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Current contact: Dave Futer or Matthew Stover
The Seminar usually takes place on Wednesdays at 2:30 PM in Room 617 on the sixth floor of Wachman Hall.
Michael Landry, Yale University
Let \(M\) be a closed hyperbolic 3-manifold which fibers over \(S^1\), and let \(F\) be a fibered face of the unit ball of the Thurston norm on \(H^1(M;R)\). By results of Fried, there is a nice flow on \(M\) naturally associated to \(F\). We study surfaces which are almost transverse to \(F\) and give a new characterization of the set of homology directions of \(F\) using Agol’s veering triangulation of an auxiliary cusped 3-manifold.
Andrew Yarmola, Princeton University
Abstract: At the interface of discrete conformal geometry and the study of Riemann surfaces lies the Koebe-Andreev-Thurston theorem. Given a triangulation of a surface \(S\), this theorem produces a unique hyperbolic structure on \(S\) and a geometric circle packing whose dual is the given triangulation. In this talk, we explore an extension of this theorem to the space of complex projective structures - the family of maximal \(CP^1\)-atlases on \(S\) up to Möbius equivalence. Our goal is to understand the space of all circle packings on complex projective structures with a fixed dual triangulation. As it turns out, this space is no longer a unique point and evidence suggests that it is homeomorphic to Teichmüller space via uniformization - a conjecture by Kojima, Mizushima, and Tan. In joint work with Jean-Marc Schlenker, we show that this projection is proper, giving partial support for the conjectured result. Our proof relies on geometric arguments in hyperbolic ends and allows us to work with the more general notion of Delaunay circle patterns, which may be of separate interest. I will give an introductory overview of the definitions and results and demonstrate some software used to motivate the conjecture. If time permits, I will discuss additional ongoing work with Wayne Lam.
Thomas Ng, Temple University
Abstract: A group is said to have uniform exponential growth if the number of elements that can be spelled with words of bounded length is bounded below by a single exponential function over all generating sets. In 1981, Gromov asked whether all groups with exponential growing group in fact have uniform exponential growth. While this was shown not to be the case in general, it has been answered affirmatively for many natural classes of groups such as hyperbolic groups, linear groups, and the mapping class groups of a surface. In 2018, Kar-Sageev show that groups acting properly on 2-dimensional CAT(0) cube complexes by loxodromic isometries either have uniform exponential growth or are virtually abelian by explicitly exhibiting free semigroups whose generators have uniformly bounded word length whenever they exist. These free semigroups witness the uniform exponential growth of the group. I will explain how certain arrangements of hyperplane orbits can be used to build loxodromic isometries generating free semigroups and then describe how to use the convex hull of their axes and the Bowditch boundary to extend Kar and Sageev's result to CAT(0) cube complexes with isolated flats. This is joint work with Radhika Gupta and Kasia Jankiewicz.
Andrew Cooper, NC State
Given a space \(X\), the configuration space \(F(X,n)\) is the space of possible ways to place \(n\) points on \(X\), so that no two occupy the same position. But what if we allow some of the points to coincide?
The natural way to encode the allowed coincidences is as a simplicial complex \(S\). I will describe how the configuration space \(M(S,X)\) obtained in this way gives rise to polynomial and homological invariants of \(S\), how those invariants are related to the cohomology ring \(H^*(X)\), and what this has to do with the topology of spaces of maps into \(X\).
I will also mention some potential applications of this structure to problems arising from international relations and economics.
This is joint work with Vin de Silva, Radmila Sazdanovic, and Robert J Carroll.
Oleg Lazarev, Columbia University
PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
Abstract TBA
Francesco Lin, Princeton University PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
Abstract TBA
Matthew Stover, Temple University
I will survey (in)coherence of lattices in semisimple Lie groups, with a view toward open problems and connections with the geometry of locally symmetric spaces. Particular focus will be placed on rank one lattices, where I will discuss connections with reflection groups, "algebraic" fibrations of lattices, and analogies with classical low-dimensional topology.
