Write-Ups

Seminars

Graduate Student Homotopy Seminar (Fall 2013)

This semester the seminar is going to focus on higher category theory with a view towards the cobordism hypothesis. We will meet on Wednesdays at 3PM in 347 Altgeld Hall. The following is not the list of the talks yet. Some of them are going to be individual talks, but some others may get split into couple of pieces. It also may be subject to change.

0. Proposal, Nerses Aramian (18-Sep)

This was an organizational meeting where we laid out the list of topics to be covered in the seminar. The result of this discussion was the proposal above. It also contains brief introductory overview of the material.

1. Introduction to Quasicategories, Sarah Yeakel (25-Sep)

Quasicategories are models for (∞,1)-categories. Our goal is to build intuition for dealing with quasicategories by exploring some examples and some of the usual categorical constructions which apply in the ∞-categorical world. In particular, we will discuss how the ordinary category theory is really a part of the higher category theory, and how the new constructions that we give are equivalent to the old ones, when restricted to the ordinary categories. Following chapter one of Higher Topos Theory, we'll start with definitions and hopefully end up with limits and colimits.

2. Constructions for Quasicategories, Mychael Sanchez (2-Oct)

I'll talk about functors and limits in the setting of quasicategories. I'll also give examples of quasicategories and talk about homotopy coherent diagrams in simplicially enriched categories and their relation to quasicategories.

3. Complete Segal Spaces, Nerses Aramian (9-Oct)

In this talk we will introduce the notion of complete Segal spaces. This is yet another model for (∞,1)-categories, which means that it has to have a connection with quasicategories. In the talk we would like to discuss the way one can go back and forth between these two notions. Incidentally, this gives an intuitive idea of how one ought to think about complete Segal spaces.

4. n-fold Complete Segal Spaces, Nima Rasekh (16-Oct)

Following the ideas of last week, where we introduced complete Segal spaces as a model for (∞,1)-categories, we introduce a model for (infinity,n)-categories generalizing complete Segal spaces to the theory of "n-fold" complete Segal spaces. The main motivation as well as example for this generalization will be the (∞,n)-category Bordn.

5. Monoidal ∞-Categories, Sarah Yeakel (25-Oct)

In this talk we will discuss monoidal ∞-categories following DAG II. This will involve a look back at the definition of ordinary monoidal categories, and how it can be reformulated to admit a generalization to higher categories.

5. ∞-Operads and Symmetric Monoidal ∞-Categories, Nerses Aramian (1-Nov)

We begin with a brief discussion on how the definition of monoidal ∞-category can be thought of as an A-algebra in the simplicial model category of ∞-categories, CatΔ. There are a couple of things that don't work out as naturally as we would want. Firstly, the notion of an A-operad (and operads in general) is rigid. If we to take higher categorical point of view seriously, we need to come up with a notion of an operad that will not be as rigid, i.e. the actions of the symmetric groups and compositions are defined up to homotopical coherence. Secondly, the action of the symmetric group is not readily present. The second problem really stems from the fact that the category Δ does not have any automorphisms. Therefore, we replace Δ with Segal's category Fin* of pointed finite sets. Then we mimic the discussion in DAG II, where this time we try to generalize the notion of a colored operad. As a result of the discussion we obtain the notion of an ∞-operad. Examples include the commutative operad Comm≃E, the little cubes operads En, so in particular, we have the associative operad E1≃Ass. The algebras over the last operad should recover the notion of a monoidal objects in an ∞-category. Particularly, if we look at AlgAss(Cat(∞,n)), we should get the monoidal (∞,n)-categories. Most of the discussion follows HA.

6. Topological Quantum Field Theories and Their Applications

7. Cobordism Hypothesis

K-Seminar (Summer 2013)

This was a summer extension of the Graduate Student Homotopy Seminar, and it focused on motivational and constructional aspects of algebraic K-theory. We discussed different constructions of algebraic K-theory and how they are equivalent to each other. We also learned a thing or two on this theory can be applied to other areas of mathematics.

1. Lower K-Theories, Sarah Yeakel (21-May)

We will go through the definitions of K0, K1 and K2, and compute the lower K groups for some standard examples. We'll also get to Milnor's definition of higher K groups.

2. Quillen +, Q Constructions and the Definition of Higher K-theory, Nerses Aramian (28-May)

This talk is mainly concerned with the construction of algebraic K-theory for rings. Without proper motivation, this may seem a little dry, so I'll spend some time discussing on what we want from algebraic K-theory. Hopefully, from the motivation we will see that we may need a space (in fact an infinite loopspace, or, even better, E-ring for commutative rings) for K-theory. The talk will concentrate on two construction of algebraic K-theory. The first one is the Quillen +-construction. This approach is not very technical, but nevertheless, gives the right definition of higher K-theory. The problem that it suffers from is that it is only defined up to homotopy, and hence, is not rigid enough for constructions. We will show that the lower K groups agree. The second construction is the Q-construction, again by Quillen, which is more abstract and clean. Not only this construction gives a space for exact categories, but generalizations of this construction allow one to deloop this space to obtain a spectrum. At the end, we will show that these two constructions give the same answers. This is know as the ``+=Q'' theorem. The proof uses an auxiliary construction called S-1S-construction. I decided to skip this proof, since it did not strike me as enlightening. As a final comment, let me just say that the proper place for the delooping of K-theory is the Waldhausen S construction.

3. Waldhausen S Construction, Peter Nelson (5-Jun)

I'll talk about Waldhausen's construction of algebraic K-theory, and about why we would need yet another construction.

