University of Chicago Algebraic Topology Seminar

4:30 pm   Tuesday, April 7, 2009   in Eckhart 203

Algebraic Topology: Computations in motivic stable homotopy theory
by Kyle Ormsby (University of Michigan)


The Morel-Voevodsky motivic homotopy category provides a framework in which homotopy-theoretic constructions and techniques can be applied to algebraic varieties.  Until recently, though, few computations were known beyond those done by Voevodsky.  I will present some motivic versions of spectral sequences from topology.  In topology, these are used in calculations of stable homotopy groups, while their motivic analogues converge to motivic stable homotopy groups, which are already known to be deep objects related to the theory of quadratic forms and other arithmetic structures.  I will give some examples providing new information about these groups, focusing on the case in which the base field is local.


4:30 pm   Tuesday, April 14, 2009   in Eckhart 203

Algebraic Topology: Not meeting this week




4:30 pm   Tuesday, April 21, 2009   in Eckhart 203

Algebraic Topology: Algebraic Models for Rational Equivariant Spectra
by David Barnes (University of Western Ontario)


For G a finite group, we seek to understand the category of rational equivariant cohomology theories by finding an algebraic model category that is Quillen equivalent to the model category of rational equivariant spectra. In particular, we want to have a monoidal product on this algebraic category that models the smash product of spectra. We then apply this result to the case where G is the group of two-by-two orthogonal matrices.


3:00 pm   Tuesday, April 28, 2009   in Eckhart 203

Algebraic Topology: Cartesian presentations of weak n-categories
by Charles Rezk (UIUC)


I'll describe a model for (infinity, n) categories, called "n-theta spaces"; this model is one way of generalizing the notion of a "complete Segal space", which is a model for (infinity, 1) categories. The category of "n-theta spaces" has two nice properties: it is enriched over spaces, and it is cartesian closed.


3:00 pm   Tuesday, May 5, 2009   in Eckhart 203

Algebraic Topology: Stable Decompositions and Almost Commuting Elements in Lie Groups
by Jose Gomez (University of British Columbia)


Two elements x,y in a Lie group G are said to be almost commutative if [x,y] is in the center of G. The space of almost commuting elements is a good way to approach the space of commuting elements in G, a space of current interest (It appears in a number of Quantum field theories). In this talk I will explain how the space of almost commuting elements in a compact Lie group splits after one suspension. This is a joint work with Alejandro Adem and Fred Cohen.

Note that this talk begins at 3:00. Pretalk is at 1:30 in E203.


4:30 pm   Tuesday, May 5, 2009   in Eckhart 203

Algebraic Topology: Brown Representability and Rosicky Functors
by Amnon Neeman (Australian National University)


We will give a review of the best known versions of Brown representability to date. Then we will state a new result, which depends on a theorem of Rosicky's. Finally we will discuss the status of Rosicky's theorem.


1:30 pm   Tuesday, May 12, 2009   in Eckhart 203

Algebraic Topology: Localization Sequences in THH and TC
by Mike Mandell (Indiana University)


This talk is about joint work with Blumberg that constructs localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a "global" construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofiber sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of Thomason-Trobaugh in K-theory.


4:30 pm   Tuesday, May 12, 2009   in Eckhart 203

Algebraic Topology/Category Theory: Monads on Double Structures
by Geoff Cruttwell (Dalhousie University)


The need to understand monads on bicategories, and their associated algebras, has arisen in several different contexts. An example is the study of (T,V)-algebras, which unify the definitions of topology, metric space, and approach space. However, since bicategories and lax functors do not organize themselves into a 2-category, the very notion of a (lax) monad on a bicategory is difficult to interpret, and so makes the study of objects such as (T,V)-algebras a complicated one.
In this talk, we will discuss on alternate possibility: defining and working with monads on double categories, framed bicategories, and/or fc-multicategories. We will show how using monads on these double structures solves the problem of working with monads on bicategories, while also simplifying the associated constructions and proofs.
This is joint work with Mike Shulman.


4:30 pm   Tuesday, May 19, 2009   in Eckhart 203

Algebraic Topology: Not meeting this week, we're going to the TFT Workshop at Northwestern




4:30 pm   Tuesday, May 26, 2009   in Eckhart 203

Algebraic Topology: Not meeting this week, we're going to the TFT Conference at Northwestern




4:30 pm   Tuesday, June 2, 2009   in Eckhart 203

Algebraic Topology: The stable homotopy hypothesis in low dimensions
by Nick Gurski (Yale University)


The homotopy hypothesis states that there is a correspondence between homotopy n-types and n-groupoids. There is a stable version of this, with "homotopy n-types" replaced by "stable connective n-types" and "n-groupoids" replaced with "symmetric monoidal n-groupoids in which all objects are invertible". When n=0, this is the claim that spectra whose only nonvanishing homotopy group is the 0th one correspond to abelian groups. The n=1 case has been a folklore theorem for quite some time now, and I will discuss a strategy for proving it that appears to generalize to the case n=2. This is part of work in progress with Mikhail Kapranov.


