Topology Seminar
2011-12
Tuesdays 11:10 am-12:00 pm
Surge 268
April 3: Helen Wong (Carleton College): Representations of the Kauffman
Skein Algebra
The following describes joint work
with Francis Bonahon. The Kauffman skein algebra of a surface not only lies
at the core of quantum topology, but bears deep meaning in the language of
hyperbolic geometry. We’ll describe how to construct representations of the
Kauffman skein algebra and construct some invariants to help tell them
apart. The latter we do by constructing invariants of the representations
using Chebyshev polynomials and relying on numerous “miraculous
cancellations”.
April 10: Michael Menke: Gauge theory and the recent advances of Edward
Witten
April 17: Elena Pavelescu (Occidental College): Legendrian graphs with all cycles unknots of maximal
Thurston-Bennequin number.
Abstract: A Legendrian
graph in a contact structure is a graph embedded in such a way that its edges
are everywhere tangent to the contact planes. In this talk we look
at Legendrian graphs in R^3 with the standard contact structure. We extend
the invariant Thurston-Bennequin number (tb) from Legendrian knots to
Legendrian graphs. We prove that a graph can be Legendrian realized with all
its cycles Legendrian unknots with tb=-1 if and only if it does not contain K_4
as a minor. This talk is based on joint work with Danielle O'Donnol.
April 24: Michael Yoshizawa (UC Santa Barbara): Generating Examples
of High Distance Heegaard Splittings
Abstract: Given a closed orientable
3-manifold M, a surface S in M is a Heegaard surface if it separates the
manifold into two handlebodies of equal genus. This decomposition is called a
Heegaard splitting of M. The distance of this splitting is the length of
the shortest path in the curve complex of S between the disk complexes of the
two handlebodies. In 2004, Evans developed an iterative process to
construct a manifold that admits a Heegaard splitting with arbitrarily high
distance. We first provide an introduction to Hempel distance and then improve
on Evans' results.
May 1: Luca Di Cerbo (Duke University): Seiberg-Witten equations on
manifolds with cusps
Abstract: In this talk, I will
present some results concerning the Seiberg-Witten equations on finite volume
complex 4-manifolds with cusps. These results extend and complete previous work
of O. Biquard and Y. Rollin. Finally, using a Seiberg-Witten scalar curvature
estimate I will present the finite volume generalization of some well-known
theorems of C. LeBrun.
May 8: Daniel Murfet (UCLA): The 2-category of matrix factorisations
Abstract: I will explain how 2-categories naturally
arise in the study of two-dimensional topological field theories with defect
lines. An interesting example, related to topological Landau-Ginzburg models,
is the 2-category whose objects are isolated hypersurface singularities and
whose 1-morphisms are matrix factorisations. I will discuss the rich structure
of this bicategory worked out in recent joint work with Nils Carqueville.
May 15: Julie Bergner: Generalized classifying space constructions
Abstract: The construction for
classifying spaces of groups can be extended to give classifying spaces of
categories. However, two classifying
spaces can be equivalent even if the categories they came from were not. In this talk, we’ll look at two different
ways of refining the definition of classifying space to correct this problem,
leading to two of the approaches to (∞,1)-categories.
May 22: Julie Bergner: Generalized classifying space constructions,
continued
May 29: Jason McCarty (University of Virginia): A Spectral Sequence for
the Homology of Ω∞ X
Abstract: I will discuss joint work
with N. Kuhn about a Goodwillie tower spectral sequence converging to H*(
Ω∞ X), the mod 2 homology of the zeroth space of a
connected spectrum X. Dyer-Lashof operations on the spectral sequence lead to
"universal" differentials that hold for all spectra. These then lead
to an algebraic version of the spectral sequence, whose pages can be completely
described in terms of the derived functors of "destabilization"
studied by W. Singer and others. The two spectral sequences coincide until the
first non-universal, or "rogue," differential. Using this
identification, the spectral sequence for various spectra can be completely
understood, including all Eilenberg-MacLane spectra. I will finish by
discussing some examples of non-connected spectra where the spectral sequence
unexpectedly converges to the right answer, or nearly so.
June 5: No seminar
September 27: Philip Hackney – Operations in spectral sequences
October 4: Philip Hackney – Operations in spectral sequences: the sequel
October 11: Ben Ward (Purdue University) – Polytope actions on the
Hochschild complex
Abstract: The purpose of this talk
will be to illustrate the role that various families of polytopes play in different
versions of Deligne's Hochschild cohomology conjecture. For example, one
approach in the associative case is the construction of a chain model for the
little disks built from simplicies indexed by trees. In the $A_\infty$
case the additional terms in the differential give a blow-up of the $n$-simplex
to the cyclohedron W_{n+1}. For another example, I will give an alternate
description of associahedra which makes the cyclic structure of this operad
explicit. This leads to a description of a class of homotopy Frobenius
algebras and a cyclic $A_\infty$ version of the conjecture, whose proof I
will sketch if time allows.
