Notre Dame Topology Seminar
Spring 2010 Schedule
- January 26 at 1PM in HH125
- Jumpei Nogami, University of Illinois - Chicago
- On Derived K3 Surfaces
- February 2 at 2PM in DBRT118
- Justin Thomas, Northwestern University
- Kontsevich's Swiss Cheese Conjecture
- February 9 at 1:50 PM in DBRT118
- Dan Berwick-Evans, University of California - Berkeley
-
Supersymmetric Sigma Models and Topology
- February 16 at 1:50 PM in DBRT118
- Stephan Stolz, University of Notre Dame
-
Traces in balanced monoidal categories
- February 23 at 1:50 PM in DBRT118
- Corbett Redden, Michigan State University
-
String classes and 3-forms
-
March 4 at 10 AM in H258
- Jim Fowler, Ohio State University
-
Rational PD Groups and Controlled Symmetric Signatures
- March 16 at 1:50PM in DBRT118
- Chris Schommer-Pries, Harvard University
- Homological Algebra for Bicategories (with applications to the
String Group)
- March 23 at 1:50PM in DBRT118
- Larry Taylor, University of Notre Dame
- Forms and Norms and Group Actions
- March 30 at 1:50PM in DBRT118
- Marcy Robertson, University of Illinois - Chicago
- Derived Morita Theory for Enriched Symmetric Multicategories
- April 6 at 1:50PM in DBRT118
- John Lind, University of Chicago
- Infinite Loop Space Theory of Diagram Spectra
- April 13 at 1:50PM in DBRT118
- Anna Marie Bohmann, University of Chicago
- The Equivariant Generating Hypothesis
- April 20 at 1:50PM in DBRT118
- Andrew Blumberg, University of Texas at Austin
- The universal property of higher algebraic K-theory
- April 22 at 2PM in H258
- Stratos
Prassidis, Canisius College
- On Linear Groups
- April 26 at 3PM in H258
- Soren Galatius, Stanford University
- Stable topology of moduli spaces of curves
- April 27 at 1:50PM in DBRT118
- Diarmuid Crowley, Hausdorff Research Institute for Mathematics
- An introduction to the Manifold Atlas Project
The additivity of the ρ-invariant and periodicity in
topological surgery
- April 29 at 3PM in HH229
- Gun Sunyeekhan, University of Notre Dame
- Geometric approach to the (equivariant) fixed point theory
- May 4 at 10:30AM in HH229
- Qayum Khan, University of Notre Dame
- On isovariant rigidity of CAT(0) manifolds
Fall 2009 Topology Seminar
Graduate Topology/Geometry Seminar
Notre
Dame Department of
Mathematics
Questions? Contact
Kate Ponto
Spring 2010 Abstracts
January 26, 2010: Jumpei Nogami
On Derived K3 Surfaces
We will discuss a generalization of K3 surfaces in derived algebraic geometry and their applications to stable homotopy theory.
February 2, 2010: Justin Thomas
Kontsevich's Swiss Cheese Conjecture
Given a vector space A, the universal algebra acting on A is End(A). This is a special case of the notion of an $E_d$ algebra acting on an $E_{d-1}$ algebra, defined by Kontsevich using Voronov's swiss cheese operad. The conjecture is that Hoch(A) is the universal $E_d$ algebra acting on the $E_{d-1}$ algebra A. This is a refinement of Deligne's conjecture. We will define the notion of $E_d$ algebra, introduce the swiss cheese operad, and prove the conjecture of Kontsevich.
February 9, 2010: Dan Berwick-Evans
Supersymmetric Sigma Models and Topology
Mathematics has always been a convenient language for physics, but recently it has been noted that
quantum field theories provide a powerful language for geometry and topology. A rich class of
examples are furnished by supersymmetric sigma models. I will give several examples where the
partition functions of these theories give rise to familiar topological invariants, and point toward some
conjectures relating sigma models to more exotic invariants.
