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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.
Samuel Taylor (Temple University)
Our goal will be to describe some work-in-progress (joint with Landry and Minsky) that classifies transverse surfaces to pseudo-Anosov flows on 3—manifolds. This is an introductory talk and there’ll be lots of background and examples.
Thomas Ng, Brandeis University
Abstract: Quotients of free products are natural combinations of groups that have been exploited to study embedding problems. These groups have seen a resurgence of attention from a more geometric point of view following celebrated work of Haglund--Wise and Agol. I will discuss a geometric model for studying quotients of free products. We will use this model to adapt ideas from Gromov's density model to this new class of quotients, their actions on CAT(0) cube complexes, and combination theorems for residual finiteness. Results discussed will be based on ongoing work with Einstein, Krishna MS, Montee, and Steenbock.
Anubhav Mukherjee, Princeton University
PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
Abstract: The groundbreaking research by Freedman, Kreck, Perron, and Quinn provided valuable insights into the topological mapping class group of closed simply connected 4-manifolds. However, the development of gauge theory revealed the exotic nature of the smooth mapping class group of 4-manifolds in general. While gauge theory can at times obstruct smooth isotopy between two diffeomorphisms, it falls short of offering a comprehensive understanding of the existence of diffeomorphisms that are topologically isotopic but not smoothly so. In this talk, I will elucidate some fundamental principles and delve into the origins of such exotic diffeomorphisms. This is my upcoming work joint with Slava Krushkal, Mark Powell, and Terrin Warren.
In the morning background talk (at 10am), I will give an overview of mapping class groups of 4-manifolds.
Jean-Pierre Mutanguha, Princeton University and IAS
PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
Abstract: Free-by-cyclic groups can be defined as mapping tori of free group automorphisms. I will discuss various dynamical properties of automorphisms that turn out to be group invariants of the corresponding free-by-cyclic groups (e.g. growth type). In particular, certain dynamical hierarchical decompositions of an automorphism determine canonical hierarchical decompositions of its mapping torus. In the intro talk, I will discuss how Bass-Serre theory (actions on simplicial trees) gives us a grip on these groups.
In the morning background talk, at 11:30am, I will introduce free-by-cyclic groups from a tree's point of view.
Chi Cheuk Tsang (UQAM)
The minimum dilatation problem asks for the minimum value of the dilatation among all pseudo-Anosov maps defined on a fixed surface. This value can be thought of as the smallest amount of mixing one can perform on the surface while still doing something topologically interesting. In this talk, we will present some recent progress on the fully-punctured version of this problem. The strategy for proving these results involves something called standardly embedded train tracks. We will explain what these are and formulate some future directions that may be tackled using this technology. This is joint work with Eriko Hironaka, and with Erwan Lanneau and Livio Liechti.
Rylee Lyman
Rutgers University, Newark
Abstract: Coxeter groups, defined by a labeled simplicial graph, are a beautiful family of groups which are in a certain precise sense generated by reflections. With Elizabeth Field, Radhika Gupta and Emily Stark, we study the family of Coxeter groups whose defining graph is complete with all edges labels at least three. We show that they fall into infinitely many quasi-isometry classes. These groups were previously studied by Haulmark, Hruska and Sathaye, who showed that generically they all have visual boundary the Menger curve and posed the question of quasi-isometric classification. Along the way to our proof, we show that these groups have a geometrically finite action on a CAT(-1) space, whose geometry can be studied by reasoning about hyperbolic 3-space. I think this space and the geometric reasoning about it are really pretty — I'd like to spend most of the talk focusing on it.
Allison Miller, Swarthmore College
Abstract: The collection of knots in the 3-sphere modulo a 4-dimensionally defined equivalence relation called "concordance", is not just a set but a group and a metric space as well. Via the satellite construction, every knot in a solid torus induces a self-map of the concordance set/ group/ metric space. In this talk, we'll survey what is known about these functions: When are they injective/ surjective/ bijective? When are they group homomorphisms? How do they interact with the metric space structure? We will end by discussing recent joint work of mine with two Swarthmore students, Randall Johanningsmeier and Hillary Kim, that unexpectedly provided significant progress towards answering one of these questions.
Nir Gadish, University of Michigan
PATCH Seminar, join with Bryn Mawr, Haverford, Penn, and Swarthmore
Abstract: The classical Hopf invariant uses the linking of two generic fibers to detect elements in higher-homotopy groups of spheres. Sinha-Walter generalized this idea and used "higher linking" to completely characterize elements in the (rational) homotopy groups of any simply connected space. By extending this setup to measure fundamental groups, we arrive at a new invariant theory for groups, which we have termed letter braiding. This is effectively a 0-dimensional linking theory for letters in words, and it realizes every finite-type invariant of any group. We will discuss the topological origins of this theory, its connection to loop spaces, and will explore an application to mapping class groups of surfaces.
In the background talk (10am in room A8), I will discuss studying topological spaces using invariants of words and groups.
