Topology Seminar
2012-13
Tuesdays 11:10-12,
Surge 268
April 2: Julie Bergner, Simplicial complexes and simplicial sets
April 9: Jesse Burke (UCLA) - Matrix factorizations and complete
intersection rings
Abstract: Matrix factorizations were
introduced by Eisenbud in 1980 to study hypersurface singularities. Since their
introduction they have been used in a wide range of fields including
representation theory, knot theory and string theory. In my talk I will give an
overview of this construction, talk briefly about how they are used, and then
discuss recent work using matrix factorizations to study complete intersection
singularities.
April 16: Moritz Groth (Radboud University Nijmegen) - An introduction to
Grothendieck derivators
Abstract: The theory of derivators
(going back to Grothendieck, Heller, and others) provides an axiomatic approach
to homotopy theory. It addresses the problem that the rather crude passage from
model categories to homotopy categories results in a serious loss of
information. In the stable context, the typical defects of triangulated
categories (non-functoriality of cone construction, lack of homotopy colimits)
can be seen as a reminiscent of this fact.
The basic idea behind a derivator is
that one should form homotopy categories of diagram categories and also keep
track of the calculus of homotopy Kan extensions. In the stable context this
calculus allows one to canonically construct triangulations -- emphasizing the
idea that stable derivators provide an enhancement of triangulated categories.
Moreover, for stable, closed symmetric monoidal derivators one can establish an
additivity result for traces -- a result which is known to be false at the
level of triangulated categories.
The aim of this talk is to give a
short introduction to the theory and to (hopefully) advertise derivators as a
convenient, 'weakly terminal' approach to axiomatic homotopy theory.
April 23: Julie Bergner, Simplicial complexes and simplicial sets, part 2
April 30: Daniel Berwick-Evans (UC Berkeley) - Concordance spaces for
sheaves
Abstract: Classifying spaces encode
locally defined geometric data on manifolds such as vector bundles, principal
bundles and cohomology cocycles. The transition from the geometry of this local
data to the topology of classifying spaces typically loses data. I will define
concordance spaces and show how they make aspects of this process precise.
After explaining the relation to classical examples I will focus on a
conjecture of Stolz and Teichner relating certain supersymmetric field theories
to K-theory. This is joint work with Dmitri Pavlov and Pedro Boavida de
Brito.
May 7: Greg Chadwick � Infinite loop spaces and generalized cohomology
May 14: Dan Ramras (New Mexico State University) - Rips complexes and the
assembly map in algebraic K-theory
Abstract: Rips complexes give a
means for translating discrete metric spaces into simplicial complexes.
These complexes are especially useful for studying the word metric on a
finitely generated group G: they encode the large-scale structure of G, while
simultaneously providing cocompact approximations to the universal space
EG. I'll discuss a model for the algebraic K-theory assembly map in
this context, and applications to linear groups. This is joint work with
Tessera and Yu.
May 21: Julie Bergner � Classifying space constructions
May 28: Julie Bergner � Classifying diagrams and complete Segal spaces
June 3: No seminar
October 2: Philip Hackney, Props
October 9: Greg Chadwick, Complex cobordism
October 16: Greg Chadwick, Structured complex orientations
October 23: Reinhard Schultz - Classifying smooth G-manifolds
up to finite ambiguity
Abstract: A set of invariants
is said to classify a family of objects up to finite ambiguity if the map from
isomorphism types to invariants is finite-to-one. Results of D.
Sullivan provide such invariants for closed, simply connected manifolds of
dimension at least 5 based upon rational homotopy theory, and subsequent work
of M. Rothenberg and G. Triantafillou yields similar results for certain
manifolds with smooth finite group actions. We shall describe further
results along these lines when certain key hypotheses in the latter work do not
hold.
October 30: Zhixu Su (UC Irvine) - Rational homotopy types of
high-dimensional manifolds
Abstract: Rational surgery can be used to study existence of manifold
realizing certain rational homotopy type. There exists no manifold along the
line of projective planes above the dimension of octonions due to the
inexistence of hopf invariant 1 map in higher dimensions. I investigated the
existence of manifolds analogous to projective planes in the rational sense,
such that the rational cohomology is rank one in dimension 0, 2k and 4k and is zero
otherwise. The problem can be reduced to finding possible Pontryagin classes
satisfying the Hirzebruch signature formula and a set of congruence relations
determined by the Riemann-Roch integrality conditions, which is eventually
equivalent to solving a system of Diophantine equations. After a negative
answer in dimension 24, the first existence dimension is 32. As a joint work
with Jim Fowler (Ohio State), a family of higher dimensions can be ruled out.
November 6: David Ayala (USC) - Higher categories as sheaves on
manifolds
Abstract: Many proposed higher
categories come from geometric situations. This talk will demonstrate a
constructive connection between a homotopy theory of local invariants of
n-manifolds and that of weak n-categories in the sense of Rezk. The basic
construction is that of labeled configuration spaces, so I will draw some
pictures. This is a report on joint work with Nick Rozenblyum.
November 13: Ben Antieau (UCLA) - Brauer spaces of commutative ring
spectra
Abstract: I will describe a
topological space that encodes some arithmetic information about a connective
commutative ring spectrum. Its points are modules for Azumaya algebras. The
fundamental group at a point is related to the picard group. The higher
homotopy groups are related to the original homotopy groups of the ring. There
is a spectral sequence to compute the homotopy groups of the Brauer space, and
this will allow me to show that the Brauer group of the sphere spectrum
vanishes. This can be interpreted as a rigidity result for the stable homotopy
category.