February 16, 2010: Stephan Stolz
Traces in balanced monoidal categories
Generalizing the notion of finite rank operators in the
category of vector spaces, or the notion of nuclear operators in the
category of Banach spaces, we define `thick morphisms' in any
monoidal category (the terminology is motivated by considering a
suitable bordism category). We show that a braiding on the monoidal
category allows the construction of a trace for any thick
endomorphism of some object, provided this object satisfies a
suitable condition (which for Banach spaces is equivalent to being
able to approximate the identity uniformly on compact sets by finite
rank operators). This trace takes values in the endomorphism of the
monoidal unit of the category, and agrees with the usual trace in the
category of vector spaces or Banach spaces. The main result is the
construction of a trace pairing which associates to a pair of thick
morphisms which can be composed to an endomorphism, an endomorphism
of the monoidal unit. This trace pairing agrees with the trace of the
composition if the latter makes sense (i.e., if the condition on the
object mentioned above is satisfied).
February 23, 2010: Corbett Redden
String classes and 3-forms
A string structure is a higher analog of a spin structure or
orientation, but it is much more complicated. However, we see that
isomorphism classes are naturally in bijection with certain degree 3
cohomology classes, denoted string classes, on the total space of a
principal bundle. Using a Hodge isomorphism, we show the harmonic
representative of a string class gives rise to a canonical 3-form on
the base space and refines an associated differential character. The
3-forms will be computed in the case of homogeneous metrics on 3-
spheres, and we discuss how the cohomology theory tmf could
potentially encode obstructions to positive Ricci curvature metrics.
March 4, 2010: Jim Fowler
Rational PD Groups and Controlled Symmetric Signatures
A uniform lattice $\Gamma$, containing torsion, rationally satisfies Poincaré duality. To what extent can this rational Poincaré duality be blamed on geometry?---in other words, is $\Gamma$ a rational PD group because it is the fundamental group of a rational homology manifold having rationally acyclic universal cover? The answer is no (at least for lattices with odd torsion), as we'll see with a calculation of a controlled symmetric signature.
March 16, 2010: Chris Schommer-Pries
Homological Algebra for Bicategories (with applications to the
String Group)
Homological algebra is ubiquitous in mathematics largely due
to its numerous applications and ease of computation. In this talk we
will discuss how to categorify homological algebra in a way which
retains these basic features. As an illustration, we will use this
bicategorical homological algebra to construct a finite dimensional
model of the String group as a central extension of smooth 2-groups.
This 2-group has fascinating connections with both abstract homotopy
theory (through String Bordism and TMF) and with quantum field theory
(through the 2D SUSY non-linear sigma model). A better geometric
understanding of String geometry has the potential to offer new
interactions between these fields.
March 23, 2010: Larry Taylor
Forms and Norms and Group Actions
Coupling a theorem of Bredon's on involutions
with an observation about equivariant bilinear forms
produces new results on group actions on manifolds.
An easy to state corollary is that if a finite group
acts freely on $M^{2n}$ then the middle Wu class is
not only invariant but it is a norm.
March 30, 2010: Marcy Robertson
Derived Morita Theory for Enriched Symmetric Multicategories
Operads, multicategories, and their representations (also called operadic/multicategorical algebras) play a key role in organizing hierarchies of higher homotopies in any category with a good notion of homotopy theory. In this talk we show how one can generalize work of Toen and Rezk to provide a description of the derived category of any multicategorical algebra. Time permitting, we will discuss applications of this theory to problems in combinatorial representation theory.
We do not assume prior knowledge of the theory of operads and multicategories.
April 6, 2010: John Lind
Infinite Loop Space Theory of Diagram Spectra
I will discuss some models for the category of spaces that are
particularly useful in doing stable homotopy theory. It turns out
that each flavor of the modern models for spectra has a corresponding
model for spaces that captures the infinite loop space underlying a
spectrum. This allows us to work with the multiplicative structure of
a diagram ring spectrum at the level of spaces.
April 13, 2010: Anna Marie Bohmann
The Equivariant Generating Hypothesis
Freyd's generating hypothesis is a long-standing conjecture in stable
homotopy theory. The conjecture says that if a map between finite spectra
induces the zero map on homotopy groups, then it must actually be
nullhomotopic. We formulate the appropriate version of this conjecture in
the equivariant setting. We then give some results about this equivariant
version and compare them to the nonequivariant results. In particular, we
show that the rational version of this conjecture holds when the group of
equivariance is finite, but fails when the group is S^1. This result uses
Greenlees's description of the rational S^1-equivariant homotopy category.
