Eighth Annual Graduate Student Topology and Geometry Conference


Speakers

There will be two keynote lectures by:

Douglas Ravenel (University of Rochester)
Alan Reid (University of Texas, Austin)

New this year, there will be three open problem sessions led by:

Moon Duchin (University of Michigan, Ann Arbor) - Geometric Group Theory
Tom Fiore (University of Michigan, Dearborn) - Homotopy Theory
Benjamin Schmidt (Michigan State University) - Differential Geometry


Abstracts of Talks

The time and location of all talks can also be found on the schedule page.

Keynote Speakers

Click here to download a .pdf version of the keynote and young faculty abstracts.

 

The Arf-Kervaire invariant problem
Douglas Ravenel - University of Rochester
Sa 4:40-5:40, 1040 Dana Building

Mike Hill, Mike Hopkins and I recently solved the 50 year old Arf-Kervaire invariant problem in algebraic topology. The talk will describe the background and history of the problem and give a brief overview of the proof of our main theorem. More information can be found online at http://www.math.rochester.edu/u/faculty/doug/kervaire.html.

 

Geometric properties of arithmetic hyperbolic 3-manifolds
Alan Reid - UT Austin
Sa 9:30-10:30, 1040 Dana Building

Arithmetic hyperbolic 3-manifolds are an important subclass of hyperbolic 3-manifolds of finite volume which arise from number theoretic constructions. This talk will focus on how one might try to understand these manifolds geometrically, and study their geometric properties.

 

 

Young Faculty Speakers

 

Problems in the large-scale geometry of groups
Moon Duchin - University of Michigan, Ann Arbor
Sa 2:00-3:00, 1024 Dana Building

It has been extremely productive in group theory to view things from far away, say by looking at the boundary at infinity, the asymptotic cone, or long-term dynamics. I'll introduce some classes of groups everyone should know and love and use those to illustrate open problems in geometric group theory, ranging from fiendishly hard to alluringly approachable.

 

Higher Operads
Tom Fiore - University of Michigan, Dearborn
Sa 2:00-3:00, 1040 Dana Building

Operads and their many variants are important tools for topologists because of their simplifying power and wide variety of applications. This talk will be an introduction to the theory of operads and its applications, with a chronological tour highlighting: the homotopy associative structure on loop spaces, the renaissance in the 90's, Batanin's operads in higher category theory, the quasi-operads of Moerdijk-Weiss, and the in finity-operads of Lurie. Open problems and exercises will be mentioned along the way.

 

Questions about geodesics in Riemannian manifolds
Benjamin Schmidt - Michigan State University
Sa 2:00-3:00, 1046 Dana Building

In a Riemannian manifold, geodesics are paths that locally minimize length. Although their local behavior admits this simple description, their global behavior can be much more complicated. I'll discuss several questions concerning geodesics, many of which aim to metrically characterize the most symmetric spaces.

 

 

Student Speakers

Click here to download a .pdf version of these abstracts.

Graphs and free groups
Muhammad Adeel - University of Utah
Su 12:00-12:30, 1024 Dana Building

The free group derives its importance from the fact that every group is a quotient of some free group. The topological treatment of free group not only results in a theory which has geometric contents but also in elegant proofs of many important theorems. In this talk, the theory of free groups will be explore via graphs and covering space theory. We will also look at the graph product of graph of groups which generalizes free product and HNN extension.

 

Linear flows of line bundles over a Riemann surface as solutions to Lax pairs
Mio Alter - UT Austin
Su 11:20-11:50, 1046 Dana Building

Matrix polynomial functions $A$ and $B$ which satisfy $\frac{dA}{dt}=[A,B]$ are called a Lax pair. I will present a construction of Hitchin's which gives a concrete way of interpreting linear flows on the space of line bundles over a Riemann surface as solutions to Lax pairs.

