Date | Speaker | Title | Abstract |
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Jan 17 | Sam Ballas | Smooth manifolds through the lens of algebraic geometry |
One of the tenets algebraic geometry is that one should try to understand a space by understanding the functions on that space. In many situations one can associate to a space a certain class of functions endowed with an algebraic structure. Furthermore, it is often possible to recover the original space from its space of functions. A simple example of this phenomenon is seen in finite dimensional vector spaces, where one can associate to a vector space its space of linear maps to the ground field (i.e. its dual space) and given the dual space one can recover the original vector space (up to isomorphism). In this talk, we will focus on smooth manifolds. In this setting one can associate to a manifold its algebra of smooth real valued functions. In this context, we will focus on two questions: how can one recover a manifold from its algebra of smooth functions and which algebras arise as smooth functions on some manifold. |
Jan 24 | Sam Ballas | Smooth manifolds through the lens of algebraic geometry: II | Last time we saw how given a sufficiently nice algebra we could construct a "dual space" on which the algebra was a space of real valued functions. We endowed this dual space with a topology and showed that for the algebra of smooth functions on R^n the dual is the R^n with its standard topology. This time we will examine some special properties of smooth functions and ultimately arrive at the algebraic properties an algebra must possess for its dual space to be a smooth manifold. |
Jan 31 | Sam Ballas | Smooth manifolds through the lens of algebraic geometry: III | Last time we introduced the notion of a smooth algebra, and showed that the algebra of smooth functions on a smooth manifold is an object of this type. Today we will describe how to recover a smooth manifold from a smooth algebra and how algebra homomorphism between smooth algebras give rise to smooth functions between the corresponding manifold. Said more succinctly, this provides an equivalence of categories between the category of smooth manifolds with smooth maps and the category of smooth algebras with algebra homomorphisms. |
Feb 7 | |||
Feb 14 | Florian Stecker | Projective triangle groups and Anosov representations | I will talk about groups generated by three projective reflections, configured so that the product of any two of them has finite order. There is only a 1-dimensional space of these groups, and some of them (called Anosov representations) show nice hyperbolic plane like dynamics. We identify when this happens and where this property suddenly breaks down. This is joint work with Gye-Seon Lee and Jaejeong Lee. |
Feb 21 | |||
Feb 28 | Phil Bowers | Geometry and Probability I | Type Problems: I will begin with the classical conformal type problem for simply connected Riemann surfaces and then walk through several discrete versions of the problem in the settings of circle packings, equilateral surfaces, negatively curved graphs, and Riemannian manifolds. I will then pivot to probability theory and consider type problems in the settings of random walks on graphs and Brownian motion on manifolds, and describe the connections between the geometric and the probabilistic settings. |
Mar 7 | Phil Bowers | Geometry and Probability II | |
Mar 21 | Phil Bowers | Geometry and Probability III | Conformally Invariant Critical Processes in the Plane: Brownian motion as the scaling limit of random walks has been understood since Donsker’s 1952 proof. In 2D, Brownian motion has the added benefit that it is conformally invariant. This is the archetype of the movement from the discrete setting to the continuous setting. Since the 1930’s physicists have modeled various processes in statistical mechanics on discrete lattices and then attempted to find scaling limits to continuous processes. Though they have developed a trove of conjectures based on numerical results and have applied QFT calculations to derive formulae, they have been hampered by the lack of mathematical tools to verify rigorously their conjectures and QFT calculations. This changed in 2000 when Oded Schramm defined SLE and used it to develop mathematical tools to attack these problems in statistical mechanics. |
Mar 28 | |||
Apr 4 | |||
Apr 11 | |||
Apr 18 | Anindya Chanda | Mixing, Counting, and Equidistribution in Geometry and Dynamics | In this talk we will introduce the concepts of mixing, counting and equidistribution in geometry and dynamics. The talk will be based on a pioneering paper by Alex Eskin and Curt McMullen. |
Apr 25 | Tom Needham | Topological Data Analysis and 2-Categories | The field of Topological Data Analysis (TDA) provides a collection of techniques for processing and analyzing datasets using tools from algebraic topology. Much of the theory can be expressed categorically; in particular, there is a family of pseudometrics, called interleaving distances, which are used to compare certain functors arising in data science applications. In this talk, I’ll present a generalization of this concept which is formulated in the language of 2-categories, with a view toward proving new Lipschitz stability results in TDA and providing a conceptual connection between TDA and well-known concepts in geometric shape analysis. This is joint work-in-progress with Patrick McFaddin (Fordham University). |
Previous semesters
Date | Speaker | Title | Abstract |
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Aug 30 | Sam Ballas | A gentle introduction to arithmetic hyperbolc manifolds | Hyperbolic manifolds are in important class of manifolds, and are extremely prevalent, especially in low dimensions. However, how do we know these important objects exist in every dimension? It turns out that essentially all known examples of high dimensional hyperbolic manifolds are constructed via arithmetic means. In this talk I will give an overview of what arithmetic manifolds are and how they can be constructed. No knowledge beyond elementary field theory will be necessary to follow along. |
Sept 6 | |||
Sept 13 | Sam Ballas | A gentle introduction to arithmetic hyperbolc manifolds: II | In this lecture, we will extend some of the ideas intruduced last time to produce examples of compact hyperbolic manifolds in dimension 2 and 3. |
Sept 20 | Sam Ballas | A gentle introduction to arithmetic hyperbolc manifolds: III | In this final lecture, we will show how to extend the techniques from the first lectures to produce examples of hyperbolic manifolds in all dimensions. As an important step we will describe a model and corresponding isometry group for hyperbolic space in arbitrary dimensions. |
Sept 27 | |||