Carolyn Abbott, University of California Berkley Imagine you are standing at the point 0 on a number line, and you take a step forward or a step backwards, each with probability 1/2. If you take a large number of steps, is it likely that you will end up back where you started? What if you are standing at a vertex of an 4-valent tree, and you take a step in each of the 4 possible directions with probability 1/4? This process is special case of what is called a random walk on a space. If the space you choose is the Cayley graph of a group (as these examples are), then a random walk allows you to choose a "random" or "generic" element of the group by taking a large number of steps and considering the label of the vertex where you end up. One can ask what properties a generic element of the group is likely to have: for example, is it likely that the element you land on has infinite order? In this talk, I will focus on the class of the class of so-called acylindrically hyperbolic groups, which contains many interesting groups, such as mapping class groups, outer automorphism groups of free groups, and right-angled Artin and Coxeter groups, among many others. I will discuss the algebraic and geometric properties of subgroups generated by a random element and a fixed subgroup.
Davide Spiriano, ETH Zurich
In a Gromov hyperbolic space, geodesics satisfies the so-called Morse property. This means that if a geodesic and a quasi-geodesic share endpoints, then their Hausdorff distance is uniformly bounded. Remarkably, this is an equivalent characterization of hyperbolic spaces, meaning that all consequences of hyperbolicity can be ascribed to this property. Using this observation to understand hyperbolic-like behaviour in spaces which are not Gromov hyperbolic has been a very successful idea, which led to the definition of important geometric objects such as the Morse boundary and stable subgroups. Another strong consequence of hyperbolicity is the fact that local quasi-geodesics are global quasi-geodesics. This allows detecting global properties on a local scale, which has far-reaching consequences. The goal of this talk is twofold. Firstly, we will prove results that are known for hyperbolic groups in a class of spaces satisfying generalizations of the above properties. Secondly, we show that the set of such spaces is large and contains several examples of interest, i.e. CAT(0) spaces and hierarchically hyperbolic spaces.
Nicholas Vlamis, CUNY Queen's College
A classical theorem of Powell (with roots in the work of Mumford and Birman) states that the pure mapping class group of a connected, orientable, finite-type surface of genus at least 3 is perfect, that is, it has trivial abelianization. We will discuss how this fails for infinite-genus surfaces and give a complete characterization of all homomorphisms from pure mapping class groups of infinite-genus surfaces to the integers. This characterization yields a direct connection between algebraic invariants of pure mapping class groups and topological invariants of the underlying surface. This is joint work with Javier Aramayona and Priyam Patel.
Christian Millichap, Furman University PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn
Abstract: Fully augmented links (FALs) are a large class of links whose complements admit hyperbolic structures that can be explicitly described in terms of combinatorial information coming from their respective link diagrams. In this talk, we will examine an infinite subclass of FALs that are constructed by fully augmenting pretzel links and describe how to build their hyperbolic structures. We will then discuss how we can use the geometries of these link complements to analyze arithmetic properties and commensurability classes of these links. This is joint work with Jeff Meyer (CSSB) and Rollie Trapp (CSSB).
The morning background talk, at 9:30am, will be an exploration of hyperbolic structures on link complements.
Dan Rutherford, Ball State University PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn
Abstract: This is joint work with Y. Pan that applies previous joint work with M. Sullivan. Let \(\Lambda \subset \mathbb{R}^{3}\) be a Legendrian knot with respect to the standard contact structure. The Legendrian contact homology (LCH) DG-algebra, \(\mathcal{A}(\Lambda)\), of \(\Lambda\) is functorial for exact Lagrangian cobordisms in the symplectization of \(\mathbb{R}^3\), i.e. a cobordism \(L \subset \mathit{Symp}(\mathbb{R}^3)\) from \(\Lambda_-\) to \(\Lambda_+\) induces a DG-algebra map, \(f_L:\mathcal{A}(\Lambda_+) \rightarrow \mathcal{A}(\Lambda_-).\) In particular, if \(L\) is an exact Lagrangian filling (\(\Lambda_-= \emptyset\)) the induced map is an augmentation \(\epsilon_L: \mathcal{A}(\Lambda_+) \rightarrow \mathbb{Z}/2.\)
In this talk, I will discuss an extension of this construction to the case of immersed, exact Lagrangian cobordisms based on considering the Legendrian lift \(\Sigma\) of \(L\). When \(L\) is an immersed, exact Lagrangian filling a choice of augmentation \(\alpha\) for \(\Sigma\) produces an induced augmentation \(\epsilon_{(L, \alpha)}\) for \(\Lambda_+\). Using the cellular formulation of LCH, we are able to show that any augmentation of \(\Lambda\) may be induced by such a filling.
In the morning background talk, at 11:00am, I will cover augmentations and immersed Lagrangian fillings.
