Date | Speaker | Title | Abstract |
---|---|---|---|
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 |
Previous semesters
Date | Speaker | Title | Abstract |
---|---|---|---|
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. |