Topology Seminar

The seminar has talks on a variety of topics in topology, including algebraic geometry, chromatic homotopy theory, continuum theory, model categories, Nielsen fixed-point theory, simplicial sets, span theory, symmetric spectra, and topological groups.

For more information contact Daniel Davis.

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Fridays at 1:00 in MDD 208

• 6 February 2015
  No seminar due to the Mathematics Colloquium at 1:10.
• 13 February 2015
  Time and location change due to holiday: 10:00 MDD 202
  An overview of Bousfield localization for spectra
  Daniel Davis
• 20 February 2015
  Instead of a seminar, there is a Mathematics Colloquium in topology:
  Finite-type invariants and Taylor towers for spaces of knots and links
  Robin Koytcheff (University of Victoria)
• 27 February 2015
  Part I: a little more on Bousfield localization; Part II: Lindner's paper on Mackey functors
  Daniel Davis
• 6 March 2015
  A 'Mackey functor' (which is a certain pair of functors) really is a functor!
  Daniel Davis
• 13 March 2015
  No seminar: The 2014 Lloyd Roeling Conference/2015 SRAC starts today.
• 20 March 2015
  More on Mackey functors: finishing Lindner's theorem and the example for finite G-sets
  Daniel Davis
• 27 March 2015
  An Introduction to Sheaves
  Christopher Ryan
  Abstract: The notion of a sheaf is introduced, and some basic properties of sheaves are given along with the origins and motivation of sheaf theory.
• 17 April 2015
  Motivic K-theory symmetric ring spectrum
  Youngsoo Kim (Tuskegee University)
  The construction of a motivic symmetric ring spectrum representing algebraic K-theory will be presented. We will discuss the associativity of the multiplicative structure, and also discuss how it can be solved using standard vector bundles.
• 24 April 2015
  Modules and splittings
  Scott Bailey (Clayton State University)
  Abstract: In this talk, we will discuss past, present, and future work in the classification of stable isomorphism classes of B-modules (where B is a sub-Hopf algebra of the Steenrod algebra). Past, present, and future applications to the splitting of the Tate spectra of v_n-periodic cohomology theories will also be discussed.
• 1 May 2015
  Resolutions of the $K(2)$-local sphere spectrum
  Irina Bobkova (University of Rochester)
  Abstract: Computing the stable homotopy groups of spheres is a long-standing problem in algebraic topology. I will begin by introducing the subject of chromatic homotopy theory which describes the homotopy of the p-local sphere spectrum S through a family of localizations $L_{K(n)}S$ with respect to Morava K-theories $K(n)$. I will discuss computational tools which arise from the theory of formal group laws and their deformations. Then I will specialize to the $K(2)$-local category and talk about finite resolutions of the $K(2)$-local sphere spectrum by a sequence of spectra and some recent computations.

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Information about the last few semesters is provided below.

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Fridays at 1:00 in MDD 208

• 19 September 2014
  Cohomology: A Mirror of Homotopy
  Agnes Beaudry (University of Chicago)
  Abstract: The philosophy of chromatic homotopy theory is that the stable homotopy groups of the sphere $S$ can be reassembled from the homotopy groups of a family of spectra $L_{K(n)}S$. Roughly, $L_{K(n)}S$ is the $n$-th chromatic layer of $S$. There are spectral sequences whose input is the cohomology of a group, the Morava Stabilizer group, and whose output is the homotopy of the $n$-th chromatic layer. In this talk, I will illustrate how some of these spectral sequences mirror the homotopy groups of $L_{K(n)}S$ and of $S$.
• 3 October 2014
  Model categories and the example of simplicial sets
  Daniel Davis
• 17 October 2014
  Model categories: some results and homotopy-theoretic maneuvers
  Daniel Davis
• 31 October 2014
  What is a simplicial category? A simplicial model category?
  Daniel Davis
• 7 November 2014
  Face maps, degeneracies, \Delta[3], and left homotopy
  Daniel Davis
• 14 November 2014
  A canonical presentation of an arbitrary simplicial set and the example of \Delta[2]
  Daniel Davis
• 21 November 2014
  Cofibrant objects, projectives, and left homotopy
  Daniel Davis
  
• 5 December 2014
  Left homotopy in general and for simplicial sets, and Whitehead's theorem
  Daniel Davis
  