 

Heegaard what?. A gentle introduction to Heegaard Splittings via Complexes of Curves.
Carlos Barrera-Rodriguez - University of California Davis
Sa 11:20-11:50, 1028 Dana Building

In 1978 William J. Harvey introduced a simplicial structure for the collection of homotopy classes of essential simple closed curves in a surface $S$, commonly called Curve Complex of $S$, to study the geometry of the action of the modular group of $S$ on the Teichm\"uller space of $S$. Later, in 1997, John Hempel discovered intrinsic relations between Heegaard Splittings of 3-manifolds with splitting surface $S$ and the curve complex of $S$, work which inspired many topologists and geometrists to study the topic more carefully. In this talk I will discuss the intrinsic relation between curve and pants complexes and the geometry and topology of 3-dimensional manifolds. If time permits I will be discussing more specific and recent development in the area and my work on new curve complexes.

 

The S^1 Equivariant Generating Hypothesis
Anna Marie Bohmann - University of Chicago
Su 12:00-12:30, 1040 Dana Building

The Freyd generating hypothesis is a long-standing conjecture in stable homotopy theory. An analogous conjecture can be formulated in any triangulated category with a set of compact generators. We formulate the appropriate conjecture in the equivariant stable homotopy category of a compact Lie group G. We then use work of Greenlees to show that this conjecture fails in the rational S^1 stable homotopy category. In fact, we find an explicit counterexample using familiar spaces.

 

Broken Lefschetz fibration and their examples
Ka Choi - UC Berkeley
Sa 4:00-4:30, 1046 Dana Building

Lefschetz fibration gives a topological way to understand symplectic 4-manifold. It turns out every closed oriented 4-manifold admits a broken Lefschetz fibration which is a kind of generalization of Lefschetz fibration. We will look at some examples and see what people have done with it.

 

An Intuitive Introduction to Homotopy Colimits
Priyavrat Deshpande - University of Western Ontario
Sa 11:20-11:50, 1040 Dana Building

Homotopy colimit is an important idea originating in homotopy theory, that was developed by Quillen, Bousfield, Kan and others. It has not only reached remarkable extension and depth but it has also proved to be an versatile tool in a lot of other areas of mathematics. The aim of my talk is to motivate and explain the construction of homotopy colimits together with some applications. I will do it using examples and pictures avoiding technical jargon from category theory.

 

Dilatations of pseudo-Anosovs in the point-pushing subgroup
Spencer Dowdall - University of Chicago
Su 11:20-11:50, 1024 Dana Building

Given a loop in the fundamental group $\pi_1(S,p)$ of a surface $S$, one may construct an element of the based mapping class group $\Mod(S,p)$ by pushing the basepoint $p$ around the loop. There is a nice geometric characterization, due to Kra, of when such a point-pushing mapping class is pseudo-Anosov. Every pseudo-Anosov mapping class has a unique number, called the dilatation, which can be thought of as its "stretching factor." We establish a lower bound on the dilatation of a point-pushing pseudo-Anosov in terms of the self-intersection number of the pushing curve. Our method uses properties of the Bass-Serre tree to estimate intersection numbers in hyperbolic space.

 

Closed geodesics on compact developable orbifolds.
George Dragomir - McMaster
Su 12:00-12:30, 1046 Dana Building

The study of existence of closed geodesics is an old and beautiful topic in classical Riemannian geometry. The importance of closed geodesics comes not only from their applications to the Riemannian geometry in the large, but also from their relationship to other branches of mathematics. One of the reasons for the interest in orbifolds is that they have similar geometric properties to manifolds. It is well known that any compact Riemannian manifold admits at least one nontrivial closed geodesic. However, the question of existence of closed geodesics of positive length on all compact Riemannian orbifolds is still not answered. In the classical case the solution to this problem involves the study of the critical points of the energy function on some version of the space of closed curves. Guruprasad and Haefliger have adapted the same method to the orbifold setting and obtained the existence of closed geodesics for a large class of compact orbifolds, including all the non-developable ones. In this talk I will describe an approach fit for the developable case which not only gives us alternative elementary proofs for the known results but also further reduces the existence problem to a very particular type of developable orbifolds.