Katherine St. John
City University of New York & American Museum of Natural History
Trees are a canonical structure for representing evolutionary histories. Many popular criteria used to infer optimal trees are computationally hard, and the number of possible tree shapes grows super-exponentially in the number of taxa. The underlying structure of the spaces of trees yields rich insights that can improve the search for optimal trees, both in accuracy and running time, and the analysis and visualization of results. We review the past work on analyzing and comparing trees by their shape as well as recent work that incorporates trees with weighted branch lengths. This talk will highlight some of the elegant questions that arise from improving search and visualizing the results in this highly structured space. All are welcome.
Kate Petersen, Florida State University
Regina Rotman, University of Toronto and IAS
PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn
Abstract: I will talk about periodic geodesics, geodesic loops, and geodesic nets on Riemannian manifolds. More specifically, I will discuss some curvature-free upper bounds for compact manifolds and the existence results for non-compact manifolds. In particular, geodesic nets turn out to be useful for proving results about geodesic loops and periodic geodesics.
Alex Nabutovsky, University of Toronto and IAS
PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn
Abstract: The Uryson \(k\)-width of a metric space \(X\) measures how close \(X\) is to being \(k\)-dimensional. Several years ago Larry Guth proved that if \(M\) is a closed \(n\)-dimensional manifold, and the volume of each ball of radius 1 in \(M\) does not exceed a certain small constant \(e(n)\), then the Uryson \((n-1)\)-width of \(M\) is less than 1. This result is a significant generalization of the famous Gromov inequality relating the volume and the filling radius that plays a central role in systolic geometry.
Guth asked if a much stronger and more general result holds true: Is there a constant \(e(m)>0\) such that each compact metric space with \(m\)-dimensional Hausdorff content less than \( e(m)\) always has \((m-1)\)-dimensional Uryson width less than 1? Note that here the dimension of the metric space is not assumed to be \(m\), and is allowed to be arbitrary.
Such a result immediately leads to interesting new inequalities even for closed Riemannian manifolds. In my talk I am are going to discuss a joint project with Yevgeny Liokumovich, Boris Lishak and Regina Rotman towards the positive resolution of Guth's problem.
Edgar Bering, Temple University
In 1982 Thurston stated the "virtual conjectures" for 3-manifolds: that every hyperbolic 3-manifold has finite covers that are Haken and fibered, with large Betti numbers. These conjectures were resolved by Agol and Wise in 2012, using the machinery of special cube complexes. Even before the work of Agol and Wise, but especially after, mathematicians have been interested in understanding the degree of these covers in terms of a manifold's invariants.
In joint work, David Futer and I give the first steps of a quantitative answer to this question in the setting of alternating link complements. Given an alternating link with n crossings we construct a special cover of degree less than n!. As a corollary, we bound the degree of a cover with Betti number at least k.
Marissa Loving, University of Illinois at Urbana Champaign
When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask which curves’ lengths do we really need to know? It is a classical result of Fricke that a hyperbolic metric on a surface is determined by its marked simple length spectrum. More recently, Duchin–Leininger–Rafi proved that a flat metric induced by a unit-norm quadratic differential is also determined by its marked simple length spectrum. In this talk, I will describe a generalization of the notion of simple curves to that of q-simple curves, for any positive integer q, and show that the lengths of q-simple curves suffice to determine a non-positively curved Euclidean cone metric induced by a q-differential metric.
Mark Hagen, University of Bristol
Abstract: Masur and Minsky's work on the geometry of mapping class groups, combined with more recent results about the geometry of CAT(0) cube complexes, motivated the introduction of the class of hierarchically hyperbolic spaces. A metric space \(X\) is hierarchically hyperbolic if there is a set of (uniformly) Gromov-hyperbolic spaces \(U\), each equipped with a projection from \(X\) to \(U\), satisfying various axioms that amount to saying that the geometry of \(X\) is recoverable, up to quasi-isometry, from this projection data. Working in this context often allows one to promote facts about hyperbolic spaces to conclusions about highly non-hyperbolic spaces: mapping class groups, Teichmuller space, "most" 3-manifold groups, etc. In particular, many CAT(0) cube complexes -- including those associated to right-angled Artin and Coxeter groups -- are hierarchically hyperbolic.