October 18: Chris Carlson – Extending vector fields using the
Poincaré-Hopf Theorem
October 25: Erica Flapan (Pomona College) – Topological symmetry groups
Abstract: Chemists have defined the point group of a molecule as
the group of rigid symmetries of its molecular graph in R3. While this group
is useful for analyzing the symmetries of rigid molecules, it does not include all of the symmetries of
molecules which are flexible or can rotate around one or more bonds. To study
the symmetries of such molecules, we define the topological symmetry group of a graph
embedded in R3 to be the subgroup of the automorphism group of the
abstract graph that is induced by homeomorphisms of R3. This group gives
us a way to understand not only the symmetries of non-rigid
molecular graphs, but the symmetries of any graph embedded in R3. The study of
such symmetries is a natural extension of the study of symmetries of
knots. In this talk we will present a survey of results about the topological
symmetry group and how it can play a role in analyzing the symmetries of
non-rigid molecules.
November 1: Claire Tomesch (University of Chicago) - Categories of Cohesion
and 'Discretized' Model Categories
Abstract: The purpose of this talk
is to describe a notion of category of cohesion -- a concept of Lawvere
introduced to describe 'relative discreteness' -- and its role in defining and
understanding model structure on Simpson-Tamsamani style versions of weak
n-categories. The main payoff of this approach is an iterable
construction of a model structure which takes into account the special role of
'discrete' objects.
November 8: Michael Williams – Introduction to low-dimensional hyperbolic
geometry
November 15: Shinpei Baba (Caltech) – Grafting complex algebraic
structures
Abstract: A projective structure is a
refinement of a Riemann surface: an atlas modeled on the Riemann sphere
with transition maps in PSL(2,C). Thus a
projective structure enjoys a representation of the fundamental group of the
base surface into PSL(2,C). We discuss about a certain surgery operation,
called grafting, that makes many different projective structures with the same
holonomy representation.
November 22: Oliver Thistlethwaite – Boolean formulae, hypergraphs, and
combinatorial topology
Abstract: With a view toward
studying the homotopy type of spaces of Boolean formulae, we introduce a simplicial
complex, called the theta complex, associated to any hypergraph, which is the
Alexander dual of the more well-known independence complex. In particular, the
set of satisfiable formulae in k-conjunctive normal form with n variables has
the homotopy type of THETA(Cube(n, n-k)), where Cube(n, n-k) is a hypergraph
associated to the (n - k)-skeleton of an n-cube. We make partial progress in
calculating the homotopy type of theta for these cubical hypergraphs, and we
also give calculations, and examples for other hypergraphs as well. Our main
interest in defining and pursuing this construction is the hope that
topologycan be brought to bear on the famous P/NP question of computer science.
However, studying the theta complex of hypergraphs is an interesting problem in
its own right.
November 29: Michael
Shulman (UC San Diego) - The additivity and multiplicativity of fixed-point
invariants
Abstract: The Lefschetz number of
a continuous map is a classical invariant that gives information about its
fixed points. More refined invariants such as the Nielsen number and the
Reidemeister trace give more information. These invariants often satisfy
additivity and multiplicativity theorems with respect to quotients and fiber
bundles, which provide important calculational tools. I will describe an
abstract framework which automatically implies very general additivity and
multiplicativity theorems in a purely formal way, using the machinery of
bicategorical traces. This is joint work with Kate Ponto.
January 10: Stefano Vidussi – LERF, vRFRS, and other laments a theory says
while it is dying
January 17: Jeremy Miller (Stanford University) - Homology of spaces of
rational J-holomorphic curves in CP^2
Abstract: In a well known work, Graeme
Segal proved that the space of holomorphic maps from a Riemann surface to a
projective space is homology equivalent to the corresponding continuous mapping
space through a range of dimensions increasing with degree. One could ask if a
result similar to that of Segal holds when other (not necessarily integrable)
almost complex structures are put on projective space. Under supervision of my
advisor Ralph Cohen, I obtained the following partial result; the inclusion of
the space of based degree k J-holomorphic maps from CP^1 to CP^2 into the based
twofold loop space is a homology surjection for dimensions j < 3k-2. The
proof involves constructing a "gluing map" in the sense of Taubes
gluing of instantons and comparing it to a gluing map induced by the little
2-disks operad action on 2-fold loop spaces.