Oct 4 | Jared Miller | Fock-Goncharov Coordinates and Cluster Varieties | Points in Teichmuller space can be seen as conjugacy classes of representations of the fundamental group of a surface into the group PSL(2, R) satisfying several conditions including discreetness and faithfulness. In this talk we will introduce the space of framed representations of a surface group into PGL(m, C) and describe Fock-Goncharov coordinates for such a representation. We will discuss the set of positive representations as a subset of framed representations and discuss connections to cluster mutations. Time permitting, we may mention Bonahon-Dreyer coordinates for closed surfaces, which build on Fock-Goncharov's work. |
Oct 11 | Abdullah Malik | The combinatorial approach to simplicial sets and their applications | Simplicial Sets model weak homotopy types of topological spaces and ∞-groupoids, thanks to their particularly simple but powerful functorial definition. By their very nature, these models are very combinatorial. However, much of the combinatorics are hidden, and therefore, mostly forgotten in the above models. In this talk, we will delve into the details of the combinatorics of a simplicial set itself, working our way through the combinatorics of a simplicial complex and delta set, with their gluing data, thus motivating the need for a simplicial set in the first place. We will introduce terminology that lets us handle all this (infinite) data simultaneously, effectively bringing the combinatorics to the fore. Finally, time permitting, we will talk about the combinatorics of Kan complexes and simplicial homotopy. |
Oct 18 | Abdullah Malik | The combinatorial approach to simplicial sets and their applications (Part II) | Simplicial Sets are combinatorial gadgets that sit at the interface of Categories and Topological Spaces. However, much of the combinatorics are hidden, and therefore, mostly forgotten in the typical definition of a simplicial set. In Part I, we delved into the combinatorics of (abstract) simplicial complexes and delta sets, with their gluing data. Based on intuition developed from these, we will now move on to defining simplicial sets and then introduce terminology that lets us handle all its (infinite) data simultaneously, effectively bringing the combinatorics to the fore. We will show that simplicial sets of any dimension can efficiently be viewed as graphs in a functorial way, which we use as our data structures. |
Oct 25 | |||
Nov 1 | Thang Nguyen | Marked length spectrum rigidity for relatively hyperbolic groups | Burns and Katok asked, among homeomorphic manifolds of negative sectional curvature, whether the lengths of the family of marked geodesic loops determine the geometry of a manifold. I will state a coarse version of this question for finitely generated groups. After going over some previously known results, we'll focus our attention on the case of relatively hyperbolic groups. This is based on a joint work with Shi Wang. |
Nov 8 | |||
Nov 15 | Mitul Islam (Heidelberg) | Relative hyperbolicity in real convex projective geometry | Real convex projective structures provide an avenue for generalizing the notion of convex co-compactness beyond hyperbolic geometry. This generalization produces many new and interesting examples of convex co-compact groups, for instance via Coxeter reflection groups, deforming and doubling 3-manifold groups, etc. In particular, non-Gromov hyperbolic groups arise as examples (which sets it apart from hyperbolic geometry). I will discuss results (joint with A. Zimmer) that provide a complete geometric characterization of relatively hyperbolic convex co-compact groups. |
Nov 29 | Anindya Chanda | Recent Developments in Classification of Quasigeodesic Anosov Flow | A flow is called quasigeodesic if the flowlines of the lifted flow in the universal cover are quasigeodesic. In this talk we will discuss which Anosov flows are quasigeodesic? The complete answer is not known yet, but some recent results complete the classification on some specific manifolds. We will talk about the results and possible future directions. |
Date | Speaker | Title | Abstract |
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Jan 11 | |||
Jan 18 | Ling Zhou (Ohio State) | Other persistence invariants: homotopy and the cohomology ring | In topological data analysis, persistent homology has been a major tool used for measuring both geometric and topological features of datasets. In this talk, we discuss the notions of persistent homotopy groups and persistent cohomology rings. This is motivated by the fact that homotopy groups and cohomology rings are in general stronger invariants than homology.
For persistent homotopy, we pay particular attention to persistent fundamental groups for which we obtained a precise structural description via dendrograms, which induces an ultrametric on the standard fundamental group. Also, we describe the notion of persistent rational homotopy groups, which is easier to handle but still contains extra information compared to persistent homology. In the case of persistent cohomology, we consider a certain persistent graded ring structure induced by the cup product. We then lift the standard cup-length to obtain a persistent invariant which can be computed efficiently and, in analogy with the case of persistent homotopy, also complements the information captured by persistent homology. |
Jan 25 | Mario Gomez Florez (Ohio State) | Curvature Sets Over Persistence Diagrams | We study an invariant of compact metric spaces inspired by the Curvature Sets defined by Gromov. The (n,k)-Persistence Set of X is the collection of k-dimensional VR persistence diagrams of any subset of X with n or less points. This research seeks to provide a cheaper persistence-like invariant for metric spaces, which we hope will aid practical computations as well as provide insights for theoretical characterizations of VR complexes. I'll focus on theoretical results in the case n=2k+2, where we can find a geometric formula for the persistence diagram of a space with n points. We explore the application of this formula to the characterization of persistence sets of several spaces, including circles, higher dimensional spheres, and surfaces with constant curvature. We also show that persistence sets can detect the homotopy type of a certain family of graphs. |
Feb 1 | |||
Feb 8 | |||
Feb 15 | |||
Feb 22 | |||
Mar 1 | |||
Mar 8 | Leandro Lichtenfelz (Wake Forest) | Fibrations of S^3 by simple closed curves. | We show that the moduli space of all smooth fibrations of a 3-sphere by oriented simple closed curves has the homotopy type of a disjoint union of a pair of 2-spheres, which coincides with the homotopy type of the finite-dimensional subspace of Hopf fibrations. In the course of the proof, we present a pair of entangled fiber bundles in which the diffeomorphism group of the 3-sphere is the total space of the first bundle, whose fiber is the total space of the second bundle, whose base space is the diffeomorphism group of the 2-sphere. This is joint work with D. DeTurck, H. Gluck, M. Merling, J. Yang and Y. Wang. |