April 20, 2010: Andrew Blumberg
The universal property of higher algebraic K-theory
In this talk I will present joint work with David Gepner and Goncalo
Tabuada characterizing the algebraic K-theory spectrum as a functor of
small stable categories (e.g., module categories). As an application, I
will show how to prove that the cyclotomic trace is essentially the unique
natural transformation from K-theory to THH.
April 22, 2010: Stratos Prassidis
On Linear Groups
A group is linear if it admits a finite dimensional faithful representation. The characterization of linear groups is a central problem in group theory. We will review the basic theorems in this direction. The rest of the talk will be on the linearity of poly-free groups, like pure braid groups and fundamental groups of complements of hyperplane arrangements. The work is based on joint work of the speaker with Fred Cohen, Jonathan Lopez and Vasilis Metaftsis.
April 26, 2010: Soren Galatius
Stable topology of moduli spaces of curves
I will talk about joint work with Eliashberg, in
which we attempt to extend the work of Madsen and Weiss on the moduli
space of Riemann surfaces. Madsen and Weiss constructed a map from
Mg to an infinite loop space Ω∞ MTSO(2) and
proved it induces an isomorphism in Hk(-;Z) for k < (g-1)/2. My
talk will be about a similar construction for the compactified moduli
space Mg. In this case there is a similar map
to an infinite loop space which, when restricted to the subspace of
irreducible curves, induces an isomorphism in Hk(-;Z)$ for large
g.
April 27, 2010: Diarmuid Crowley
An introduction to the Manifold Atlas Project:
the Manifold Atlas Project
The Manifold Atlas is a scientific Wiki about manifolds with
plans to become a sort of on-line journal. In this short talk I will
introduce the Atlas by outlining it's goals, its structure and some
aspects of using the Atlas.
The additivity of the ρ-invariant and periodicity in
topological surgery
The ρ-invariant is a powerful invariant of a closed odd-
dimensional topological manifold M. It is based on the signature
defect associated to the G-signature of even dimensional manifolds
with boundary. Taking the difference of the ρ-invariants of the
source and target of a manifold structure N → M defines a function,
ρ : S(M) → R,
where S(M), the topological structure set of M, is know to admit a
canonical abelian group structure and R is a certain rational vector
space.
In this talk I will report on our recent proof that ρ is a
homomorphism of abelian groups.
Since the group structure on S(M) remains geometrically mysterious the
proof uses the Siebenmann periodicity map which relates S(M) to S(M
× D4k), where it is easy to understand the abelian group
structure. Another aspect of our work is to give a detailed account
of Hutt's construction of the Cappell-Weinberger map from S(M) to S(M
× D4k), a geometric realisation of the periodicity map in
topological surgery.
April 29, 2010: Gun Sunyeekhan
Geometric approach to the (equivariant) fixed point theory.
Let f be a self-map of a compact smooth manifold M. We may ask if f is homotopic to a fixed point free map.
Wecken (1942) and Jiang (1981) showed that if dim(M) > 2 then there is an invariant in a framed bordism group which vanishes if and only if f is homotopic to a fixed point free map.
The next question we may ask if this invariant is not trivial, what can we say about the fixed point set? In this talk, I will discuss that we can extract important information from this invariant by using the geometric points of view. The key ingredients in the proof of the main theorem are transversality, homotopy embedding and classifying spaces which in general do not hold for the equivariant setting. I’ll discuss the conditions to achieve the equivariant transaversality and equivariant homotopy embedding.
May 4, 2010: Qayum Khan
On isovariant rigidity of CAT(0) manifolds
We discuss Quinn's equivariant generalization of the Borel Conjecture.
This concerns cocompact proper actions of a discrete group Γ on a
Hadamard manifold X. We give a complete solution when the action of
Γ is pseudofree and when X more generally is a CAT(0) manifold.
Here, pseudofree means that the singular set is discrete. A
rich class of examples is obtained from crystallographic groups Γ
made out of isometric spherical space form groups G.
If Γ has no elements of order two, then we obtain equivariant
topological rigidity of the pair (X,Γ). Hence, if Γ is
torsionfree, then we generalize a recent theorem of A. Bartels and W.
Lück, which validates the classical Borel Conjecture for CAT(0)
fundamental groups. Otherwise, if Γ has elements of order two,
we show how to parameterize all possible counterexamples, in terms of
Cappell's UNil summands of the L-theory of infinite dihedral groups.
This ongoing project is joint work with Frank Connolly.