 

Anosov representations and length functions
Guillaume Dreyer - University of Southern California
Su 9:30-10:0, 1028 Dana Building

Let S be a closed surface of negative Euler characteristic. We consider the space Rep_n(S) of homomorphisms from the fundamental group of S to SL_n(R), with n > 2. Hitchin gives a complete description of the connected components of the space Rep_n(S). A particularly interesting component is the one containing the Fuchsian representations, which is called the Hitchin component. Given a curve c on S and a representation r in the Hitchin component of Rep_n(S), we can consider the eigenvalues of r(c). We show how to extend these eigenvalue functions to length functions on the space of measured laminations on S, or more generally to the space of Holder geodesic currents. This is based on Labourie's dynamical characterization of those representations which are in the Hitchin component.

 

Hey dude, where's the bathroom?
Timothy Emerick - University of Virginia
Su 9:30-10:00, 1024 Dana Building

Buildings are classically viewed as simplicial complexes which are built in terms of smaller complexes called apartments. However, the more modern approach to buildings is actually quite different; instead of viewing the building as a simplicial complex, you can view the building as a set in possession of a sort of floor plan (called a W-metric) which tells you how to 'navigate' the building. This W-metric approach will be the subject of the talk. No prior exposure to buildings is assumed.

 

MCG, Out(F_n), and Linear Groups: These Are a Few of My Favorite Things
Greg Fein - Rutgers Newark
Sa 10:40-11:10, 1024 Dana Building

You may have heard about these things (in particular the mapping class group of a surface and the outer automorphism group of a free group) that tons of geometric group theorists like to gab about. If, however, all you know about these wonderments are their names, then perhaps now is the time to learn a bit more and find out what all the fuss is about. I'll give a tour of some of the similarities and differences between the two groups, focusing in a bit on the classical classifications. We'll see how a lot about them can be explained via analogy with the (perhaps more familiar) linear groups and hyperbolic isometry groups. I promise there will be tons of examples to help us along the way.

 

Fibered category of Beck modules
Martin Frankland - MIT
Su 10:10-10:40, 1040 Dana Building

Beck modules are a convenient notion of module over an object, which recovers the usual notion in familiar settings (groups, commutative rings, Lie algebras...) and is well suited to provide coefficients for cohomology theories. Instead of looking at modules over one object, what if we look at all modules over all objects, relating modules over different objects via pullbacks? This yields a category fibered over our original category. In this talk, I will present the construction in more detail and explain why I like to think of it as some kind of tangent bundle of the category.

 

Invariants of Legendrian Knots using Chekanov's Differential Graded Algebra
Whitney George - University of Georgia
Su 10:10-10:40, 1028 Dana Building

We define a Legendrian knot and its associated 3 classical invariants; the topological knot type, the rotation number, and the Thurston-Bennequin number. In the mid-90s Eliashberg and Fraser proved that these three invariants classified the Legendrian unknot up to Legendrian isotopy. Later, Etnyre and Honda showed the classical invariants classified all torus knots and the figure eight knot. However, it was unclear if the knot, 5_2, could be classified by these three invariants. Chekanov developed a Differential Graded Algebra, which is the focus of this paper, to show that knot 5_2 cannot be classified by these invariants.

 

Topology of Group Embeddings
Richard Gonzales - University of Western Ontario
Sa 10:40-11:10, 1046 Dana Building

Let G be a reductive group. A GxG-variety X is called an equivariant compactification of G if X is normal, projective, and contains G as an open and dense orbit. Regular compactifications and reductive embeddings are the main source of examples. My goal is to give an overview of the theory of group embeddings and their relations to GKM theory.

 

Genera from Deformation Quantization
Ryan Grady - University of Notre Dame
Sa 1:20-1:50, 1040 Dana Building

I will recall the \hat{A} and Witten genera of a manifold and give a few reasons why they are important (Atiyah-Singer Index Theorem, positive scalar/ricci curvature, etc). I then will overview the work of Fedosov, Bressler, Nest, Tsygan, Felder, et al. on algebraic index theorems in deformation quantization. If time permits I would like to mention some recent work of Costello on the Witten genus.

 

Knot Genera
Kate Kearney - Indiana University
Sa 10:40-11:10, 1028 Dana Building

The three genus, four genus, and concordance genus of knots are independent invariants that all give a measure of the types of surfaces bounded by the knot. We will define these and look at several examples to highlight the differences between these. We will discuss further notions of genus, including the stable four genus and stable concordance genus. This talk will assume a basic knowledge of knot theory, but will give all relevant definitions.