Mar 22 | |||
Mar 29 | |||
Apr 5 | |||
Apr 12 | |||
Apr 19 |
Date | Speaker | Title | Abstract |
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Aug 31 | |||
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Sept 14 | |||
Sept 21 | Teddy Weisman (UT Austin) | Relative Anosov representations and convex projective structures | Anosov representations are a higher-rank generalization of convex cocompact subgroups of rank-one Lie groups. They are only defined for word-hyperbolic groups, but recently Kapovich-Leeb and Zhu have suggested possible definitions for an Anosov representation of a relatively hyperbolic group - aiming to give a higher-rank generalization of geometrical finiteness. In this talk, we will introduce a more general version of relative Anosov representation, defined in terms of the topological dynamics of a relatively hyperbolic group acting on its Bowditch boundary. The definition includes many examples coming from the theory of convex projective structures - in particular, it allows for deformations of cusped convex projective manifolds (including hyperbolic manifolds) in which the cusp groups change in nontrivial ways. Moreover, these representations always satisfy a relative stability property in the representation variety, which allows us to find new examples of discrete faithful representations of relatively hyperbolic groups. |
Sept 28 | |||
Oct 5 | |||
Oct 12 | |||
Oct 19 | |||
Oct 26 | |||
Nov 2 | |||
Nov 9 | |||
Nov 16 | |||
Nov 30 |
Date | Speaker | Title | Abstract |
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Jan 12 | Christian Zickert (Maryland) 3:35p | Polylogarithms | We give a survey of polylogarithms and their appearances in various areas of mathematics including hyperbolic geometry, invariants of 3-manifolds, motivic cohomology, and cluster algebras. |
Jan 19 | Ignat Soroko (LSU) | Groups of type FP: their quasi-isometry classes and homological Dehn functions | There are only countably many isomorphism classes of finitely presented groups, i.e. groups of type \(F_2\). Considering a homological analog of finite presentability, we get the class of groups \(FP_2\). Ian Leary proved that there are uncountably many isomorphism classes of groups of type \(FP_2\) (and even of finer class FP). R.Kropholler, Leary and I proved that there are uncountably many classes of groups of type FP even up to quasi-isometries. Since `almost all' of these groups are infinitely presented, the usual Dehn function makes no sense for them, but the homological Dehn function is well-defined. In an on-going project with N.Brady, R.Kropholler and myself, we show that for any even integer \(k\ge4\) there exist uncountably many quasi-isometry classes of groups of type FP with a homological Dehn function \(n^k\). In particular there exists an FP group with the quartic homological Dehn function and the unsolvable word problem. In this talk I will give the relevant definitions and describe the construction of these groups. Time permitting, I will describe the connection of these groups to the Relation Gap Problem. |
Jan 26 | |||
Feb 2 | Jason DeBlois (Pittsburgh) | High-density packings of hyperbolic surfaces | I will give a survey-ish talk about some problems related to packing complete hyperbolic surfaces of finite area with equal-radius metric disks. For example: what is the maximal density of such packings, and how does one characterize, count, and otherwise understand the geometry of those surfaces that admit maximal-density packings? I will list several open problems and attempt to draw connections to things that other people are interested in. |
Feb 9 | Daniele Alessandrini (Columbia) | Non commutative cluster coordinates for Higher Teichmüller Spaces | In higher Teichmuller theory we study subsets of the character varieties of surface groups that are higher rank analogs of Teichmuller spaces, e.g. the Hitchin components and the spaces of maximal representations. Fock-Goncharov generalized Thurston's shear coordinates and Penner's Lambda-lengths to the Hitchin components, showing that they have a beautiful structure of cluster variety. Here we apply similar ideas to Maximal Representations and we find new coordinates on these spaces that give them a structure of non-commutative cluster varieties, in the sense defined by Berenstein-Rethak. This is joint work with Guichard, Rogozinnikov and Wienhard. |
Feb 16 | Yu-Chan Chang (Emory) | Dehn functions and abelian splittings of Bestvina--Brady groups from their defining graphs. | The Dehn function of a finitely generated group gives an upper bound of the complexity of the word problem on that group, and the Bestvina--Brady groups have been proved to satisfy quartic Dehn functions. In the first part of the talk, I will discuss a class of Bestvina--Brady groups whose Dehn functions can be identified from their defining graphs. One of the cases within our discussion of the Dehn functions is when the Bestvina--Brady groups split over \(\mathbb{Z}\). In the second part of the talk, I will discuss some non-trivial splittings of Bestvina--Brady groups over abelian subgroups. |
Feb 23 | Irene Pasquinelli (Bristol) | Deligne-Mostow lattices and branched covers of line arrangements | We will talk about lattices in the group PU(n,1) of holomorphic isometries of complex hyperbolic space. Constructing lattices in PU(n,1) has been one of the major challenges of the last decades. In particular, we will talk about a well known class that is that of the Deligne-Mostow lattices. In the first part of the talk, I will introduce the complex hyperbolic space and its space of isometries. I will then explain some equivalent ways of constructing the Deligne-Mostow lattices. Among these, I will concentrate on the construction by Bartel-Hirzebruch-Hoefer, which uses branched covers, ramified over a line arrangement in complex projective 2-space. Finally I will explain to you how this interpretation gives hope towards a complex equivalent of the hybridisation technique. |
Mar 2 | Darren Long (UCSB) | Zariski dense surface groups in SL(2k+1,Z) | I'll introduce some of the history of thin groups and discuss a proof that there are Zariski dense surface groups in SL(2k+1,Z). |
Mar 9 | |||
Mar 16 | Corey Bregman (Southern Maine) 2:05p | Outer Space for Right-angled Artin Groups | Right-angled Artin groups (RAAGs) span a range of groups from free groups to free abelian groups. Thus, their (outer) automorphism groups range from Out(F_n) to GL(n,Z). Automorphism groups of RAAGs have been well-studied over the past decade from a purely algebraic viewpoint. To allow for a more geometric approach, one needs to construct a contractible space with a proper action of the group. I will present joint work with Ruth Charney and Karen Vogtmann in which we construct an analogue of Culler-Vogtmann’s Outer Space for arbitrary RAAGs, and discuss further directions of study. |