The relationship between CAT(0) cube complexes and hierarchically hyperbolic spaces is intriguing. Just as, in a hyperbolic space, a collection of n points has quasiconvex hull quasi-isometric to a finite tree (i.e. 1-dimensional CAT(0) cube complex), in a hierarchically hyperbolic space, there is a natural notion of the quasiconvex hull of a set of n points, and it is quasi-isometric to a CAT(0) cube complex, by a result of Behrstock-Hagen-Sisto. The quasi-isometry constants depend on n in general. However, when each hyperbolic space U is quasi-isometric to a tree, it turns out that this dependence disappears. From this one deduces that, if \(X\) is a metric space that is hierarchically hyperbolic with respect to quasi-trees, then \(X\) is quasi-isometric to a CAT(0) cube complex. I will discuss this theorem and some of its group-theoretic consequences. This is joint work with Harry Petyt.
Jeff Danciger, University of Texas at Austin
We study properly convex real projective structures on closed 3-manifolds. A hyperbolic structure is one special example, and in some cases the hyperbolic structure may be deformed non-trivially as a convex projective structure. However, such deformations seem to be exceedingly rare. By contrast, we show that many closed hyperbolic manifolds admit a second convex projective structure not obtained through deformation. We find these examples through a theory of properly convex projective Dehn filling, generalizing Thurston’s picture of hyperbolic Dehn surgery space. Joint work with Sam Ballas, Gye-Seon Lee, and Ludovic Marquis.
Mark Pengitore, The Ohio State University
This talk will be an introduction to separability of finitely generated groups. The premise is that we can detect membership of interesting subsets of finitely generated groups such as the identity subgroup, finitely generated subgroups, and conjugacy classes via surjective group morphisms to finite groups. This idea can be interpreted in many distinct ways such as lifting of closed loops of manifolds to finite covers, topological properties of a totally disconnected compact topological group, and well approximation of elements in a metric space. One of the many applications of these ideas is a quantitative solution to the word problem, conjugacy problem, and other decision problems. In a more topological direction, another application is the constructing lifts of an immersed submanifold to an embedded submanifold in a finite cover. This talk will be expository and will explain connections between all of above ideas and motivate interest in separability.
Mark Pengitore, The Ohio State University
In this talk, we introduce quantitative approaches to the study of separability in nilpotent and solvable groups. In particular, we will describe effective residual finiteness, effective subgroup separability, and effective conjugacy separability and discuss various results for asymptotic lower bounds of these properties for these classes of groups. Moreover, we introduce the algebraic, number theoretic, and geometric methods used in the construction of these lower bounds.
Matthew Stover, Temple University Hyperbolic manifolds, n≥3, that are arithmetic were characterized by Borel and Margulis as being infinite index in their commensurator. One can use this to show that an arithmetic hyperbolic n-manifold either contains no totally geodesic hypersurfaces or they are everywhere dense. Reid and McMullen (for n= 3) asked whether having infinitely many totally geodesic hypersurfaces conversely implies arithmeticity. I will discuss work with Bader, Fisher, and Miller that answers this question in the positive.
Mona Merling, University of Pennsylvania The "stable parametrized h-cobordism theorem" provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. For a compact smooth manifold M it gives a characterization of the stable h-cobordism space of M in terms of Waldhausen's algebraic K-theory of M. I will talk about joint work with Malkiewich on this story when M is a smooth compact G-manifold.
Tarik Aougab, Haverford College
Given two finite degree regular covers (not necessarily of the same degree) Y, Z of a surface S, suppose that for any closed curve gamma on S, gamma lifts to a simple closed curve on Y if and only if it does to Z. We prove that Y and Z must be equivalent covers. The proof uses some Teichmuller theory and the curve complex. This represents joint work with Max Lahn, Marissa Loving, and Sunny Yang Xiao.
Dan Margalit, Georgia Tech
PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn About 100 years ago, Artin showed that any homomorphism from the braid group \(B_n\) to the symmetric group \(S_n\) is either cyclic or conjugate to the standard homomorphism. Much more recently, Castel showed that any endomorphism of \(B_n\) is either cyclic or conjugate to (a transvection of) the identity map. With Lei Chen and Kevin Kordek, we extend Castel's result by showing that any homomorphism from \(B_n\) to \(B_{2n}\) is either cyclic or conjugate to (a transvection of) one of the standard maps.
In the morning background talk (9:30am in room 527) I will review braid groups, mapping class groups, canonical reduction systems, and totally symmetric sets.