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Fridays at 1:00 in MDD 208

• 31 January 2014
  Bigraded modules, exact couples, and spectral sequences
  Daniel Davis
• 7 February 2014
  The tail of quantum spin networks
  Mustafa Hajij (graduate student, LSU)
  Abstract: We study the tail, a q-power series invariant of a sequence of admissible trivalent graphs with edges colored n or 2n. We use local skein relations to understand and compute the tail of these graphs. For many quantum spin networks they turn out to be interesting number-theoretic q-series. In particular, certain quantum spin networks give a skein-theoretic proof for the Andrews-Gordon identities for the two-variable Ramanujan theta function as well to corresponding identities for the false theta function. Finally, we also give a product formula that the tail of such graphs satisfies.
• 14 February 2014
  A nugget in the tool chest of a working homotopy theorist: affine group schemes
  Daniel Davis
  
• Note
  The speaker originally scheduled for 14 February, Scott Bailey (Clayton State University), was not able to make it due to the ice storm in Atlanta. He had planned to speak on: Modules and splittings
  Abstract: In this talk, we will discuss past, present, and future work in the classification of stable isomorphism classes of B-modules (where B is a sub-Hopf algebra of the Steenrod algebra). Past, present, and future applications to the splitting of the Tate spectra of v_n-periodic cohomology theories will also be discussed.
• 21 February 2014
  Invariant contact structures on 7-dimensional nilmanifolds
  Sergii Kutsak (Florida Institute of Technology)
  Abstract: I will give the list of all 7-dimensional nilpotent real Lie algebras that admit a contact structure. Based on this list, I will describe all 7-dimensional nilmanifolds that admit an invariant contact structure. Also I will give countably infinitely many examples of 7-dimensional nilmanifolds $N$ such that $N$ admits an invariant contact structure and $N \times S^1$ cannot admit a symplectic structure.
• 28 February 2014
  Vojislav Petrovic (graduate student)
  (Note: Due to the preparations for Mardi Gras, Voja's talk has been rescheduled for 2 May 2014.)
• 7 March 2014
  A computation in stable homotopy theory using topological modular forms
  Don Larson (Penn State Altoona)
  Abstract: In this talk we will discuss a computation related to a very important object in stable homotopy theory known as the (3-primary) $K(2)$-local sphere. We will first put this computation in the context of a broader problem---namely, computing the stable homotopy groups of spheres---and then describe some surprising connections with classical number-theoretic objects like elliptic curves and modular forms. Finally, we will talk a bit about the computation itself.
• 14 March 2014
  Finite subgroups of a formal group of height 2 over F_9
  Yifei Zhu (Northwestern University)
  Abstract: As an algebraic invariant attached to topological spaces, an elliptic cohomology theory records information about elliptic curves and integral modular forms. In particular, power operations in such a cohomology theory encode moduli problems of elliptic curves, specifically cyclic isogenies of the corresponding power. In this talk I'll discuss an explicit example.
• 21 March 2014
  Spaces of commuting elements in Lie groups
  Mentor Stafa (Tulane University)
  Abstract: We study the spaces of commuting n-tuples in a compact and connected Lie group G, denoted Hom(Z^k,G). We introduce an infinite dimensional topological space denoted Comm(G), reminiscent of a Stiefel variety, that assembles the spaces Hom(Z^k,G) into a single space. This construction admits stable decompositions which allow the study of the spaces Hom(Z^k,G) and the Hilbert-Poincare series is also calculated using Molien's theorem. The cohomology of Comm(G) is given in terms of the tensor algebra generated by the reduced homology of the maximal torus. This is joint work with Fred Cohen.
• 28 March 2014
  Zariski's intrinsic description of nonsingular affine varieties
  Chris Ryan (University of Louisiana at Lafayette)
  Abstract: This is an expository talk on the subject of classical algebraic geometry. We will look at a fundamental theorem of Zariski, which states that an affine variety is nonsingular at a point P if and only if the local ring of P on Y is a regular local ring.
• 4 April 2014
  Knot and link invariants for vector fields
  Rafal Komendarczyk (Tulane University)
  Abstract: In 1979, V. I. Arnold showed that the fundamental invariant of 2--component links i.e. the linking number, can be generalized to an invariant of volume preserving vector fields. In this talk, the Arnold's construction will be outlined, as well as its various applications. Further, more recent results concerning generalizations of this construction to Vassiliev invariants of knots, will be discussed (joint work with Ismar Volic).
• 11 April 2014
  A Projective Model Structure on Pro-categories, and the Relative Étale Homotopy Type
  Tomer Schlank (M.I.T.)
  Abstract: Isaksen showed that a proper model category $C$, induces a model structure on the pro-category $Pro(C)$. In this talk I will present a new method for defining a model structure on the pro-category $Pro(C)$. This method requires $C$ to satisfy a much weaker condition than having a model structure. The main application will be a novel model structure on pro-simplicial sheaves. We see that in this model structure a "topological lift" of Artin and Mazur's Étale homotopy type is naturally obtained as an application of some natural derived functor to the terminal object of the étale topos. This definition can be naturally generalized to a relative setting, namely, given a map of topoi T \to S, we get a notion of a relative homotopy type of T over S which is a Pro-simplicial object in S. This definition turns out to be useful for the study of rational points on algebraic varieties. This is a joint work with Ilan Barnea.
• 2 May 2014
  Derived functors of inverse limits and profinite G-modules
  Vojislav Petrovic (graduate student)

