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Current contacts: Dave Futer, Matthew Stover, or Sam Taylor.
The Seminar usually takes place on Wednesdays at 2:30 PM in Room 617 of Wachman Hall.
Motivated by counting problems for closed geodesics on hyperbolic surfaces, I will present a family of new results describing the dynamics of mapping class groups on Teichmuller spaces and spaces of closed curves of closed surfaces.
Random groups are one way to study "typical" behavior of groups. In the Gromov density model, we often find a threshold density above which a property is satisfied with probability 1, and below which it is satisfied with probability 0. Two properties of random groups that have been well studied are cubulation (or more generally, acting cocompactly on a CAT(0) cube complex without global fixed point) and Property (T). In this setting these are mutually exclusive properties, but the threshold densities are not known. In this talk I'll present a method to show that random groups with density less than 3/14 act on a CAT(0) cube complex, and discuss how this might be extended to densities up to 1/4. This extends results of Ollivier-Wise and Mackay-Przytycki at densities less than 1/5 and 5/24, respectively.
Given a codimension 1 foliation on a 3-manifold, we say that its leaf space has branching if the lift of the foliation to the universal cover fails to be a product foliation. I'll talk about some examples of this phenomenon, and explain how a detailed understanding of the branching of some foliations can help us produce left-orderings of fundamental groups.
There are two commonly-used presentations for the braid group. In Artin's original presentation, we linearly order the n strands and use n-1 half-twists between adjacent strands to generate the group. The dual presentation, defined by Birman, Ko, and Lee in 1998, introduces additional symmetry by using the larger generating set of all half-twists between any pair of strands. Each presentation has an associated cell complex which is a classifying space for the braid group: the Salvetti complex for the standard presentation and the dual braid complex for the dual presentation. In this talk, I will present a combinatorial perspective for complex polynomials which comes from the dual presentation and describe how this leads to a cell structure for the spaceof complex polynomials which arises from the dual braid complex. This is joint work with Jon McCammond.
PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
Abstract:
From an Anosov flow on a 3-manifold, one can extract an action of the fundamental group of the manifold on a plane preserving a pair of transverse foliations, and on a compactification of the plane by an ideal circle. My talks will give an introduction to this picture and show a recent application, joint with Thomas Barthelme and Steven Frankel on the classification problem for Anosov flows. By proving rigidity results about group actions on planes and circles, we show that transitive (pseudo-)Anosov flows are determined (up to orbit equivalence) by the algebraic data of the set of free homotopy classes of closed orbits.
In the morning background talk (at 9:30am),
PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
Abstract: Homotopy theory has proven to be a robust tool for studying non-homotopical questions about manifolds; for example, surgery theory addresses manifold classification questions using homotopy theory. In joint work with Sarah Yeakel, we are developing a program to study manifold topology via isovariant homotopy theory. I'll explain what isovariant homotopy theory is and how it relates to the study of manifolds via their configuration spaces, and talk about an application to fixed point theory.
In the morning background talk (at 11:00am), I will talk about
Thesis: Left-orderability of Dehn surgeries on knot complements
Date: Monday, April 4, 2022
Time: 3:30 p.m. - 4:30 p.m.
Place: 617 Wachman Hall, 6^{th} floor
All faculty are invited to attend the exam, ask questions and participate in the follow up discussions.
Tarik Aougab, Haverford College
Mirzakhani's beautiful work allows one to count closed geodesics in a specified orbit of the mapping class group on a hyperbolic surface. Later work of Erlandsson-Souto and Rafi-Soutoreproves these counting results while avoiding some of the most difficult aspects of Mirzakhani's proofs by recasting the problem as a convergence statement for a certain family of measures on the space of geodesic currents.We will follow this approach to count harmonic graphs, graphs that arise as the image of a harmonic map from a weighted graph into a hyperbolic surface. To do this, we define currents with corners, ageneralization of a geodesic current that allows for singularities which we think of as corresponding to the vertices of a graph. This represents joint work with Jayadev Athreya and Ryokichi Tanaka.