November 20: Michele Intermont (Kalamazoo College) - The Shape of Data
Abstract: Increasingly, topological
ideas are being applied to large data sets in an attempt to organize and gain
new insight into the data.� In this talk,
we�ll give an introduction to the field of Topological Data Analysis, and talk
about some current work in bacterial genomics.
November 27: Jonathan Campbell (Stanford University) - �Topological Hochschild Homology and
Koszul Duality
Abstract: Topological Hochschild
Homology (THH) is an invariant of ring spectra related both to K-theory and
topological field theories. In this talk I'll state and prove a theorem
concerning the relationship between THH and Koszul duality. I'll introduce the
necessary definitions, and in particular say what I mean by "Koszul
duality". I will also introduce some (∞,1)-categorical background
that will be necessary for the proof. Finally, I'll discuss some related
results that I believe to be true, and applications of the work above to
topological field theories.
December 4: Viraj Navkal (UCLA) - G-Theory of a Local Ring of Finite CM
Type
Abstract: Let R be a complete local
ring of finite CM type -- for example, the complete local ring of
functions at a simple complex hypersurface singularity. A classical
theorem of Auslander and Reiten describes a presentation of the Grothendieck
group of the category mod(R) of finitely generated R-modules. I will
explain this theorem and describe how the group presentation arises from a
homotopy fiber sequence of certain K-theory spectra. I will also
show how the homotopy fiber sequence can be used to obtain quite
explicit descriptions of the higher K-groups of mod(R).
January 8: Bruce Corrigan-Salter (SUNY Buffalo) - The Rigidification of
Homotopy Algebras over Finite Product Sketches
Abstract: Multi-sorted algebraic theories provide a formalism for describing
various structures on spaces that are of interest in homotopy theory. The results of Badzioch and Bergner showed
that an interesting feature of this formalism is the following rigidification property. Every multi-sorted
algebraic theory defines a category of homotopy algebras, i.e. a category of spaces equipped with certain
structure that is to some extent homotopy invariant. Each such homotopy algebra can be replaced by a weakly
equivalent strict algebra which is a
purely algebraic structure on a space. The equivalence between the homotopy
categories of loop spaces and
topological groups is a special instance of this result. In this talk we will
introduce the notion of a
multi-sorted semi-theory which is a useful generalization of a multi-sorted
algebraic theory. We will show that in the
setting of multi-sorted semi-theories we can still obtain results paralleling
these of Badzioch and Bergner,
although a rigidification of a homotopy algebra over a multi-sorted semi-theory
is given by a strict algebra
over a certain resolution of that semi-theory. This extends the result obtained
by Badzioch for
single-sorted semi-theories. We will finish the talk by showing that these
results can be extended for a rigidification of homotopy algebras over finite product sketches.
January 15: Kate Poirier (UC Berkeley) - Compactifying String Topology
Abstract: String topology studies
the algebraic topology of the space of loops and paths in a manifold. Previous
treatments of string topology describe algebraic structures on the homology of
this space and operations parameterized by the moduli space of Riemann
surfaces. One perspective is that these structures should be a shadow of a
richer structure at the chain level and that the space parametrizing the
operations should be compactified. In this talk, we describe a compact space of
graphs giving string topology operations on the singular chains of the space of
loops and paths which induce known operations on homology. This is joint work
with Gabriel C. Drummond-Cole and Nathaniel Rounds.
January 22: Julie Bergner � Group actions on Γ-spaces
January 29: Julie Bergner � Simplicial localization
February 5: Greg Chadwick � Genera in algebraic topology
February 12: Philip Hackney � Hochschild homology of structured algebras
February 19: Rena Levitt (Pomona College) - The word problem for quandles
Abstract: In 1911 Max Dehn stated his now famous word
problem: given a finitely generated group G is there an algorithm to determine
if two words in the generators represent the same element in G? While stated in
group theoretic terms, Dehn's motivation for the word problem came from the
study knot groups and surface groups. In this talk, I will discuss a natural
generalization of Dehn's word problem to finitely generated quandles, and show
that the word problem is solvable for both free and knot-like quandles. The
algorithm we define is similar to Dehn's original method for the fundamental
groups of surfaces with genus at least two. This is joint work with Sam Nelson.
February 26: Philip Hackney � Hochschild homology of structure algebras, 2
March 5: David Rose (USC) - Quantum link invariants and (higher)
representation theory via skew Howe duality
Abstract: Quantum link invariants
(e.g. the Jones polynomial) arise due to structures on the category of
finite-dimensional representations of quantum groups. These categories often
have diagrammatic descriptions which give the skein-theoretic definitions of
the link invariants. We will discuss a recent result of
Cautis-Kamnitzer-Morrison which gives a relation between the diagrammatic
framework and skew Howe duality, a representation-theoretic construction which
is intimately connected to link invariants. Time permitting, we'll also discuss
recent work of the speaker (joint with A. Lauda and H. Queffelec) where we sort
out the categorified version of this picture, showing that Khovanov homology is
a 2-representation of the categorified quantum group.
March 12: Mike Williams - A nonseparating torus in a tunnel number one
knot complement in S1 x S2
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