Mar 23 | Harry Bray (George Mason) | Volume-entropy rigidity for convex projective manifolds | I will discuss joint work with Constantine, building on joint work with Adeboye and Constantine, on a volume-entropy rigidity result for finite volume strictly convex projective manifolds in dimension at least 3. The result is a Besson-Courtois-Gallot type theorem, using the barycenter method. As an application, we get a uniform lower bound on the Hilbert volume of a finite volume strictly convex projective manifold of dimension at least 3. |
Mar 30 | Christian El Emam (Luxembourg) | Immersions of surfaces into SL(2,C) as an approach to transition geometry | We will discuss SL(2,C) equipped with its global complex killing form, which turns out to be a very interesting and regular space: in fact, SL(2,C) can be seen as a complex analog of the Riemannian notion of "space form". Moreover, every 3-dimensional pseudo-Riemannian space form of constant curvature -1, such as H^3, AdS^3, and -S^3, embed in a canonical isometric way into SL(2,C), providing a complex formalism to "transit" from one geometry to the other. As an application, the theory of immersions of surfaces into SL(2,C) generalizes the usual theory of immersions into pseudo-Riemannian space forms, and a holomorphic variation of the immersion data into SL(2,C) - passing for instance from an immersion into H^3 to one into AdS^3 - provides a holomorphic variation of its monodromy. This is joint work with Francesco Bonsante. |
Apr 6 | Anindya Chanda (FSU) | Pujal’s Conjecture And Its recent Development | A monster problem in the study of partially hyperbolic dynamics is to classify partially hyperbolic maps. Pujal’s conjecture served as one the principal motivation towards the classification problem for the last twenty years. Though the conjecture has been proved to be false in the general case, it is still relevant in its new form in more specific contexts. In our talk we will survey the recent developments, open problems and new examples of partially hyperbolic maps in dimension 3 made in the last few years. |
Apr 13 |
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Aug 25 | |||
Sep 1 | |||
Sep 8 | |||
Sep 15 | Dani Kaufman (Maryland) | Markov Numbers, Teichmüller Theory and Cluster Algebra Invariants. | The Markov numbers have been studied in connection with quadratic forms and diophantine approximation since the 1880s. These numbers satisfy a simple diophantine equation and have a particularly nice way of generating them via exchanges. More recently, connections between these numbers and the hyperbolic geometry of the once punctured torus, S_{1,1}, have been found. The key relationship is that the Markov numbers appear as the coordinates of a particular point in the Teichmüller space of S_{1,1}, and the exchanges correspond to changes of coordinate systems. I will give an introduction to these types of coordinates on Teichmüller spaces and an introduction to the cluster algebra structures underlying them. I can then abstract away the geometry from this picture to give new examples of diophantine equations with solutions coming from exchanges via the notion of a "cluster algebra invariant" |
Sep 22 | |||
Sep 29 | |||
Oct 6 (9:30a) | Gye-Seon Lee (Sungkyunkwan University) | Convex real projective Dehn filling | Hyperbolic Dehn filling theorem proven by Thurston is a fundamental theorem of hyperbolic 3-manifold theory, but it is not true anymore in dimension > 3. Since hyperbolic geometry is a sub-geometry of convex real projective geometry, it is natural to ask whether Thurston’s Dehn filling theory for hyperbolic 3-manifolds can generalize to convex real projective manifolds in any dimension. In this talk, I will give evidence towards a positive answer to the question. Joint work with Suhyoung Choi and Ludovic Marquis. |
Oct 13 | Sara Maloni (Virginia) | Convex hulls of quasicircles in hyperbolic and anti-de Sitter space. | Thurston conjectured that quasi-Fuchsian manifolds are determined by the induced hyperbolic metrics on the boundary of their convex core and Mess generalized those conjectures to the context of globally hyperbolic AdS spacetimes. In this talk I will discuss a universal version of these conjectures (and prove the existence part) by considering convex sets spanning quasicircles in the boundary at infinity of hyperbolic and anti-de Sitter space. This work generalizes Alexandrov and Pogorelov’s results about the characterization metrics induced on the boundary of a compact convex subset of hyperbolic space. Time permitting, we will discuss why in hyperbolic space quasicircles can't be characterized by the width of their convex hulls, except when the convex hulls have small width. This is different than the anti-de Sitter setting, as Bonsante and Schlenker showed. (This is joint work with Bonsante, Danciger and Schlenker.) |
Oct 20 (2:30p) | Ludovic Marquis (IRMAR) | Strongly convex-cocompact reflection group | I will present the notion of strongly convex-cocompact subgroup of PGL_d+1 (R), which generalizes the notion of convex-cocompact subgroup of the isometry group of the hyperbolic space. This notion has connections with projective Anosov groups. Reflection groups are image of Coxeter groups under some representation introduced by Vinberg in the 60's. We will characterize which reflection groups acts as strongly convex-cocompact group and present all the aforementioned notions. In the end, we will build several new projective Anosov groups. This is joint work with Jeff Danciger, François Guéritaud, Fanny Kassel and Gye-Seon Lee. |
Oct 27 | John Bowers (James Madison) | Obtaining Koebe-Andre’ev-Thurston packings via flow from tangency packings | Recently, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation, flip-and-flow, that allows two tangency packings whose contact graphs differ by a combinatorial edge flip to be continuously deformed from one to the other while maintaining tangencies across all of their common edges. Starting from a canonical tangency circle packing with the desired number of circles a finite sequence of flip-and-flow operations may be applied to obtain a circle packing for any desired (proper) contact graph with the same number of circles. In this talk, I will show how to extend the Connelly-Gortler method to allow circles to overlap by angles up to π/2. This results in a new proof of the general Koebe-Andre’ev-Thurston theorem for disk packings on S^2 with overlaps and a numerical algorithm for computing them. The development makes use of the correspondence between circles and disks on S^2 and hyperplanes and half-spaces in the 4-dimensional Minkowski spacetime R^(1,3). Along the way I will generalize a notion of convexity of circle polyhedra that has recently been used to prove the global rigidity of certain circle packings and use this view to show that all convex circle polyhedra are infinitesimally rigid, generalizing a recent related result. |