January 24: Bill Kronholm (Whittier College) – Computing equivariant
homology
Abstract: Peter May, et. al., developed
the theory of RO(G)-graded equivariant cohomology which simultaneously extends
classical cohomology and Bredon cohomology. Until recently, however, very few
explicit computations have been performed. In this talk, we'll explore current
techniques for computing cohomology in the case where the group is of order
two, view some explicit computations, and see the difficulties which stand in
the way of further computations.
January 31: Sam Nelson (Claremont McKenna College) - Enhancements of
Counting Invariants
Abstract: Counting invariants of
knots and links are invariants defined by counting labelings of diagrams using
various algebraic and combinatorial structures. An enhancement of a counting
invariant is a stronger invariant which determines the counting invariant,
generally obtained using invariants of labeled diagrams.
February 7: Kristine Pelatt (University of Oregon) - A non-trivial class
in the homology of spaces of knots
Abstract: In classical knot theory,
one studies the components of the space of knots. More generally, one can
consider the topology of the space of embeddings into higher dimensional
Euclidean space. By resolving knots with k double points, Cattaneo,
Cotta-Ramusino and Longoni produced explicit, non-trivial k(d-3)-dimensional
cycles. We generalize these results to resolutions of singular knots with
triple points, producing a non-trivial 3(d-8)-dimensional cycle. This
extends and corrects the results in a preprint of Longoni. The techniques
we use are closely related to the combinatorics of the embedding calculus
homology spectral sequence due to Sinha, suggesting that they may lead to
recipes for geometric representatives for all of the cycles in that spectral
sequence.
February 14: Alex Coward (UC Davis)
- Unknotting crossing
changes and circular Heegaard splittings
Abstract: Twenty years ago
Scharlemann and Thompson used deep results from sutured manifold theory to
prove that a genus reducing crossing change on a knot may be realized as
untwisting a Hopf band plumbed onto a minimal genus Seifert surface. This gives
a hint that understanding genus reducing crossing changes is closely related to
understanding how a compact surface in S^3 changes when it is twisted. In this
talk we use modern technology from the theory of Heegaard splittings to show
that understanding when two surfaces are related by a single twist implies the
existence of an algorithm to determine when two (hyperbolic or fibered) knots
of different genus are related by a single crossing change.
February 21: Marcy Robertson (University of Western Ontario) -Spaces of Operad
Structures
Abstract: What exactly is the
connection between cohomology operations and deformation theory of algebras?
The purpose of this talk is to describe several well know examples, such as the
existence of quantized deformations, and explain how all of these examples can
be understood via a model of the derived mapping space between two operads or
multi categories. No previous knowledge of operads will be assumed.
February 28: Tye Lidman (UCLA) - Left-Orderability and Graph Manifolds
Abstract: Recently, a relationship
has between developing between the existence of a left-invariant order on
the fundamental group of a three-manifold and the structure of certain
Floer homology groups. We will discuss what is currently known and
how this pertains to the topology of the three-manifold. In
particular, we will study this for graph manifold integer homology
spheres. This is joint work with Adam Clay and Liam Watson.
March 6: Heather Russell (USC) - Odd cohomology of type A Springer
varieties
Abstract: Springer varieties are of interest in geometric
represenation theory because their cohomology carries an action of the
symmetric group. The top degree cohomology is an irreducible representation of
Sn. Springer varieties are also related to Khovanov's
categorification of the Jones polynomial. A new categorified knot invariant
called odd Khovanov homology suggests the existence of a parallel odd theory of
Springer varieties. In this talk we will review a simple algebraic description
of the cohomology of Springer varieties. We will then describe joint work with
Aaron Lauda defining the odd cohomology of Springer varieties. (No prior
knowledge of Springer varieties will be required.)
March 13: Sean Tilson (Wayne State University) - Power operations and the
Kunneth spectral Sequence
Abstract: Power operations have been
constructed and successfully utilized in the Adams and Homological Homotopy
Fixed Point Spectral Sequences by Bruner and Bruner-Rognes. It was thought that
such results were not specific to the spectral sequence, but rather that they
arose because highly structured ring spectra are involved. In this talk, we
show that while the Kunneth Spectral Sequence enjoys some nice multiplicative
properties, there are no non-zero operations on the E2 page of the
spectral sequence. Despite the negative results we are able to use old
computations of Steinberger’s with our current work to compute operations in the
homotopy of some relative smash products.
Back to Julie Bergner’s research page
http://www.math.ucr.edu/~jbergner/topologysem1112.htm Last updated: 8 June 2012