4. Motivating Algebraic K-Theory through Algebraic Geometry, Nathan Fieldsteel (13-Jun)

In hoping to answer the question "why do algebraic geometers care about K-Theory?", we will discuss an approach to intersection theory by using K-theory to define the intersection product on Chow rings in a way which agrees with Serre's "Tor formula". Time permitting, we may branch into other applications that bring the higher K groups into play.

[Break: 3 Weeks]

5. Towards Motivic Cohomology, Nerses Aramian (10-Jul)

As I was trying to learn about motivic homotopy theory I was crushed by the difficulty of the subject. However, it wasn't all for nothing. There is a pretty inspiring story of how this subject got kicked off. It has its roots in number theory, specifically Weil conjectures. Furthermore, the story demonstrates on concrete basis how the techniques of complex geometry can give useful insight into arithmetic. Motivic cohomology is the thing that's lurking behind this story, and in fact, is the thing responsible for all the observed phenomena. At the end, so that the talk does not sound too romantic, I'll give a definition of motivic cohomology, and mention some connections with K-theory.

6. THH and TC, Sarah Yeakel (17-Jul)

Hopefully you've gotten the impression that K-theory is hard to compute. Much of the modern work in K-theory is trying to understand some easier theories and the maps connecting them. Topological Hochschild and cyclic homologies are examples of more computable theories. In this talk, we'll define these theories and some of the maps connecting them. I'll also try to explain why these theories are "close" to K-theory, or maybe compute something if there's time.

Sarah compiled a pretty good list of references, so we include it here.



Graduate Student Homotopy Seminar (Spring 2013)

This was seminar primarily focusing on covering introductory material on chromatic homotopy theory. The purpose of the seminar was the preparation for the Spring 2013 Talbot workshop. The following is the list of talks.

1. Symmetric Spectra, Sarah Yeakel (30-Jan)

We will discuss the stable homotopy category, Brown representability, ring spectra, and the smash product of symmetric spectra, paying extra attention to complex cobordism theory MU.

2. Bousfield Localization, Nerses Aramian (6-Feb)

This talk is concerned with the construction of Bousfield localization functors. In the first part, I'll make an attempt to motivate and introduce the notion of categorical localization. The formulation of Bousfield localization does not require any advanced categorical notions. The existence of these localizations is a subtle issue. In the second part, I'll show the existence of localization functors on the homotopy category of spectra, since these are the primary objects that we are interested in. One can generalize this proof to model categories if certain appropriate hypotheses are made.

3.1. Complex Cobordism, Formal Groups, Brown-Peterson Spectrum, Peter Nelson (13-Feb)

I'll talk about formal group laws. Many examples and constructions that are relevant to topology will be given, and I'll try to say something about BP. Time permitting, I'll also talk about a coordinate-free perspective.

3.2. Stacks and Formal Groups, Nerses Aramian (20-Feb)

I will start off by defining what we mean by a stack over a general Grothendieck site. After a couple of basic examples we will focus on the main example: the moduli stack of formal groups over the Zariski site for affine schemes. We will explore some of its properties. Ultimately, we will prove Landweber exact functor theorem. If we have time, I'll also talk about stacky perspective on the Adams spectral sequence.

3.3. The Moduli Stack of Formal Groups, Landweber Exact Functor Theorem (27-Feb)

I am going to continue last week's lecture on stacks. We will start off with the moduli stack of formal groups and explore some its basic properties. We will phrase the Landweber exact functor theorem in the language of stacks: it is essentially saying that flat modules over the moduli stack of formal groups produce cohomology theories. Then we will state Landweber's criterion, which gives one way to understand whether a module over the Lazard ring is flat over the moduli stack of formal groups or not.

4. Lubin-Tate Deformation Theory, Morava E-Theory, Zhen Huan (6-Mar)

The theory of Lubin and Tate gives a functor from a category of finite height formal group laws to the category of complete local rings. By Hopkins-Miller theorem, the entire theory can actually be lifted to commutative S-algebra in an essentially unique way. I will mainly talk about the construction of Morava E-theory and may also talk about some properties of it and its Bousfield class.

5. Adams and Adams-Novikov Spectral Sequences, Martin Frankland (13-Mar)

6. Formal Geometry and Chromatic Group Cohomology, Peter Nelson (3-Apr)

I'll briefly run through some formal algebraic geometry (the stuff I didn't have time for in my last talk) and then talk about how this relates to the E-cohomology of finite groups. I'll probably even say something about Hopkins-Kuhn-Ravenel character theory.

7. S-Modules, Nerses Aramian (10-Apr)

In this talk, I'll work towards constructing the spectrum K(n) following EKMM. This will require a review of EKMM itself. As a motivation, I will give a quick historical background of how this model for stable homotopy (known as S-modules) came to be, since that would make some of the construction more lucid. This model has the virtue of having a symmetric monoidal smash product -- a very convenient tool for performing algebraic manipulations on spectra. In fact, the machinery allows one to construct not only K(n), but the Brown-Peterson spectrum BP with its truncated versions BPs, the Johnson-Wilson spectra E(n), the connective Morava K-theory k(n).

8. Construction of Morava K-theories, Nerses Aramian (17-Apr)

This is the continuation of the previous week's talk. In this part I'll show how to construct homotopy rings starting with an E-ring spectrum, where by homotopy ring I mean a ring spectrum where all the ring properties hold only up to homotopy, i.e. a ring in the homotopy category. In particular, if we apply the theory to p-local MU, we can recover all the desired theories.