 

Refocusing of Null-Geodesics in Lorentz Manifolds
Paul Kinlaw - Dartmouth College
Sa 1:20-1:50, 1046 Dana Building

Lorentz manifolds provide a purely mathematical model for spacetime which is used in the general theory of relativity. The null-geodesic curves represent light rays. We will start with an introduction to Lorentz metrics and a brief comparison with the more familiar Riemannian metrics. We will then cover several results on refocusing properties of null-geodesics in Lorentz manifolds, which are the subject of my thesis.

 

Dehn function and finiteness property
Sang Rae Lee - University of Oklahoma
Su 10:10-10:40, 1024 Dana Building

The free group derives its importance from the fact that every group is a quotient of some free group. The topological treatment of free group not only results in a theory which has geometric contents but also in elegant proofs of many important theorems. In this talk, the theory of free groups will be explore via graphs and covering space theory. We will also look at the graph product of graph of groups which generalizes free product and HNN extension.

 

2-Vector Bundles
John Lind - University of Chicago
Sa 4:00-4:30, 1040 Dana Building

A vector bundle is a vector space parametrized by a base space B. A 2-vector space is a module category over the category of vector spaces parametrized by B. A 2-vector bundle of rank 1 is essentially a (S^1-)gerbe, in the sense of geometry. These are inherently 2-categorical objects. I will discuss their K-theory and connection to stable homotopy theory. I may also discuss their applications in higher guage theory.

 

Homotopy of closed curves on a 2-disc
Yevgeniy Liokumovich - University of Toronto
Su 10:10-10:40, 1046 Dana Building

Frankel and Katz in 1993 answered negatively a question of Gromov: if we consider the set of Riemannian 2-discs of uniformly bounded diameter, is it possible to place a uniform bound on how much a closed curve needs to stretch when it is homotoped from the boundary to a point? I will include a quick overview of their construction and discuss how the situation changes when we also bound the volume, the curvature, or both.

 

Knot groups onto hyperbolic knot groups
Yi Liu - UC Berkeley
Sa 4:00-4:30, 1024 Dana Building

In the 1970s, John Simon asked whether a knot group maps onto at most finitely many knot groups. This is affirmatively answered if the target knots are restricted to be hyperbolic. This is a joint work with Ian Agol.

 

Immersed surfaces in the modular orbifold
Joel Louwsma - Cal Tech
Sa 11:20-11:50, 1024 Dana Building

A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the following stability theorem: for every hyperbolic element of the modular group, the product of this element with a sufficiently large power of a parabolic element is represented by a geodesic that virtually bounds an immersed surface. This is joint work with Danny Calegari.

 

Towards an Unstable Chromatic Filtration
Dustin Mulcahey - CUNY Graduate Center
Sa 10:40-11:10, 1040 Dana Building

In this talk, I will explore one direction of finding an unstable variant of the Morava change of rings isomorphism. After some background, I will begin by considering the general problem of finding sufficient conditions for a change of Ext between two comonads. I will then specialize to the case of the categories of unstable BP comodules and unstable K(n) comodules, respectively.

 

Incompressible surfaces in handlebodies and boundary reducible 3-manifolds
Joao Miguel Nogueira - UT Austin
Su 11:20-11:50, 1028 Dana Building

In this talk, we prove that for every compact surface with boundary, ori- entable or not, there is an incompressible embedding of the surface into the genus two handlebody. In the orientable case the embedding can be either sep- arating or non-separating. We also consider the case in which the genus two handlebody is replaced by an orientable 3-manifold with a compressible bound- ary component of genus greater than or equal to two.

 

The Quantum Teichmuller Space and Invariants of Surface Diffeomorphisms
Julien Roger - University of Southern California
Sa 1:20-1:50, 1028 Dana Building

The quantum Teichmuller space is a deformation of the algebra of rational functions on the classical Teichmuller space of a surface with punctures. I will describe its construction and then explain how to build quantum invariants of surface diffeomorphisms from its representation theory.