Daniel Woodhouse
A spectre is haunting Geometric Group Theory -- the spectre of a generalized Leighton's Graph Covering Theorem. The original theorem states that any two graphs with common universal cover have a common finite cover. Haglund conjectured that this should generalize to all compact special cube complexes. I will talk about recent progress on this, my own contributions alongside others. I will discuss the implications for quasi-isometric rigidity, and for hyperbolic groups in particular. I will give some conjectures and explain why they should be true and very loosely how (other people) will likely one day prove them.
Hannah Schwartz, Princeton University
PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
Abstract: Recent work of both Gabai and Schneiderman-Teichner on the smooth isotopy of homotopic surfaces with a common dual has reinvigorated the study of concordance invariants defined by Freedman and Quinn in the 90's, along with homotopy theoretic isotopy invariants of Dax from the 70's. We will outline, give context to, and discuss techniques used to prove these so called "light bulb theorems", and present new light bulb theorems for disks rather than spheres.
At 9:30am, there will be a background talk on picture-based geometric interpretations of the Freedman-Quinn and Dax invariants.
Franco Vargas Pallete, Yale University
PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
Abstract: It is a consequence of a well-known result of Ahlfors and Bers that the \(PSL(2, C)\) character associated to a convex co-compact hyperbolic 3-manifold is determined by its peripheral data. In this talk we will show how this map extends to a birational isomorphism of the corresponding \(PSL(2, C)\) character varieties, so in particular it is generically a 1-to-1 map. Analogous results were proven by Dunfield in the single cusp case, and by Klaff and Tillmann for finite volume hyperbolic 3-manifolds. This is joint work with Ian Agol.
At 11:00am, there will be a background talk on volume in hyperbolic geometry, including volume rigidity and the Bonahon-Schlafli formula.
When proving that geodesic triangles in a given space are delta-thin, the hardest step is often simply defining the geodesics. The guessing geodesics lemma allows us to skip that step entirely, by replacing true geodesics with paths that are "good enough". In this expository talk, I'll give a proof of this lemma and demonstrate its use in a very elegant proof about the curve complex of a surface.
Abstract:
Abstract:
Invariant random subgroups (IRS) are conjugation invariant probability measures on the space of subgroups in a given group G. They arise as point stabilizers of probability measure preserving actions. Invariant random subgroups can be regarded as a generalization both of normal subgroups and of lattices lattices. As such, it is interesting to extend results from the theories of normal subgroups and of lattices to the IRS setting. A more general notion is a stationary random subgroup (SRS) where the measure on the space of subgroups is no longer required to be conjugation invariant, but only stationary with respect to some random walk. SRS are useful in studying IRS which are in themselves useful for studying lattices.
Jointly with Arie Levit, we prove such a result: the critical exponent (exponential growth rate) of an infinite IRS in an isometry group of a Gromov hyperbolic space (such as a rank 1 symmetric space, or a hyperbolic group) is almost surely greater than half the Hausdorff dimension of the boundary. We prove a related bound for SRS, with "half" replaced by entropy divided by drift of the random walk.
PATCH Seminar, joint with Bryn Mawr, Haverford, Swarthmore, and Penn
Abstract:
In the morning background talk (at 9:30am), I will discuss the definitions of many different knot invariants, the linking pairing on the homology of the double branched cover of S^3 branched over a knot, and, if time permits, some basics of Kirby calculus that will be useful.
PATCH Seminar, joint with Bryn Mawr, Haverford, Swarthmore, and Penn
Abstract:
In the morning background talk, at 11:30, I will introduce the Nielsen realization problem for group actions on manifolds and explain some of its connections to geometry, topology, and dynamics.
The proof involves a mixture of geometric constructions and subgroup separability tools. One of the separability tools is a new theorem about separating a peripheral subgroup from every conjugate of a coset. I will try to give a glimpse into both the geometry and the separability. This is joint work with Emily Hamilton and Neil Hoffman.
PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
Abstract: Knot Floer homology is a knot invariant which comes in the form of a graded vector space. The algebra of this vector space determines much of the geometry of the knot, including its genus and whether it is fibered. For fibered knots, knot Floer homology also provides a great deal of geometric information about the monodromy of the fibration. The goal of this talk will be to describe some applications of this theory to knot detection problems. The strategy relies heavily on the theory of train tracks for surface automorphisms, which I will describe in the intro talk (at 9:15am).
PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
Abstract: There are two commonly-used presentations for the braid group. In Artin's original presentation, we linearly order the n strands and use $n-1$ half-twists between adjacent strands to generate the group. The dual presentation, defined by Birman, Ko, and Lee in 1998, introduces additional symmetry by using the larger generating set of all half-twists between any pair of strands. Each presentation has an associated cell complex which is a classifying space for the braid group: the Salvetti complex for the standard presentation and the dual braid complex for the dual presentation. In this talk, I will present a combinatorial perspective for complex polynomials which comes from the dual presentation and describe how this leads to a new cell structure for the space of complex polynomials with distinct roots. This is joint work with Jon McCammond.
Abstract: I will talk about mapping class group action on the curve complex with the following conjecture in mind: an element with small stable translation length is a normal generator. This conjecture is motivated by a similar statement in the case of the action on the Teichmüller space proved by Lanier-Margalit. I will discuss various partial results (based on joint works with various subsets of {Dongryul Kim, Hyunshik Shin, Philippe Tranchida, Chenxi Wu}).
Abstract:
Abstract: Motivated by questions from number theory, we consider the set of slopes of saddle connections on a fixed translation surface $(X,\omega)$ of length at most $N$. How do the slopes distribute as $N$ tends toward infinity? By a result of Veech we know that the directions of saddle connections equidistribute as $N$ goes to infinity, which suggests that they appear quite “random” or uniformly distributed. However, we may consider finer notions of randomness, such as the distribution of the sizes of (renormalized) gaps between slopes as $N$ tends toward infinity. Athreya-Cheung show that finding this gap distribution for a translation surface can be translated into a problem of computing the return times of the horocycle flow to an appropriate transversal on \(\rm{SL}_2(R)/\rm{SL}(X,\omega)\), where $\rm{SL}(X,\omega)$ is the Veech group of $(X,\omega)$.
In contrast to the distribution of slopes, the distribution of slope \emph{gaps} of a translation surface appears to be highly non-random, and for any Veech surface (i.e. translation surface with a lattice Veech group) the distribution has no support at 0, quadratic tail, and is continuous and piecewise real-analytic with finitely many points of non-analyticity (Athreya-Chaika, Athreya-Chaika-Lelievre, Uyanik-Work, Kumanduri-Sanchez-Wang). Work of Uyanik-Work and Kumanduri-Sanchez-Wang provides a practical method of computing such distributions, but this has been carried out in only a small number of cases, and many questions about the general behavior of slope gap distributions remain open. In this talk, we will discuss some of the motivation for this area of study, as well as the method of turning the problem of slope gap distributions into a problem in dynamics. Finally, we will discuss the slope gap distributions for the family of translation surfaces given by regular $2n$-gons with opposite sides glued. We provide linear upper- and lower-bounds on the number of points of non-analyticity in terms of $n$, providing the first example of a family of slope gap distributions with unbounded number of points of non-analyticity.
PATCH seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
Abstract:
ball \(B^n\) are Steklov eigenfunctions with eigenvalue 1. For many embedded FBMS in \(B^3\) we show its first Steklov eigenspace coincides with the span of its coordinate functions, affirming a conjecture of Fraser & Li in an even stronger form. One corollary is a partial resolution of the Fraser-Schoen conjecture: the critical catenoid is the unique embedded FBM annulus in \(B^3\) with antipodal symmetry. This is joint work with Peter McGrath.
PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
Abstract:
A hidden symmetry of a finite volume hyperbolic 3-manifold M is an isometry between two of its finite covers which is not a lift of a self-isometry of M. This talk will center around the study of hidden symmetries for hyperbolic knot complements guided by the following question of Neumann and Reid: Is there a hyperbolic knot except for the figure eight knot and the two dodecahedral knots of Aitchison and Rubinstein, whose complement has a hidden symmetry?
Hyperbolic knot complements that we will consider in our talk originate from hyperbolic link complements that are tetrahedral manifolds, i.e., they have a decomposition into regular ideal tetrahedra. The Fominykh-Garoufalidis-Goerner-Tarkaev-Vesnin census provides abundant examples of tetrahedral manifolds. In this talk, we will concentrate on investigating the existence of hidden symmetries in hyperbolic knot complements obtained from Dehn filling all but one cusp of the members of an infinite family of tetrahedral link complements, all of which cyclically cover the complement of a single tetrahedral link from this census.
2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023