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Fridays at 11:00 in MDD 311

• 13 September 2013
  How do we regard a set of interesting morphisms in a category as being isomorphisms?
  Daniel Davis
• 20 September 2013
  The homotopy category of a homotopical category
  Daniel Davis
• 27 September 2013
  A presentation of the morphisms in a homotopy category by type
  Daniel Davis
• 4 October 2013
  A 3-arrow calculus for a homotopical category and saturation
  Daniel Davis
• 18 October 2013
  Boolean spaces
  Maciej Niebrzydowski
• 25 October 2013
  Representation theorems for Boolean algebras
  Maciej Niebrzydowski
• 1 November 2013
  A homotopical version of uniqueness in a category
  Daniel Davis
• 8 November 2013
  Lloyd Roeling Topology Conference
• 15 November 2013
  On poset polynomials
  Maciej Niebrzydowski
• 22 November 2013
  Boolean quotients and Boolean derivatives
  Maciej Niebrzydowski
• 6 December 2013
  Continuous group cohomology of discrete G-modules and $\delta$-functors
  Vojislav Petrovic (graduate student)

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Fridays at 1:00 in MDD 214

• 1 February 2013
  An introduction to double categories
  Maciej Niebrzydowski
• 8 February 2013
  No seminar -- Mardi Gras break
• 15 February 2013
  Symmetric spectra: the objects that give rise to generalized cohomology theories
  Daniel Davis
• 22 February 2013
  The symmetric monoidal category of symmetric spectra
  Daniel Davis
• 29 February 2013
  no seminar this week
• 8 March 2013
  Symmetric spectra: some examples and the notion of stable equivalence
  Daniel Davis
• 15 March 2013
  The Temperley-Lieb category and its applications
  Maciej Niebrzydowski
• 22 March 2013
  On some models in statistical mechanics and their connections with topology
  Maciej Niebrzydowski
• 12 April 2013
  On n-ary algebras and their applications in knot theory
  Maciej Niebrzydowski
• 19 April 2013
  On stable equivalences and connective symmetric spectra
  Daniel Davis
• 26 April 2013
  Finite projective geometry
  Vic Schneider

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Fridays at noon in MDD 214

• 7 September 2012
  Entropic operations in knot theory
  Maciej Niebrzydowski
• 14 September 2012
  The Cuntz semigroup of low dimensional spaces
  Leonel Robert
  Abstract: I will first define the Cuntz semigroup of a topological space (this object is of interest to C*-algebraists). I'll then describe the Cuntz semigroup for spaces of dimensions 0,1,2, and 3.
• 21 September 2012 (ROOM CHANGE MDD 311)
  Progress on the "Hit Problem"
  Shaun Ault (Valdosta State University)
  