Nov 3 | Braulio Molina-Gonzalez (FSU) | Obstruction to dynamical coherence in partially hyperbolic dynamics of 3-manifolds | Partially hyperbolic dynamics is a (relative) new area of dynamical systems, which seeks to understand what properties of hyperbolic dynamics, extend to more general classes of systems. In this talk, I will introduce partially hyperbolic diffeomorphisms, and examples of manifolds that support and do not support these systems. I will then introduce dynamical coherence and talk about a stronger notion of partial hyperbolicity on the 3-torus which guarantees dynamical coherence. Lastly, I will proceed to a result by Hertz, Hertz, Ures which provides an obstruction for a partially hyperbolic diffeomorphism of 3 manifolds to be dynamical coherent. |
Nov 10 | Jeff Danciger (Texas) | Exotic real projective Dehn surgery space | 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 3-manifolds admit a second convex projective structure not obtained through deformation. We find these structures 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. |
Nov 17 | Anindya Chanda (FSU) | An Example of A 2-dimensional Foliation with Both Funnel and Non-Funnel Leaves | A funnel leaf of a two-dimensional foliation is a hyperbolic leaf foliated by quasi-geodesics, emitting from or merging at a common point on the ideal boundary of the leaf. In this project we produce an example of a two-dimensional foliation F with Gromov-hyperbolic leaves on a 3-manifold M such that each leaf of F is foliated by quasi-geodesics and F has both funnel and non-funnel leaves. To construct such a foliation, we start with the Franks-Williams flow and consider a foliation which is transverse to both weak-stable and weak-unstable foliation given by the flow. |
Dec 1 |
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Feb 4 | Suhyoung Choi (KAIST) | Lens-shaped totally geodesic ends of convex real projective manifolds | We will study totally geodesic ends of convex real projective n-manifolds. These are ends that we can compactify by totally geodesic manifolds of codimension-one. A sufficient condition for lens-shaped end-neighborhoods to exist for a totally geodesic end is the condition of the uniform-middle-eigenvalues on the end-holonomy-group. Every affine deformation of a discrete dividing linear group satisfying this condition acts on properly convex domains in the affine n-space. We also discuss the relationship to the globally hyperbolic space-times in flat Lorentz geometry. Finally, we show that lens-shaped radial ends are dual to lens-shaped totally geodesic ends. |
Feb 11 | |||
Feb 18 | |||
Feb 25 | |||
Mar 3 | Lorenzo Ruffoni | Generalized Bestvina-Brady groups over block graphs | Given a finite simplicial graph, the right-angled Artin group (RAAG) associated to it is the group generated by its vertices, in which two generators commute when they are connected by an edge; so RAAGs interpolate between free and free abelian groups. Bestvina and Brady studied the subgroup of a RAAG given by the kernel of the map obtained by sending every generator to the integer number 1, and discussed its finiteness property in terms of the combinatorics of the underlying graph G. More generally any labeling of the vertices by integer numbers provides an integral character of the group, and in this talk we will discuss how choosing the labeling affects the rank of the kernel, for a specific class of graphs. The topological motivation consists in the fact that RAAGs defined over trees arise as fundamental groups of certain non-compact 3-manifolds, and different integral characters correspond to different fibrations over the circle. This is a joint work with M. Barquinero and K. Ye. |
Mar 10 | Jared Miller | Positive Representations and Fock-Goncharov Coordinates | In this talk we will discuss representations of the fundamental group of a punctured surface into PGL(m, C). We will define flags and discuss how to use invariants of triples and quadruples of flags to construct coordinates on a surface. These coordinates, called Fock-Goncharov coordinates, give a structure on the space of representations. We will then discuss the converse, starting with coordinates on a surface and constructing elements of PGL(m, C) to associate to each element of the fundamental group. Most of the focus will be on the \(m=2\) case, which is related to hyperbolic structures. |
Mar 24 | Anindya Chanda | ||
Mar 31 | Ben Prather | ||
Apr 7 | Phil Bowers | ||
Apr 14 | John Bergschneider | ||
Apr 21 | Opal Graham |
Date | Speaker | Title | Abstract |
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Sep 3 | Ben Prather | Dueling Metric Spaces | We consider smooth manifolds endowed with two metrics and the resulting geometric structures. A motivating example come from classical tests of general relativity, where an empirical Galilean metric is useful for making measurements while a theoretical metric provides fundamental insight. The equations of motion using one metric can be interpreted as motion in a refractive medium measured using the other metric. Some of these structures have a passing resemblance to tetrad fields. The difference between these structures will be discussed. |
Sep 10 | Alex Casella | Tame CP1 structure on the thrice punctures sphere with triangular holonomy. | CP1 structures are geometric structures modelled on the complex projective line, acted on by the projective group PSL(2,C). These structures are not as rigid as Riemmannian structures (like Euclidean, hyperbolic or spherical), nor as flexible as conformal structures. For example they still allow the notion of circles and therefore can be used to study circle packings. In this talk we show that all CP1 structures on the thrice punctured sphere -with triangular holonomy- that are tame (i.e. the developing map extends continuously to the ends) can be constructed by elementary cutting and gluing (i.e. grafting) on simple triangular structures. The talk will be accessible to non-experts, with minimal background in topology. |
Sep 17 | Sam Ballas | Arithmetic hyperbolic manifolds in low dimensions I | I will explain how we can use techniques from algebraic number theory to construct examples of low dimensional hyperbolic manifolds. Outside of some elementary Galois theory, no background in number theory will be necessary to follow along. |
Sep 24 | Sam Ballas | Arithmetic hyperbolic manifolds in low dimensions II | In this talk I will discuss how we can extend the techniques from last time to produce examples of compact hyperbolic manifolds in all dimensions. |