 

Categorification and Knot Homology
David Rose - Duke University
Su 11:20-11:50, 1040 Dana Building

Categorification can be viewed as the process of lifting scalar and polynomial invariants to homology theories having those invariants as (graded) Euler characteristics. In this (expository) talk, we will discuss categorification in general and as manifested in Khovanov homology and other knot homology theories. Examples will be given showing how the categorified invariants are stronger and often more useful than the original invariants.

 

Combinatorics of Grassmannians
Matthew Samuel - Rutgers New Brunswick
Su 9:30-10:00, 1040 Dana Building

We present various combinatorial results by others and of our own describing aspects of Grassmannians of Lie types A, B, C, and D. We describe CW structures whose cells are indexed by various kinds of box diagrams and discuss notions of "positivity" of multiplicative structure constants in ordinary cohomology, equivariant cohomology, and algebraic K-theory. We will also discuss various methods for computing these structure constants in a positive, combinatorial fashion.

 

Box-Dot Diagrams for "Regular" Rational Tangles
Greg Schneider - University at Buffalo, SUNY
Sa 4:00-4:30, 1028 Dana Building

We introduce a new presentation for rational tangles which illustrates a geometric connection to the number theory of positive regular continued fractions. This presentation also admits a suitable extension to the contact setting, allowing us to define a natural Legendrian embedding of a particular class of rational tangles into the standard contact Euclidean 3-space. We will briefly discuss how these box-dot diagrams, along with an associated construction, can be used to determine when the Legendrian flyping operation yields tangles which are not Legendrian isotopic, further refining an earlier result of Traynor.

 

Real Places and Surface Bundles
Jonah Sinick - University of Illinois at Urbana Champaign
Su 12:00-12:30, 1028 Dana Building

A finite volume hyperbolic 3-manifold has an associated finite extension of the rational numbers called its *trace field.* In a 2006 paper, Danny Calegari proved that the trace field associated to a hyperbolic surface bundle with fiber the once punctured torus has no real places. We present some new findings concerning hyperbolic surface bundles with fibers other than the once punctured torus, complementing Calegari's result.

 

Non-Bounded Generation of Word-Hyperbolic Groups
Chunyi Sun - Yale
Sa 1:20-1:50, 1024 Dana Building

A finitely generated group $\Gamma$ is boundedly generated if there is a finite ordered set $S = \{g_1 , g_2,\ldots,g_k \}$ such that every element $g$ in $\Gamma$ can be written as a product of powers of elements in $S$, i.e. $g = g_1^{n_1}\cdot g_2^{n_2} \cdot \cdots \cdot g_k^{n_k}$ . For example, for $n \geq 3$, $\mathrm{SL}(n, \mathbb{Z})$ is boundedly generated by elementary matrices but $\mathrm{SL}(2, \mathbb{Z})$ is not bounded generated by any finite set $S$. I would like to show that non-elementary word-hyperbolic groups are not boundedly generated with two different arguments. One uses known answers of Burnside problem of word-hyperbolic groups, and the other generating a contradiction with the exponential growth of word-hyperbolic group.

 

Converse of Lefschetz Fixed point theorem
Gun Sunyeekh - University of Notre Dame
Sa 11:20-11:50, 1046 Dana Building

The famous Lefschetz fixed point theorem states that if a smooth self map f of a compact smooth manifold M is homotopic to a fixed point free map, then the Lefschetz number L(f) = 0. In general, the converse does not hold. I'll show that the fixed points of f determine an element in a framed bordism group of a twisted free loop space of M. Then I'll show that if dim(M)>2 such an element vanishes if and only if f is homotopic to a fixed point free map.

 

An Introduction to Diffeological Spaces
Enxin Wu - University of Western Ontario
Su 9:30-10:00, 1046 Dana Building

Manifolds are very nice objects in modern mathematics. However, the category of manifolds is not that pleasant. Many generalizations of manifolds are proposed around 1980's. Diffeological spaces are one of them, which were first defined by J. Souriau in 1980, and later on systematically developed by P. Iglesias-Zemmour, J. Baez, A. Hoffnung and others. In this talk, some known results on the basic properties of diffeological spaces and some of their differential geometric and topological aspects will be described. Some new results on the general topological aspects and categorical aspects will be presented at the end.