale University, Random walks and random group extensions
Let us consider a group \(G\) of isometries of a \(\delta\)-hyperbolic metric space \(X\), which is not necessarily proper (e.g. it could be a locally infinite graph). We can define a random walk by picking random products of elements of \(G\), and projecting this sample path to \(X\).
We show that such a random walk converges almost surely to the Gromov boundary of \(X\), and with positive speed.
As an application, we prove that a random k-generated subgroup of the mapping class group is convex cocompact, and a similar statement holds for \(Out(F_n)\).
This is joint work, partially with J. Maher and partially with S. Taylor.
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Priyam Patel, Purdue University, Separability Properties of Right-Angled Artin Groups.
Right-Angled Artin groups (RAAGs) and their separability properties played an important role in the recent resolutions of some outstanding conjectures in low-dimensional topology and geometry. We begin this talk by defining two separability properties of RAAGs, residual finiteness and subgroup separability, and provide a topological reformulation of each. We then discuss joint work with K. Bou-Rabee and M.F. Hagen regarding quantifications of these properties for RAAGs and the implications of our results for the class of virtually special groups.
Ser-Wei Fu, Temple University, The earthquake deformation of hyperbolic structures
Earthquakes are deformations of a hyperbolic surface introduced by Thurston as generalized Dehn twists. I will describe the earthquake flow on moduli space and discuss some dynamical properties. In particular, there is a cusp excursion result for the once-punctured torus that can be obtained by methods in the study of logarithm laws.
Renato Bettiol, University of Pennsylvania, On the Singular Yamabe Problem on Spheres
The solution to the Yamabe problem of finding a constant scalar curvature metric in a prescribed conformal class on a closed manifold was a major achievement in Geometric Analysis. Among several interesting generalizations to open manifolds, great attention has been devoted to the so-called "singular Yamabe problem". Given a closed Riemannian manifold \(M\) and a submanifold \(S\), this problem consists of finding a complete metric on the complement of \(S\) in \(M\) that has constant scalar curvature and is conformal to the original metric. In other words, these are solutions to the Yamabe problem on \(M\) that blow up along \(S\). A particularly interesting case is the one in which \(M\) is a round sphere and \(S\) is a great circle. In this talk, I will describe how bifurcation techniques and spectral theory of hyperbolic surfaces can be used to prove the existence of uncountably many nontrivial solutions to this problem. This is based on joint work with B. Santoro and P. Piccione.
Saul Schleimer, University of Warwick, Recognizing three-manifolds
Through the eyes of a topologist, manifolds have no local properties: every point has a small neighborhood that looks like Euclidean space. Accordingly, as initiated by Poincaré, the classification of manifolds is one of the central problems in topology. The ``homeomorphism problem'' is somewhat easier: given a pair of manifolds, we are asked to decide if they are homeomorphic.
These problems are solved for zero-, one-, and two-manifolds. Even better, the solutions are ``effective'': there are complete topological invariants that we can compute in polynomial time. On the other hand, in dimensions four and higher the homeomorphism problem is logically undecidable.
This leaves the provocative third dimension. Work of Haken, Rubenstein, Casson, Manning, Perelman, and others shows that theseproblems are decidable. Sometimes we can do better: for example, if one of the manifolds is the three-sphere then I showed that the homeomorphism problem lies in the complexity class NP. In joint work with Marc Lackenby, we show that recognizing spherical space forms also lies in NP. If time permits, we'll discuss the standing of the other seven Thurston geometries.
Will Worden, Temple University, Hidden symmetries and commensurability of 2-bridge link complements
The canonical triangulations and symmetry groups of 2-bridge link complements are well understood and relatively easy to describe. We exploit this fact to show that non-arithmetic 2-bridge link complements have no hidden symmetries (i.e., symmetries of a finite cover that do not descend to symmetries of the link complement itself), and are pairwise incommensurable. Much of the talk will focus on understanding 2-bridge links, the canonical triangulations of their complements, and their symmetry groups. From there we will give a sketch of the proof that hidden symmetries do not exist, and touch on the question of pairwise incommensurability.
This is joint work with Christian Millichap.
Amos Nevo, Technion, Actions of arithmetic groups and effective Diophantine approximation
We will describe some recent new developments in Diophantine approximation on algebraic varieties, focusing on some familiar natural examples. The approach we describe utilizes harmonic analysis and ergodic theory on semisimple Lie groups, and provides the best possible solution to many Diophantine approximation problems which were not accessible by previous techniques.
Based on joint work with Alex Gorodnik and on joint work with Anish Ghosh and Alex Gorodnik.
Anastasiia Tsvietkova, UC Davis, The number of surfaces of fixed genus in an alternating link complement
Let \(L\) be a prime alternating link with \(n\) crossings. We show that for each fixed \(g\), the number of genus \(g\) incompressible surfaces in the complement of \(L\) is bounded by a polynomial in \(n\). Previous bounds were exponential in \(n\). This is joint work with Joel Hass and Abigail Thompson.
PATCH seminar, joint with Bryn Mawr, Haverford, and Penn
Effie Kalfagianni, Michigan State University, Non-orientable knot genus and the Jones polynomial.
