Algebraic topology from a geometric perspective.
The materials below are recordings of remote lectures, along with the associated whiteboards
and other supporting materials. These lectures started on March 30, 2020.
There are also office hours and perhaps other opportunties to learn together.
If you are interested in joining send an email to dps **at*** uoregon ++DOT+++ edu.
 Welcome lecture and corresponding
PDF of the whiteboard.
 An overview of the mathematics, and a brief personal introduction.
 Catchup lecture and
clips from the whiteboard.
 An introduction to Thom cochains, which will play a role throughout, and
application to the cohomology of loop spaces and cohomology of configuration spaces.
Friedman, Medina and I plan to post a paper on the more elementary aspects of Thom cochains
soon. The material on homology and cohomology of configuration spaces is treated in
this expository paper.
A much more extensive introduction to configuration spaces is
here.
 First ten minutes and
last forty five minutes of
the first lecture, along with
clips from the whiteboard.
 A treatment of Hopf invariants, which are explicit linking invariants preimages of
submanifolds under maps from spheres. Remarkably, the geometry is perfectly governed
by a Quillen functor which we'll discuss in the next lecture. The material in this
lecture is the subject of this paper.
 Second lecture and
clips from the whiteboard.
 We continue to develop Hopf invariants, this time giving a full definition as well as defining
the Lie coalgebraic bar construction. We state the main theorem, that Hopf invariants are
sharp, and share a brief outline of proof.
As was the case for the last lecture, the material in this
lecture is the subject of this paper.
 Third lecture and
clips from the whiteboard.

We finish our story of Hopf invariants (except for the fundamental group) by connecting
them with previous work, starting with Hopf, Freudenthal, Whitehead, Hilton and
BoardmanSteer and then especially the approaches to rational homotopy theory of
Chen, Quillen and Sullivan. We end with a (vague) conjecture about rational homotopy classes
of maps in general. The above linked paper is still the primary source, but there is a
large literature which introduces rational homotopy theory more broadly. One paper that an
audience member shared which I hadn't seen but is quite relevant to the perspective from
commutative algebra is "Through the looking glass: a dictionary between rational
homotopy theory and local algebra." by Avramov and Halperin.
 Fourth lecture and
whiteboards.

New topic: classifying spaces. These are of central importance in algebraic topology  associating a homotopy type canonically to a group (algebraic topology!). In this introduction we try to bring together key definitions/ perspectives: the simplicial BG, the homotoptical characterization, and natural geometric models. We use the simplicial construction to give concrete models for EilenbergMacLane spaces.
 Fifth lecture and
whiteboards.

We develop characteristic classes through Thom cochains, establishing cases of the Whitney
sum formula as well as all other axioms geometrically. We also develop both homology and
cohomology of infinite Grassmannians and understand the pairing between them. Along the
way we touch on related topics such as the Thom isomorphism theorem, a proof by Bruno Harris of
Bott Periodicity,
and restriction maps in group cohomology.
 Sixth lecture and
whiteboards.

After addressing a technical issue about the Thom cochains proposed in lectur five, we discuss
the Wu formula, wich relates characteristic classes to Steenrod operations. We then turn
to the homology of EilenbergMacLane spaces, developing a Hopf ring structure and giving
Wilson's concise description.
 Additional lecture, on linking of letters
and
whiteboards.

We develop Hopf invariants for the fundamental group of wedges of circles (that is, free groups),
which combinatorially correspond to letter linking. These give a new approach to determining
the representative of a word in the lower central series subquotients for free groups (which
constitute a free Lie algebra). The results are newly written up in
this paper.
 Seventh lecture
and
whiteboards.

We start iterated loop space theory. The central theorems of this theory are
 the little disks operad acts on iterated loop spaces
 thus the homology of an iterated loop space is a graded Poisson algebra and DyerLashof operations act on their homology
 (recognition) an Hspace with inverses up to homotopy on which the Fulton MacPherson operad in dimension n acts is an nfold loop space
 (free models) the group completion of the free little ndisks algebra on X provides a model for nfold loops on the nfold suspension of X.
Today we discussed the n=1 case of these theorems, defined the little disks operad and developed the Poisson algebra structure on homology of an nfold loop space. Next time: the FultonMacPherson operad, the recognition theorem, factorization homology and DyerLashof operations.
 Eigth lecture
and
whiteboards.