• 28 September 2012
  An introduction to nonabelian continuous cohomology
  Daniel Davis
• 5 October 2012
  A non-short, but not long, exact sequence in nonabelian continuous cohomology
  Daniel Davis
• 19 October 2012
  How does one build a Thom spectrum?
  Daniel Davis
• 26 October 2012
  An introduction to tolerance space theory
  Maciej Niebrzydowski
• 2 November 2012
  Finite geometries
  Vic Schneider
• 9 November 2012
  Bicomplex from degenerate elements of a weak simplicial module
  Jozef Przytycki (The George Washington University)
  Abstract: Knot Theory motivated homology of quandles and racks, these in turn motivated the speaker to introduce distributive homology and to show that they can be described by a weak simplicial module. The degenerate part is not acyclic, however, it splits and its (right handed) filtration leads to a bicomplex, which is very approachable in the case of distributive homology.
• 16 November 2012
  Some remarks on differential groupoids
  Maciej Niebrzydowski
• 30 November 2012
  Finite geometries, part 2
  Vic Schneider

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• 27 January 2012
  Introduction to Haar integrals
  Vic Schneider
• 3 February 2012
  Introduction to Haar integrals, part 2
  Vic Schneider
• 10 February 2012
  The ends of groups
  Maciej Niebrzydowski
• 17 February 2012
  Ordered sets in combinatorics and topology
  Maciej Niebrzydowski
• 24 February 2012
  A discussion of inverse limits without a formal definition
  Brian Hill (graduate student)
• 2 March 2012
  NO SEMINAR
  
• 9 March 2012
  Lusternik-Schnirelmann theory - old and new
  Yuli Rudyak (University of Florida)
  
• 16 March 2012
  A discussion of inverse limits without a formal definition, part 2
  Brian Hill (graduate student)
• 23 March 2012
  Ordered sets in combinatorics and topology, part 2
  Maciej Niebrzydowski
• 30 March 2012
  Tent maps and topological conjugacy
  Brian Hill (graduate student)
• 20 April 2012
  Pro-objects in a category and their morphisms
  Daniel Davis
• 27 April 2012
  Ordered sets in combinatorics and topology, part 3
  Maciej Niebrzydowski

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• 2 September 2011
  On some Cayley type theorems
  Maciej Niebrzydowski
• 9 September 2011
  On some Cayley type theorems, part 2
  Maciej Niebrzydowski
• 16 September 2011
  Selfcoincidences of Mappings between Spheres
  Duane Randall (Loyola University, New Orleans)
  Abstract: The concepts of loose and also loose by small deformation are defined for mappings between spheres. The relationship in certain dimensions between mappings which are loose, but not loose by small deformation, with the existence or non-existence of Kervaire invariant one elements will be explained.
• 23 September 2011
  Discrete groups and manifolds in S²xR
  Jozsef Z. Farkas
  Abstract: Thurston's geometrization conjecture played a crucial role for example in proving the Poincare conjecture. There are eight homogeneous simply connected geometries which give rise to compact three-manifolds. One of the simplest of the non-constant curvature ones is the space S²xR, which as its name suggests, is the direct product of the sphere with the real line. Similarly to the "Euclidean strategy" we classify the crystallographic groups in S²xR. We find 134 equivalence classes of space groups up to similarity. These in turn give rise to the 4 well-known compact manifolds admitting S²xR geometry.
• 30 September 2011
  Obstruction theory for E_infty maps
  Niles Johnson (University of Georgia)
  Abstract: We take an obstruction-theoretic approach to the question of algebraic structure on spectra. At its heart, this is an application of the Bousfield-Kan spectral sequence adapted for general operadic structure in a range of topological categories. This talk will focus on examples from rational homotopy theory which illustrate the obstructions to rigidifying homotopy algebra maps between differential graded algebras to strict algebra maps. In the topological context, these provide explicit examples of H_infty maps which cannot be rigidified to E_infty maps.
• 7 October 2011
  An introduction to operads and algebras over operads
  Daniel Davis
  
• 14 October 2011
  Algebras over operads and some canonical examples
  Daniel Davis
  
• 28 October 2011
  S²xR space groups: generalized Coxeter groups and ball packings
  Jozsef Farkas
  
• 4 November 2011
  Fibered categories, fibers, and groupoids
  Daniel Davis
  
• 11 November 2011
  Categories fibered in groupoids, monoids, and classifying spaces
  Daniel Davis
  
• 2 December 2011
  Profinite Groups and Discrete G-Sets
  Brian Hill (graduate student)
  Abstract: Some interactions between profinite groups and discrete G-sets will be explored. In addition, we will examine the sets of morphisms between discrete G-sets, as well as the Hom-set functor.