1:30 pm   Thursday, June 11, 2009   in Eckhart 203

Algebraic Topology: The Triad as Place and Action: a Transformational Perspective on Stability
by Thomas Noll (Escola Superior de Musica de Catalunya)


Recent transformational approaches to the study of triads are based on group actions on the set of the major and minor triads. A particular music-theoretical interest in this subject is driven by the possibility of parsimonious voice leadings between certain triads. To each triad X, say X = {C, E, G}, (considered modulo octave) there are three triads P(X)= {C, Eb, G}, L(X)= {E, G, B}, R(X)= {A, C, E}, each sharing two tones with X. What distinguishes triads from arbitrary 3-chords is the small amount by which the third tone has to be displaced: In the case of P(X) it is an augment prime (E -> Eb), in the case of L(X) it is a minor second (C-> B) and in the case of R(X) it is a major second (G -> A). Richard Cohn therefore speaks of the "over-determined triad", as - traditionally - the music-theoretical prominence of the triad is explained in terms of consonance.

My pre-talk is dedicated to yet another property which can be added to the list of over-determining decorations of the triad. This property provides a conceptual link between (the discussion about) Hugo Riemann's concept of consonance on the one hand and the Neo-Hugo- Riemannian transformations P, L, R as mentioned above, on the other. This approach is based on a transformational investigation of the intervallic constitution of the triads. Each triad is studied as a sub-action of a monoid-action of an 8-element monoid on Z12. Each transformation is a Twelve-Tone-Operation (an affine endomorphism of Z12) which stabilizes the triad in question and which extrapolates an association of an internal interval of the triad with its fifth. With this approach I hope to make a contribution to an abandoned discourse between Hugo Riemann and Carl Stumpf and in particular to an elaboration of Stumpf's concept of the triad as a concord of consonances.

Mathematically the approach is an application of elementary topos theory. The Neo-Riemannian transformations P, L, R can be studied as equivariant maps between monoid actions (i.e. as arrows in an associated topos). The structure of the sub-object classifier and its Lawvere-Tierney topolgies allow to draw link between the different qualities of tones in the complement of a triad as such, and the roles of these tones as images of proper triad tones under the transformations P,L, R on the other. In a way, this approach is an attempt to actualize Hugo Riemann's vision of the theory of harmony as a "musical logic".

I will refer to two musical examples and related discussions in music theory: the first movement of Schubert's sonata in Bb (D 960) as discussed by Richard Cohn and by Balz Trümpy and the last study of Alexander Skriabin Op. 65 No. 3 as discussed by Clifton Callender.

This is the Pretalk, see also the abstract below for the talk at 3:00.


3:00 pm   Thursday, June 11, 2009   in Eckhart 203

Algebraic Topology: Diatonic and Tetractys Modes as Instances of Christoffel Duality
by Thomas Noll (Escola Superior de Musica de Catalunya)


A recent development in the mathematical and music-theoretical study of so called well-formed scales is closely related to research directions within the field of algebraic combinatorics on words, namely Sturmian words and their finite analogues, i.e. Christoffel words and their conjugates. I'm going to report on joint work with David Clampitt (Ohio), Karst de Jong (Barcelona) and Manuel Dominguez (Madrid). There is a general music-theoretical desire to understand the principles which guide or constrain the constitution of musical tone relations. Well-formed scales are generated by a fixed interval modulo some period (typically the octave) and thereby embody a concept of tone kinship - given by the generation order of the scale tones. A second concept of tone relation is given by the pitch height order of the scale (step order). The well-formedness condition (as introduced and studied by Carey and Clampitt 1989) requires that the conversion from step order to generation order is a linear automorphism of Zn (where n is the number of tones). A refinement of this condition for musical modes (instead of scales) leads to Christoffel words and their conjugates and to a related concept of duality.

This word-theoretic approach is tightly connected with traditional accounts of tone kinship. This connection is given through the projection from the free non-commutative group F_2 with two generators to the commutative group Z^2. Traditionally, the mathematical investigation of tone kinship is based on the free commutative group over two (or more) musical intervals as linearly independent generators. In the case of the two generators fifth and octave this group is called the Pythagorean tone lattice P. Pitch height enters into this picture as a linear form h: P -> R. It is insightful to embed P into the real plane R^2 with respect to a basis, which is constituted by the gradient (pitch height axis) and the kernel (pitch width axis) of this linear form. Word theory enters here through the approximation of the pitch width axis by polygons in P, i.e. by elements of F_2. For each Christoffel prefix of the infinite "Pythagorean word" (along the pitch width axis) there is a natural candidate for a metric on P, respectively R^2.

I will summarize actual results about the interdependence of regions (such as the Guidonian hexachord aabaa) and standard modes (such as the authentic Ionian mode aaba|aab). The underlying duality has a clear expression in terms of Standard morphisms. The generalization of the diatonic case does not only include larger and/or musically counter-factual modes, but also simpler ones, such as the modes of the three note tetractys scale. I will conclude my talk with remarks on a "Neo-Riemannian" approach to the analysis of fundament progressions.

Note the unusual time of this seminar talk: 3:00. See also the Pretalk abstract above, at 1:30.