Oct 1 | Lorenzo Ruffoni | Moving branch points on complex projective structures | We consider CP1-structures on closed surfaces of genus at least 2, which are geometric structures locally modeled on the geometry of Möbius transformations of the Riemann sphere. Every Riemann surface admits a CP1-structure by uniformization, and vice versa a CP1-structure induces a complex structure on the surface. In the unbranched case a CP1-structure is uniquely determined by the underlying complex structure together with another invariant which takes the form of a representation of the fundamental group into PSL(2,C). If we allow branch points, then these structures admit non-trivial deformations which preserve this representation, and in this talk we consider the problem of understanding if the induced deformation of the underlying complex structure is trivial, in terms of conditions on the collection of branch points. We will present some old and new results, and possibly discuss some (motivating) interactions with the theory of a certain class of ODEs. This is joint work with S. Francaviglia. |
Oct 8 | Wolfgang Heil | 2-stratifolds and 3-manifolds | 2-stratifolds are a generalization of 2-manifolds in that there are disjoint simple closed branch curves. Since every 3-manifold has some 2-complex as a spine one may ask which 3-manifolds have 2-stratifold spines. In this talk we will define 2-stratifolds and obtain a list of all closed 3-manifolds with 2-stratifold spines. This is joint work with J.C. Gomez-Larranaga and F. Gonzalez-Acuna |
Oct 15 | Phil Bowers | Polyhedra from Steiner to Rivin: Rigidity, Inscription, and Existence | Our primary objective is a presentation of Rivin's characterization of ideal hyperbolic polyhedra obtained in the period 1988-1996 and its application to completely resolve a question of Steiner’s from 1832 on the inscribability of Euclidean polyhedra in the 2-sphere. We will take a historical tour from Steiner to Rivin and beyond, with side trips to visit Cauchy (1813), Dehn (1916), Steinitz (1928), Aleksandrov (1950’s), Andre’ev (1970), Gluck (1975), Connelly (1977), Thurston (1978), Shulte (1985), Schramm (1991), Bao-Bonahon (2003), Chen-Schlenker (2017), and Bowers-B-Pratt (2018), as time permits. |
Oct 22 | John Bergschneider | Trivalent 2-Stratifolds | 2-stratifolds are compact topological spaces such that any point has a neighborhood homeomorphic to n-sheets meeting. They are a generalization of surfaces as graphs are a generalization of 1-manifolds. Unlike surfaces, 2-stratifolds are not uniquely determined by their fundamental group and have no general classification. In this talk, we will explore how to classify trivalent 2-stratifolds by their fundamental group. This work is supervised by Wolfgang Heil. |
Oct 29 | Anindya Chanda | Zimmer's Conjecture | The Conjecture emerged for Robert Zimmer in the period between late 1970s to early 1980s. Zimmer’s conjecture concerns special kinds of symmetries known as higher-rank lattices. It asks if the dimension of a geometric space limits whether or not those types of symmetries apply. The authors of the new work — Aaron Brown and Sebastian Hurtado-Salazar of the University of Chicago and David Fisher of Indiana University showed that below a certain dimension, these special symmetries can’t be found. They proved Zimmer’s conjecture true. This is what Quanta magazine says about the proof : 'Their proof stands as one of the biggest mathematical achievements in recent years.' In our introductory talk we will discuss about the statement of the conjecture and various rigidities on dynamics of groups. |
Nov 5 | Opal Graham | Rigidity of Points and Planes in Hyperbolic Space | Points on the ideal boundary of hyperbolic space are infinitely far away from points and planes in hyperbolic space. This makes it difficult to uniquely place an ideal point based upon its position to objects in hyperbolic space. I define a new conformal invariant we can use to analyze rigidity of collections of intermingled hyperbolic points, planes, and ideal points up to hyperbolic isometry. |
Nov 12 | Jared Miller | Existence of Branched Coverings | A branched covering is a map between closed, connected surfaces which is a covering map everywhere except for a nowhere dense set known as a branch set. Several necessary conditions are known for the existence of a branched covering between surfaces, but in general these conditions are not sufficient for a branched covering to be realized. In this talk we will discuss when a candidate branched covering is actually realized. |
Nov 19 | Facundo Memoli (Ohio State) | The Gromov-Hausdorff distance between ultrametric spaces | We study a variant of the Gromov-Hausdorff distance which is fine tuned to ultrametric spaces. This distance, denoted uGH, is itself an ultrametric on the collection of all compact ultrametric spaces. We will discuss the topology generated by uGH and some of its structural properties. Whereas it is well understood that computing the standard Gromov-Hausdorff distance between finite (ultra) metric spaces is difficult (it is an NP-hard computational problem), we find that computing uGH can be done in time which depends polynomially on the cardinality of the two given ultrametric spaces. |
Dec 3 | Thanitta Kowan | Classification on Planar Polygonal Complexes | This talk describes the metric space of isomorphism classes of rooted planar polygonal complexes and its properties. The metric space can be separated into either hyperbolic subspace or parabolic subspace by considering extremal length on planar polygonal complexes. The hyperbolic space is a union of countably many nowhere dense closed subsets forming the first category subset in the metric space, while the parabolic space is of the second category subset. |
Date | Speaker | Title | Abstract |
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Jan 15 | Alex Casella | Let’s Talk! | Communication is a fundamental skill in life. Whether you are professor, a scientist or a salesman, you will often find yourself talking in front of an audience. In this talk I will present my own experience regarding the preparation and execution of a scientific talk, covering most common DOs and DON’Ts. |
Jan 22 | Lorenzo Ruffoni | Strict hyperbolization and its applications | Gromov introduced some procedures to turn a given polyhedron into a new one endowed with a piecewise Euclidean metric of non-positive curvature, while preserving some of its original topological features. In this talk we will describe a refinement of Gromov's construction due to Charney and Davis, in which the new space carries a strictly negatively curved metric, and thus has hyperbolic fundamental group. Some applications will be discussed |
Jan 29 | Ettore Aldrovandi | Homotopy theoretic aspects of central extensions | The classification of central extensions of a group \(G\) by a
(necessarily abelian) group \(A\) is very well understood from both the
Algebra and Geometry viewpoints. Indeed, one of the most interesting
aspects is the interplay between the two.