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• 19 January 2011
  Schubert polynomials and cohomology of flag manifolds
  Leonardo Mihalcea
• 26 January 2011
  Inverse limits with subsets of IxI
  Thelma West
• 2 February 2011
  Inverse limits with subsets of IxI, Part 2
  Thelma West
• 16 February 2011
  Inverse limits with subsets of IxI, Part 3
  Thelma West
• 23 February 2011
  Inverse limits of upper semi-continuous set valued functions
  Thelma West
• 2 March 2011
  Schubert varieties revisited
  Leonardo Mihalcea
• 16 March 2011
  Towards Schubert polynomials
  Leonardo Mihalcea
• 23 March 2011
  Towards Schubert polynomials, part 2
  Leonardo Mihalcea
• 30 March 2011
  Spatial graphs and their invariants
  Maciej Niebrzydowski
• 6 April 2011
  Areas of certain quadrilaterals
  Vic Schneider
• 13 April 2011
  Areas of certain quadrilaterals, part 2
  Vic Schneider
• 27 April 2011
  Continuous group cohomology for towers of discrete G-modules
  Daniel Davis

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• 17 September 2010
  Knotted surfaces
  Maciej Niebrzydowski
• 24 September 2010
  Knotted surfaces, part 2
  Maciej Niebrzydowski
• 8 October 2010
  A Tale of Six Atriodic Continua, Part 1
  Thelma West
• 15 October 2010
  No seminar this week due to the Roeling Conference.
  
• 22 October 2010
  A Tale of Six Atriodic Continua, Part 2
  Thelma West
• 29 October 2010
  postponed
• 5 November 2010
  Introduction to inverse limits
  Thelma West
• 12 November 2010
  Quantum Schubert Calculus
  Leonardo Mihalcea
  This will be a gentle introduction to main definitions and ideas of Schubert Calculus and its "quantum" version for Grassmannians.
• 19 November 2010
  An introduction to model categories
  Daniel Davis
• 3 December 2010
  An introduction to model categories, part 2
  Daniel Davis

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• January 29th:
  Kan complexes and categories
  Daniel Davis
• February 5th:
  Kan complexes and $\infty$-categories
  Daniel Davis
• February 19th:
  $\infty$-categories and composition through horns
  Daniel Davis
• February 26th:
  Introduction to digital topology
  Maciej Niebrzydowski
• March 12th:
  Axiomatic digital topology
  Maciej Niebrzydowski
• March 19th:
  'Composition' and joins for $\infty$-categories
  Daniel Davis
• March 26th:
  Equivalent metrics and the spans of graphs, Part I
  Thelma West
• April 16th:
  Equivalent metrics and the spans of graphs, Part II
  Thelma West
• April 23rd:
  Equivalent metrics and the spans of graphs, Part III
  Thelma West
• April 30th (ROOM 208):
  $A^1$-Representability of Hermitian $K$-theory
  Girja Shanker Tripathi (graduate student, LSU)
  Abstract: I will discuss my result that in the $A^1$-homotopy category of smooth schemes over a field of characteristic not equal to 2, the Hermitian $K$-theory is representable by "orthogonal Grassmannians." This result is the Hermitian analogue of the corresponding result for algebraic $K$-theory. I will introduce some ideas (parallel to the ones in topology) from the $A^1$-homotopy theory developed by Morel and Voevodsky (1999).

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• 11 September 2009:
  Graph embeddings and chromatic numbers
  Maciej Niebrzydowski
• 18 September 2009:
  More on graph embeddings
  Maciej Niebrzydowski
• 25 September 2009:
  Sheaves of sets on a Grothendieck site
  Daniel Davis
• 9 October 2009:
  Limits and sheaves of sets on a site
  Daniel Davis
• 16 October 2009:
  Crossing numbers of graphs
  Maciej Niebrzydowski
• 23 October 2009:
  What is the Dimension of R(n)?
  Roger Waggoner
• 30 October 2009:
  Lloyd Roeling Conference
• 6 November 2009:
  Representing graphs
  Jake Sundberg (graduate student)
• 13 November 2009:
  Representing graphs, part two
  Jake Sundberg (graduate student)
• 20 November 2009:
  Path connectedness
  Vic Schneider
• 4 December 2009:
  Path connectedness, part 2
  Vic Schneider