Kyle Hayden, Columbia University
PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn
From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from the knot Floer homology concordance invariant nu, which we prove is an invariant of a simple 4-manifold associated to a knot, called the knot trace. As a corollary to the existence of exotic Mazur manifolds, we produce integer homology 3-spheres admitting two distinct \(S^1 \times S^2\) surgeries, resolving a question from Problem 1.16 in Kirby's list. We also resolve a related question about the knot concordance invariants tau and epsilon. This is joint work with Tom Mark and Lisa Piccirillo.
In the morning background talk (at 11:00am), we'll review the key constructive ingredients (Dehn surgery, handlebody structures, and cork twists), and provide some extra historical context for the results described above. In the afternoon talk, I'll explain our main results, present examples demonstrating our constructive results, and discuss the key ideas in the proofs. In particular, I'll focus on how the main arguments take tools from smooth 3- and 4-dimensional topology, hyperbolic geometry, and Heegaard Floer homology and play them off one another.
Rylee Lyman, Tufts University
Mapping class groups, \(GL(n,\mathbb{Z})\), and \(Out(F_n)\), the outer automorphism group of a free group are among some of the most well-studied infinite discrete groups. One facet they have in common is that, although finitely presented, they are "big" groups, in the sense that their elements exhibit a rich and wide array of dynamical behavior. The Nielsen–Thurston normal form, Jordan normal form and relative train track representative, respectively, all attempt to expose and present this information in an organized way to aid reasoning about this behavior.
The group of outer automorphisms of a finite free product of finite groups is closely related to \(Out(F_n)\), but is comparatively understudied. In this talk we will introduce these groups, related geometric structures they act on, and review some of the known results. We would like to argue that these groups are also "big": to this end we have shown how to extend work of Bestvina, Feighn and Handel to construct relative train track representatives for outer automorphisms of free products.
Justin Lanier, Georgia Tech
A polynomial can be viewed as a branched cover of the sphere over itself that is compatible with a complex structure. If handed a topological branched cover of the sphere, we can ask whether it can arise from a polynomial, and if so, which one? In 2006, Bartholdi and Nekrashevych used group theoretic methods to explicitly solve this problem in special cases, including Hubbard’s twisted rabbit problem. We introduce a new topological approach that draws from the theory of mapping class groups of surfaces. By iterating a lifting map on a complex of trees, we are able to certify whether or not a given branched cover arises as a polynomial. This is joint work with Jim Belk, Dan Margalit, and Becca Winarski.
Ivan Levcovitz, Technion
Associated to any simplicial graph K is the right-angled Coxeter group (RACG) whose presentation consists of an order 2 generator for each vertex of K and relations stating that two generators commute if there is an edge between the corresponding vertices of K. RACGs contain a rich class of subgroups including, up to commensurability, hyperbolic 3-manifold groups, surface groups, free groups, Coxeter groups and right-angled Artin groups to name a few. I will describe a procedure which associates a cube complex to a given subgroup of RACG. I will then present some results regarding structural and algorithmic properties of subgroups of RACGs whose proofs follow from this viewpoint. This is joint work with Pallavi Dani.
Effie Kalfagianni, Michigan State University PATCH Seminar, at UPenn
Abstract: The generalization of the Jones polynomial for links and 3-manifolds, due to Witten-Reshetiking-Turaev in the late 90’s, led to constructions of Topological Quantum Field Theory in dimensions (2+1). These theories also include representations of surface mapping class groups. The question of how much of the Thurston geometric picture of 3-manifolds is reflected in these theories is open. I will report on recent work in this direction, with emphasis on the corresponding mapping class group representations. The talk is based on joint work with R. Detcherry and G. Belletti, R. Detcherry, T. Yang.
Roberta Guadagni, University of Pennsylvania
PATCH Seminar, at UPenn
Abstract: The standard notion of concordance and cobordism of smooth knots translates into a notion of Lagrangian concordance and cobordism for Legendrian knots. A natural question is then: can we interpret the cobordism relation as a sequence of moves in the front diagrams of the knots? We will look at the elementary handle attachments that yield a "decomposable" cobordism (Ekholm-Honda-Kalman). We will then construct cobordisms and concordances that are not decomposable in the EHK sense (this is a new result) and end with some currently open questions.