A much more interesting situation is when the topology is directly part of the structure, for example if \(G\) and \(A\) are topological groups. Alternatively, since the category of topological spaces and that of simplicial sets have equivalent homotopy theories, we can assume they are simplicial groups. I will discuss some of the aspects of the classification of central extensions in this context, centered around the statement that such classification is given by homotopy classes of maps \(BG \to B^2A\) between classifying spaces. This is work in progress in collaboration with Niranjan Ramachandran (UMD) and my student Michael Niemeier. |
Feb 5 | John Bergschneider | Finite 2-Stratifold Groups | 2-Stratifolds are a generalization of closed surfaces in that they contain simple closed curves where several sheets meet. Currently, there is no general classification of these spaces or their fundamental groups. Most finitely generated Fuchsian Groups and some generalized triangle groups can be realized as the fundamental of a 2-stratifold. We explore a possible solution on how to classify finite 2-stratifold groups and how Fuchsian groups and generalized triangle groups are involved. |
Feb 12 | |||
Feb 19 | |||
Feb 26 | Kate Petersen | Representations and Hyperbolic structures on knot complements | Thurston’s hyperbolic Dehn surgery theorem gives a geometric picture of many (mostly incomplete) hyperbolic structures on a knot complement. I’ll discuss how this can be used to define representations of the knot group into SL(2,C) in a completely diagrammatic way. |
Mar 5 | |||
Mar 12 | Woojin Kim | Persistent homology for time-evolving metric/network data | Characterizing the dynamics of time-evolving data within the framework of topological data analysis has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in animals and social networks in the human sphere. A natural mathematical model for such collective behaviors is a dynamic metric space (DMS)/dynamic network (DN). We will discuss (1) how to induce a multiparameter/zigzig persistent homology as an invariant of a DMS/DN, and (2) stability of these invariants. In order to address the stability, we extend the Gromov-Hausdorff distance on metric spaces to the setting of DMSs/DNs. This is a joint work with Facundo Memoli and Zane Smith |
Mar 26 | Anindya Chanda | Classification of Partial Hyperbolic Automorphisms on 3-Manifolds | The notion of hyperbolicity of a Automorphisms on a Riemannian Manifold and it's properties were introduced around 1960-70. But the idea of partial hyperbolicity was not much explored at that time. After 1995, it was proven that partial hyperbolic systems satisfy some very strong and exceptional properties in the field of Dynamical Systems and Ergodic Theory and those results greatly motivated the study of partial hyperbolicity. But till today it is a largely open area of research and the field is not discovered in great details, specially very little is known about the dimensions greater or equal to 4. In this talk we will try to present a (partial) classification of Partial Hyperbolic Automorphisms over the dimension 3. |
Apr 2 | Sam Ballas | Gluing equations for projective structures on 3-manifolds | One of Thurston’s many amazing ideas are his hyperbolic gluing equations. Roughly speaking, given an ideally triangulated 3-manifold M, one can construct a set of complex polynomial equations whose solutions correspond to hyperbolic structures on M. In this talk I will describe some work in progress (joint with Alex Casella) on generalizing these equations in the context of real projective structures. After describing our parameters and equations, I will describe how a solution enables one to build a developing map, find a holonomy representation for a real projective structure and draw some nice pictures of the developing image. |
Apr 9 | |||
Apr 16 | Aamir Rasheed | Surface subgroups of 3-manifold groups. | A closed irreducible 3-manifold M with infinite fundamental group is uniquely determined up to homeomorphism by its fundamental group. One can understand the topology of M by studying its group structure and conversely the group structure can be understood by studying the topology. One important tool in this study is the study of embedded surfaces. The image of the fundamental groups of these surfaces (surface subgroups) in the fundamental group of M encode a lot of useful information about the topology and geometry of M. In this talk we will discuss this relationship further. In particular, we will see, how various properties such as malnormality and maximality of surface subgroups give us information about the 3-manifold itself. |
Apr 23 | Daniel Hartman | Anosov flows and contact surgery | Until a few years ago, the only know examples of contact Anosov flows were geodesic flows of Riemannian manifolds. In 2013, Patrick Foulon and Boris Hasselblatt gave a surgery method which, when performed along an E-transverse link, results in a new contact Anosov flow. This surgery method subsumes the Handel-Thurston and Goodman surgeries. The goal of the talk will be to outline the surgery |
Date | Speaker | Title | Abstract |
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Sept 4 | Ben Prather | The Symmetry of the Octonions | The octonions are noted for there exceptional symmetry. This can be expressed by examining the size of automorphism group relative to other algebras of a similar size. In particular, the automorphism group of the octonions is the 14 dimensional compact real form of the exceptional Lie group \(G_2\), while the Clifford algebras over three dimensional space have the three dimensional SO(3) as their automorphism group. |
Sept 11 | Sam Ballas | Geometric Structures via Flags | Roughly speaking a geometric structure is a recipe for gluing together pieces of a geometric space using geometry preserving maps to build a manifold with interesting topology. Some classical examples of this are spherical, hyperbolic, and Euclidean structures on surfaces. In this talk I will describe a method for constructing hyperbolic structures on punctured surfaces and if time allows describe how this can be generalized to produce interesting projective structures on these surfaces. |
Sept 18 | Alex White | Central Strips of Sibling Leaves in Laminations of the Unit Disk | Quadratic laminations of the unit disk were introduced by Thurston as a vehicle for understanding the Julia sets of quadratic polynomials and the parameter space of quadratic polynomials. The Central Strip Lemma plays a key role in Thurston's classification of gaps in quadratic laminations and in describing the corresponding parameter space. This paper will generalize the notion of "Central Strip" to laminations of all degrees and prove a Central Strip Lemma for higher degrees. |
Sept 25 | Alex White | Central Strips of Sibling Leaves in Laminations of the Unit Disk (Part II) | Quadratic laminations of the unit disk were introduced by Thurston as a vehicle for understanding the Julia sets of quadratic polynomials and the parameter space of quadratic polynomials. The Central Strip Lemma plays a key role in Thurston's classification of gaps in quadratic laminations and in describing the corresponding parameter space. This paper will generalize the notion of "Central Strip" to laminations of all degrees and prove a Central Strip Lemma for higher degrees. |
Oct 2 | |||
Oct 9 | Ettore Aldrovandi (Cancelled Due to Hurricane) | Homotopy theoretic aspects of central extensions | The classification of central extensions of a group \(G\) by a
(necessarily abelian) group \(A\) is very well understood from both the
Algebra and Geometry viewpoints. Indeed, one of the most interesting
aspects is the interplay between the two.