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• January 21st:
  Introduction to span
  Thelma West
• January 28th:
  Span, pt. II: a consideration of 'an atriodic tree-like continuum with positive span' (after Ingram)
  Thelma West
• February 6th:
  Sheaves of abelian groups on topological spaces
  Daniel Davis
• February 13th:
  Regular Sequences in unstable algebras over the Steenrod algebra
  Mara Neusel (Texas Tech)
• February 20th:
  Connections between graph theory and knot theory
  Maciej Niebrzydowski
• February 27th:
  Defining sheaves with equalizer diagrams and Grothendieck sites
  Daniel Davis
• March 6th:
  Fibered products, sieves, and pretopologies on categories
  Daniel
• March 13th:
  Connections between graph theory and knot theory, Part II
  Maciej Niebrzydowski
• March 20th:
  Obtaining Grothendieck sites via bases
  Daniel
• March 27th:
  Topological quandles
  Maciej
• April 3rd:
  Introduction to hyperspaces
  Thelma West
• April 24th:
  An Application of the snake and horseshoe lemmas to the functors Hom( - , G) and Ext( - , G), in the category Ab
  Chris Ryan (graduate student)
• May 1st:
  Size levels of arcs, continued
  Thelma West

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• September 5:
  Introduction to Khovanov homology I
  Maciej Niebrzydowski
• September 12:
  No seminar due to Hurricane Ike
• September 19:
  Introduction to Khovanov homology II
  Maciej Niebrzydowski
• September 26:
  Introduction to simplicial sets
  Daniel Davis
• October 3:
  No seminar due to Fall break
• October 10:
  No seminar due to Hurricane Ike makeup classes
• October 17:
  Homotopy theory and simplicial sets
  Daniel Davis
• October 24:
  Gram determinants in Knot Theory: skein module motivation
  Jozef Przytycki (The George Washington University)
• October 31:
  The lattice of topologies on the set
  Vic Schneider
• November 7:
  The lattice of topologies on the set, part 2
  Vic Schneider
• November 14:
  No seminar
• November 21:
  Dimension theory 101
  Roger Waggoner
• December 5:
  Elementary open problems in knot theory
  Maciej Niebrzydowski

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• January 28:
  Inverse limits of spaces and their homotopy limits, with an eye on examples in continuum theory,
  Daniel Davis.
• February 11:
  Beginning steps in understanding the relationship between algebraic geometry and complex-oriented cohomology theories,
  Daniel Davis.
• February 18:
  Quandles, racks, and related knot invariants,
  Maciej Niebrzydowski.
• February 25:
  Quandle homology theories and their connection with geometry of knots,
  Maciej Niebrzydowski.
• March 3:
  Topological groups,
  Vic Schneider.
• March 10:
  Topological groups, part two,
  Vic Schneider.
• March 17:
  The homological algebra of the continuous cohomology of topological groups increases as one restricts to profinite groups,
  Daniel Davis.
• March 31:
  Three basic examples employing inverse limits, Part I,
  Thelma West.
  Inverse limits have appeared in various ways in the seminar, but no speaker has yet really dug into the interior of an inverse limit of topological spaces and really unpacked its meaning in a particular situation. One of the purposes of this talk (and the sequel on April 21st) is to give graduate students a better feel for inverse limits.
• April 7:
  The Fixed Point Property,
  Roger Waggoner.
• April 14:
  The Nielsen number,
  Roger Waggoner.
• April 21:
  Three basic examples employing inverse limits, Part II,
  Thelma West. (This talk is a continuation of the March 31st seminar.)
• April 28:
  Higher Grothendieck-Witt groups in Algebra and Topology,
  Marco Schlichting (Louisiana State University).
  Abstract: I will motivate the study of higher Grothendieck-Witt groups (alias hermitian K-groups) of rings and schemes with two examples from topology-- cobordism categories of certain 4 manifolds (due to Giansiracusa) and an algebraic reinterpretation of 8-fold real Bott periodicity (due to Karoubi). Then I will explain a recent result of mine concerning the local-global behavior of those groups.
• May 5:
  Part one: a brief statement of the definition of elliptic spectrum (related to Q(2) - from the last result of Daniel's colloquium)
  Daniel Davis, 5 minutes;
  Part two: Elliptic curves, their associated abelian groups, and points of finite order,
  Matthew Lennon (graduate student), a 25-minute talk;
  Part three: The Nielsen number and the Jiang subgroup,
  Roger Waggoner, a 35-minute talk.