Samuel Taylor, Temple University
In 2013, Chas, Li, and Maskit produced numerical experiments on random closed geodesics on a hyperbolic pair of pants. Namely, they drew uniformly at random conjugacy classes of a given word length, and considered the hyperbolic lengths of the corresponding closed geodesic on the pair of pants. Their experiments lead to the conjecture that the length of these closed geodesics satisfies a central limit theorem. I will discuss a proof of this conjecture obtained in joint work with I. Gekhtman and G. Tiozzo, and its generalizations to all negative curved surfaces.
Rémi Coulon Université de Rennes 1
Let G be a group. Given an (outer) automorphism f of G, one can study its properties by considering the dynamics induced by the action of f on the set of conjugacy classes of G. A classical problem is to understand how the length of a conjugacy class grows under the iterations of f. For many groups (e.g, free groups, free abelian groups, surface groups, etc) one observes a strong dichotomy : the length of any conjugacy class grows either polynomially or at least exponentially. In this talk, we will explain how to build examples of outer automorphisms of finitely generated groups for which this dichotomy fail.
Talia Fernós, UNC Greensboro
Let \(G\) be a group acting on a finite dimensional CAT(0) cube complex \(X\). By studying equivariant maps from the Furstenberg-Poisson boundary to the Roller boundary, we deduce a variety of phenomena concerning the push-forward of the random walk from \(G\) to an orbit in \(X\). Under mild and natural assumptions, we deduce positivity of the drift, sublinear tracking, and the central limit theorem. Along the way we prove that regular elements are plentiful and establish a homeomorphism between the boundary of the contact graph of \(X\) with a special subset of the Roller boundary called the regular points. This is joint work with Jean Lécureux and Frédéric Mathéus.
Caglar Uyanik Yale University
Geodesic currents on surfaces are measure theoretic generalizations of closed curves on surfaces and they play an important role in the study of the Teichmuller spaces. I will talk about their analogs in the setting of free groups, and try to illustrate how the dynamics and geometry of the Out(F_N) action reflects on the algebraic structure of Out(F_N).
Bulent Tosun, University of Alabama
PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
Title: Stein domains in complex 2-plane with prescribed boundary
Abstract: A Stein manifold is a complex manifold with particularly nice convexity properties. In real dimensions above 4, existence of a Stein structure is essentially a homotopical question, but for 4-manifolds the situation is more subtle. In this talk we will consider existence of Stein structures in “ambient” setting (Stein manifolds/domains as open/compact subsets of a fixed complex manifold, e.g. complex 2-plane). In particular, I would like to discuss the following question that has been circulating among contact and symplectic topologist for some time: "which integral homology spheres embed in complex 2-plane as the boundary of a Stein domain". This question was first considered and explored in detail by Gompf. At that time, he made a fascinating conjecture that: No non-trivial Brieskorn homology sphere, with either orientation, embed in complex 2-plane as a Stein boundary. In this talk, I will survey what we know about this conjecture, and report on some closely related recent work in progress that ties to an interesting symplectic rigidity phenomena in low dimensions.
In the morning background talk (9:30am), I will talk about embedding 3-manifolds in 4-manifolds.
Jessica Purcell, Monash University
PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
Abstract: In the 1950s, Moise showed that any 3-manifold decomposes into tetrahedra. But how many tetrahedra? The triangulation complexity of a 3-manifold is the minimal number of tetrahedra in any triangulation of the manifold. While knowing this number can be useful for computing and algorithmic topology, it seems to be difficult to determine. In this talk, I will discuss recent work giving upper and lower bounds on the triangulation complexity of any closed orientable hyperbolic 3-manifold that fibres over the circle. We show that the triangulation complexity of the manifold is equal to the translation length of the monodromy action on the mapping class group of the fibre, up to a bounded factor, where the bound depends only on the genus of the fibre. This is joint work with Marc Lackenby.
In the morning background talk (11:30am), I'll give a little background on triangulation complexity, and then describe various terms in the abstract, particularly fibred manifolds, and translation length in the mapping class group. I'll discuss related work, and give an outline of why this is the "right" result in the context of various things we know about fibred manifolds and their geometry.
Alex Kontorovich Rutgers University, New Brunswick
We will discuss recent work on "crystallographic" sphere packings (defined in work with Nakamura), and the subclass of "superintegral" such. (A quintessential example is the classical Apollonian Circle Packing.) These exist in finitely many dimensions, and in fact in finitely many commensurability classes in each dimension. This is a consequence of the Arithmeticity Theorem, that such packings come from arithmetic hyperbolic reflection groups.
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