Ao Sun, Lehigh University
PATCH Seminar, joint with Bryn Mawr, Haverford, Penn, and Swarthmore
Abstract: The interpolation method is a very powerful tool to construct special solutions in geometric analysis. I will present two applications in mean curvature flow: one is constructing a new genus one self-shrinking mean curvature flow, and another one is constructing immortal mean curvature flow with higher multiplicity convergence. The talk is based on joint work with Adrian Chu (UChicago) and joint work with Jingwen Chen (UPenn).
In the morning background talk (11:30-12:30 in room A8) I will discuss singularities and solitons of mean curvature flow.
Michael Magee, Durham University
Abstract: Let $G$ be an infinite discrete group e.g. hyperbolic 3-manifold group. Finite dimensional unitary representations of $G$ in fixed dimension are usually quite hard to understand. However, there are interesting notions of convergence of such representations as the dimension tends to infinity. One notion — strong convergence — is of interest both from the point of view of $G$ alone but also through recently realized applications to spectral gaps of locally symmetric spaces. For example, this notion bypasses (unconditionally) the use of Selberg's Eigenvalue Conjecture in obtaining existence of large area hyperbolic surfaces with near-optimal spectral gaps.
The talk is a discussion on these themes, based on joint works with W. Hide, L. Louder, D. Puder, J. Thomas.
Edgar Bering, San Jose State University
Abstract: In general, the classification of finitely generated subgroups of a given group is intractable. Restricting to two-generator subgroups in a geometric setting is an exception. For example, a two-generator subgroup of a right-angled Artin group is either free or free abelian. Jaco and Shalen proved that a two-generator subgroup of the fundamental group of an orientable atoroidal irreducible 3-manifold is either free, free-abelian, or finite-index. In this talk I will present recent work proving a similar classification theorem for two generator mapping-torus groups of free group endomorphisms: every two generator subgroup is either free or conjugate to a sub-mapping-torus group. As an application we obtain an analog of the Jaco-Shalen result for free-by-cyclic groups with fully irreducible atoroidal monodromy. This is joint work with Naomi Andrew and Ilya Kapovich.
Anna Marie Bohmann (Vanderbilt)
Algebraic K-theory is an important invariant of rings (and other related mathematical things). In this talk, we'll give a little bit of background on the classical story of algebraic K-theory and then talk about some of the many ways it's developed in recent years.
Spencer Dowdall (Vanderbilt)
I will review the key features of Teichmuller space that are relevant to counting lattice points for the action of the mapping class group. Specifically, after introducing Teichmuller space I will discuss the thin regions and their inherent product structure. This will lead us to subsurface projections and Rafi's distance formula and associated combinatorial description of how Teichmuller geodesics pass through thin regions of subsurfaces. I will then introduce a new notion of "complexity length" in Teichmuller space that aims to carefully account for this motion of geodesics through product regions in a way that gives better control on the multiplicative errors and lends itself to counting problems.
Anna Marie Bohmann (Vanderbilt)
Scissors congruence, the subject of Hilbert's Third Problem, asks for invariants of polytopes under cutting and pasting operations. One such invariant is obvious: two polytopes that are scissors congruent must have the same volume, but Dehn showed in 1901 that volume is not a complete invariant. Trying to understand these invariants leads to the notion of the scissors congruence group of polytopes, first defined the 1970s. Elegant recent work of Zakharevich allows us to view this as the zeroth level of a series of higher scissors congruence groups.
In this talk, I'll discuss some of the classical story of scissors congruence and then describe a way to build the higher scissors congruence groups via K-theory of covers, a new framework for such constructions. We'll also see how to relate coinvariants and K-theory to produce concrete nontrivial elements in the higher scissors congruence groups. This work is joint with Gerhardt, Malkiewich, Merling and Zakharevich.
Spencer Dowdall (Vanderbilt)
I will discuss the growth rate of the number of elements of the mapping class group of each Nielsen-Thurston type, that is, either finite-order, reducible, or pseudo-Anosov, measured via the number of lattice points in a ball of radius $R$ in Teichm\"uller space. For the whole mapping class group of the closed genus $g$ surface, Athreya, Bufetov, Eskin, and Mirzakhani have shown this quantity is asymptotic to $e^{(6g-6)R}$ as $R$ tends to infinity. Maher has obtained the same asymptotics for those orbit points that are translates by pseudo-Anosov elements. Obtaining a count for the finite-order or reducible elements is significantly more challenging due to the fact these non-generic subsets are not perceptible to the standard dynamical techniques. I will explain a naive heuristic for why the finite-order elements should grow at the rate of $e^{(3g-3)R}$, that is, with half the exponent. While this approach presents several obstacles, our new notion of complexity length provides the tools needed to make the argument work. Time permitting, I will also explain why the reducible elements grow coarsely at the rate of $e^{(6g-7)R}$. Joint work with Howard Masur.
Hongbin Sun, Rutgers University
Abstract: We show that any arithmetric lattice $\Gamma<\text{Isom}_+(\mathbb{H}^n)$ with $n\geq 4$ is not LERF (locally extended residually finite), including type III lattices in dimension 7. One key ingredient in the proof is the existence of totally geodesic 3-dim submanifolds, which follows from the definition if $\Gamma$ is in type I or II, but is much harder to prove if $\Gamma$ is in type III. This is a joint work with Bogachev and Slavich.
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