The non-orientable genus (a.k.a crosscap number) of a knot is the smallest genus over all non-orientable surfaces spanned by the knot. In this talk, I'll describe joint work with Christine Lee, in which we obtain two-sided linear bound of the crosscap number of alternating link in terms of the Jones link polynomial. The bounds are often exact and they allow us to compute the crosscap numbers of infinite families of alternating knots as well as the crosscap number of 283 knots with up to twelve crossings that were previously unknown. Time permitting, we will also discuss generalizations to families of non-alternating links.
The proofs of the results use techniques from angled polyhedral decomposition of 3-manifolds, normal surface theory, and the geometry of augmented links. The background talk, by Jessica Purcell, will explain some of these tools and techniques.
PATCH seminar, joint with Bryn Mawr, Haverford, and Penn
Tye Lidman, Institute for Advanced Study, Floer homology and symplectic four-manifolds.
Floer homology is a powerful technique in many areas of geometric topology, such as symplectic geometry and three-manifold topology. In the background talk, I will discuss the formal structure of this invariant, as well as its relationships with other objects in low-dimensional topology, including symplectic four-manifolds.
Symplectic manifolds are pervasive objects in geometric topology which often give rise to the construction of exotic smooth four-manifolds. We give some new constraints on the topology of symplectic four-manifolds using invariants from Heegaard Floer homology. This is joint work with Jen Hom.
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Balazs Strenner, Institute of Advanced Study, Construction of pseudo-Anosov maps and a conjecture of Penner
There are many constructions of pseudo-Anosov elements of mapping class groups of surfaces. Some of them are known to generate all pseudo-Anosov mapping classes, others are known not to. In 1988, Penner gave a very general construction of pseudo-Anosov mapping classes, and he conjectured that all pseudo-Anosov mapping classes arise this way up to finite power. This conjecture was known to be true on some simple surfaces, including the torus, but has otherwise remained open. In this talk I prove that the conjecture is false for most surfaces. (This is joint work with Hyunshik Shin.)
Anja Bankovic, Boston College, Marked length spectral rigidity for flat metrics
In this talk we will introduce the set of non-positively curved Euclidean cone metrics on closed surfaces and explore the lengths of curves in those metrics. We will introduce the techniques we used to show that two such metrics assigning the same lengths to all closed curves differ by an isometry isotopic to the identity and give the idea of the proof. This is joint work with Chris Leininger.
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Daryl Cooper, UC Santa Barbara, Finite-volume hyperbolic 3-manifolds contain immersed quasi-Fuchsian surfaces
will discuss a proof that a complete, non-compact hyperbolic 3- manifold M with finite volume contains an immersed, closed, quasi-Fuchsian surface. Joint with Mark Baker.
Jonah Gaster, Boston College, Lifting curves simply
It is a corollary of a celebrated theorem of Scott that every closed curve on a hyperbolic surface \(X\) has a simple lift in a finite cover. In order to discuss a quantitative version of this statement, let the `degree' of a curve be the minimal degree of such a cover. We show: If \(X\) has no punctures, then the maximum degree among curves of length at most \(L\) is coarsely equal to (with constants depending only on the topology of \(X\) the quotient of \(L\) by the length of a systole of \(X\). Time permitting, we will discuss related questions, partial answers, and work in
Abigail Thompson, UC Davis and IAS, Surgery on fibered knots.
It is a classical result that any closed orientable 3-manifold can be obtained by an operation called surgery on a link in the 3-sphere. The link may have many components. This leads to a natural question: Which 3-manifolds can be obtained by surgery on a knot (i.e. on a 1-component link)? And on which knots? For example, Gordon and Luecke showed that non-trivial surgery on a non-trivial K can't yield the 3-sphere back again.
Which knots have surgeries yielding a lens space? A conjecture of Gordon is that only certain knots, called Berge knots, have such a surgery. The pool of potential counter-examples to this conjecture is slowly diminishing. I'll describe some of what is known so far, and show that some fibered knots can't have lens space surgeries. This is work in progress.
Charles Livingston, Indiana University, Heegaard Floer knot homology and its applications.
In 2001, Peter Ozsvath and Zoltan Szabo developed Heegaard Floer theory. Using HF theory, one can associate to each knot K in \(R^3\) a chain complex, CFK(K). From a 3-dimensional perspective, CFK(K) determines the genus of a knot and whether or not it is fibered; from a 4-dimensional perspective, it offers strong constraints on the surfaces the knot can bound in upper 4-space. Applications include new results concerning the classification of complex algebraic curves.
As an algebraic object, CFK(K) has multiple structures: it is a chain complex, it is graded and bifiltered, and it is a module over a polynomial ring. I will begin this talk with a simple example that clarifies the details of these structures. I will then illustrate how, from CFK(K), one can extract a variety of knot invariants. Finally, I will describe families of knots for which the computation of CFK(K) follows from a simple algorithm.
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Andrew Yarmola, Boston College, Basmajian's identity in higher Teichmuller-Thurston theory
We demonstrate an extension of Basmajian's identity to Hitchin representations of compact bordered surfaces. For 3-Hitchin representations, we show that this identity has a geometric interpretation for convex real projective structures analogous to Basmajian's original result. As part of our proof, we demonstrate that the limit set of an incompressible subsurface of a closed surface has measure zero in the Lebesgue measure on the Frenet curve associated to an n-Hitchin representation. This generalizes a classical result in hyperbolic geometry. Finally, we recall the Labourie-McShane extension of the McShane-Mirzakhani identity to Hitchin representations and note a close connection to Basmajian's identity in both the hyperbolic and the Hitchin setting. This is joint work with Nicholas G. Vlamis.
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