A much more interesting situation is when the topology is directly part of the structure, for example if \(G\) and \(A\) are topological groups. Alternatively, since the category of topological spaces and that of simplicial sets have equivalent homotopy theories, we can assume they are simplicial groups. I will discuss some of the aspects of the classification of central extensions in this context, centered around the statement that such classification is given by homotopy classes of maps \(BG \to B^2A\) between classifying spaces. This is work in progress in collaboration with my student Michael Niemeier. |
Oct 16 | Daniel Hartman | The h-cobordism theorem | The h-cobordism theorem is due to S. Smale and is one of the main tools in high dimensional topology. It and it's corollary, the generalized Poincar\'e conjecture earned Smale the Fields medal. The theorem is proved with the use of Morse theory, homology, and "Whitney's trick". Outlining the proof will be the goal of the talk. |
Oct 23 | Thanittha Kowan | Average Kissing Numbers For Non-Congruent Sphere Packings | I will introduce a definition of kissing numbers of a sphere packings in 3-dimensional real space. We will talk about Greg Kuperberg and Oded Schramn’s Theorem showing that the average numbers for non-congruent sphere packings is bounded. The goals is to go over the proof of the upper bound of the theorem. |
Oct 26 | Osman Okutan
(1:25pm in LOV 201) |
Metric Graph Approximations of Geodesic Spaces | A standard result in metric geometry is that every compact geodesic metric space can be approximated arbitrarily well by finite metric graphs in the Gromov-Hausdorff sense. It is well known that the first Betti number of the approximating graphs may blow up as the approximation gets finer. In our work, given a compact geodesic metric space X , we define a sequence \((\delta^ X_n )_{ n ≥ 0}\) of non-negative real numbers by \(\delta^X_n:=\inf \{d_{GH}(X,G):G {\rm\ a\ finite\ metric\ graph\ }, \beta_1(G)≤n\}.\) By construction, and the above result, this is a non-increasing sequence with limit 0. We study this sequence and its rates of decay with n. We also identify a precise relationship between the sequence and the first Vietoris-Rips persistence barcode of X . Furthermore, we specifically analyze \(\delta^X_0\) and find upper and lower bounds based on hyperbolicity and other metric invariants. As a consequence of the tools we develop, our work also provides a Gromov-Hausdorff stability result for the Reeb construction on geodesic metric spaces with respect to the specific function given by distance to a reference point. This is a joint work with Facundo Memoli. For more detail, please see arxiv preprint: https://arxiv.org/abs/1809.05566 |
Oct 30 | John Bergschneider | 2-Stratifolds | 2-Stratifolds are a generalization of surfaces where there is a family of disjoint simple closed curves where several sheets meet. They are able to be represented as bipartite graphs from which their fundamental group can be read. We will discuss the different types of regular neighborhoods of the exceptional simple closed curves and how the neighborhoods affect the fundamental group. Then we will then enumerate some highlights about what is known about 2-stratifolds. |
Nov 6 | Braulio Molina Gonzalez | Partially Hyperbolic Dynamics: An Extension of Uniform Hyperbolicity | In the talk I will introduce Uniform Hyperbolicity and give a background on its development, some examples and some mayor important results in the Theory. Then I will introduce Partially Hyperbolic Dynamics as a generalization from Uniform Hyperbolicity and if time permits I will talk about some examples/results fron Partially Hyperbolic Dynamics in 3-manifolds with solvable fundamental group. |
Nov 13 | Anindya Chanda | Gromov Boundary of Hyperbolic Groups | Geometric Group Theory is one of the most explored topic in Mathematics for last few decades. In this topic we mostly try to find connections between abstract algebraic properties of groups and geometric properties of spaces on which those groups acts 'nicely'. The idea of boundary of a Gromov-hyperbolic space is to associate the so-called 'space of infinity' and to compactify the given space. This boundary turns out to be an extremely useful tool to study the hyperbolic groups. |
Nov 27 | Opal Graham | The Rigidity of Configurations of Points and Spheres | We discuss the work of Beardon & Minda, and Crane & Short in the rigidity of points and spheres, and how their results can be improved upon when independence is introduced as an extra condition. |
Dec 4 | Aamir Rasheed | Maximal surface groups and essential embeddings and immersions of surfaces in a 3-manifold | In this talk we will discuss the relationship between maximal surface groups and embedded essential surfaces in a 3-manifold. Furthermore we will also discuss a theorem which gives a sufficient condition for two immersed surfaces in an irreducible 3-manifold to be homotopic. |