We develop the FultonMacPherson operad and use it to sketch the proof of the recognition theorem,
namely that spaces on which it acts whose components form a group are iterated loop spaces. Note
that we need to modify what was in the lecture using configurations on the sphere, which
when one passes to configurations with labels have points "vanish" at the point at infinity.
Extending
the construction from configurations in
the sphere to those in framed manifolds
gives a definition of factorization homology, and we discuss nonabelian
Poincare duality. For the material on the FultonMacPherson operad, one source is
these two
papers. This paper uses the FM operad
for construcing classifying spaces and in nonabelian Poincare duality.
 Ninth lecture
and
whiteboards.

We finish, for the moment at least, our discussion of iterated loop spaces through development
of KudoAraki(DyerLashof) operations on their homology. We then turn to modtwo homology
and cohomology of symmetric groups, both of which we take together as a single object.
Both end up being free over suitable structure, and we see those structures as
overlapping but distinct. A main focus is and will be cup product structure for the cohomology
of symmetric groups, which is necessarily complicated (many generators and relations in
minimal presentations) for individual symmetric groups but when incorporated into a Hopf
ring stucture is determined by little data. If you want to "skip ahead",
you can see more about cohomology of
symmetric groups in these slides from a talk or this
paper.
 Tenth lecture
and
whiteboards.

We give the definitions needed for the description of the cohomology of symmetric groups as a free primitively generated divided powers component Hopf ring. Then we show how these give rise to additive bases and rules for computing cup products. While the object in total is free, we start to see relations arising from Hopf ring distributivity, relations also present in classical rings of symmetric polynomials.
 Eleventh lecture
and
whiteboards.

We elaborate one line of proofs for the calculation of homology and cohomology of
symmetric groups, through subgroups. We focus on the example of S_4. We start
with the elementary fact that the modp cohomology of a group injects into the cohomology
of its pSylow subgroup. For symmetric groups, the pSylow subgroups are given by
wreath products, whose homology is given by "formal KudoAraki operations."
Switching to cohomology, we show further that the cohomology of the wreath product
(and thus S_4) is itself detected on the cohomology of two subgroups, both isomorphic to
S_2 x S_2 (but not conjugate to each other). These are the maximal elementary abelian
subgroups, and detection on such was proven by MadsenMilgram for symmetric groups.
More generally Quillen proved for general groups that these detect modulo nilpotents.
Our proof is more elementary and geometric.
We finish by discussing some open questions in this area.
 Twelfth lecture
and
whiteboards.

We develop FoxNeuwirth cell complexes, which though standardly defined in terms of configuration spaces may be viewed as the cell structure on subquotients of the iterated bar complex for the group of order two. They have been studied in many guises. We show how they illuminate a surprising connection between homology of EilenbergMacLane spaces and cohomology of symmetric groups. And we also share some ideas (some established, some speculative) as to how to use these ideas to model mapping spaces.
Some of these ideas are developed in this paper, though
note that unlike what is claimed without proof there, the cup product structure is not
modeled at the cochain level. So, in a sense there are two skyline bases for the
cohomology of symmetric groups  one which is the Hopf monomial basis and one formed by
associating representative FoxNeuwirth cycles to a skyline diagram.
 Thirteenth lecture
and
whiteboards.

We start with the BlakersMassey theorem, a fundamental theorem about the extent to which homotopy groups have a MayerVietoris sequence (or spectral sequence) in settings where there is such in homology. We apply this to spaces of embeddings, showing how Goodwillie's cutting method allows for decomposition of the homotopy types of spaces of embeddings in terms of homotopy types of simpler such spaces which are in fact configuration spaces (embeddings of zeromanifolds). These ideas are explained in much more detail
(five hundred pages vs. just an hour lecture) in this book.
 Fourteenth lecture
and
whiteboards.

In the previous lecture we related embedding spaces to simpler embedding spaces, which are
homotopy equivalent to configuration spaces. Here we explain a more direct relationship
through induced maps on configuration spaces (also known as evaluation maps or Gauss maps).
We discuss the structure of spectral sequences for homotopy and homology of spaces of knots.
This material is the focus of this paper,
and the collapse results mentioned are in this paper.
 Fifteenth lecture
and
whiteboards.

We discuss the GoodwillieWeiss tower of approximations to the space of classical knots, and the
theory of finitetype knot invariants. We present cases of and progress on the
conjecture that the tower serves as a universal such invariant.
The "quadrisecant result" is in
this paper.
Universality over the real numbers is established in
this paper.
Recent collapse results at any prime are in
this paper.
Along with results announced
here,
this establishes the universality conjecture at the prime p in degrees less than p
(but does not  yet  give new knot invariants).