Kottwitz Seminar, Winter 2011

Organizer: Mitya Boyarchenko (email me if you want to get on the seminar mailing list, or if you have any questions)
Schedule: Fridays, 3:10pm-4:30pm, in room 4088 of East Hall

Purpose of the seminar

Just as last year, the main goal of my talks will be to discuss a number of important topics in algebra
that are normally not treated in undergraduate (and often not even graduate) courses. The format of
the seminar will be slightly different: we will probably meet every week for an hour and a half. The
topics of the lectures will sometimes be more advanced than last year, although I am always happy
to provide as much background during the lectures as necessary (upon request from the audience).

The rules remain the same as last year.

Some general background

Typically the bulk of each lecture will not require many prerequisites apart from an introductory course
in algebra, some mathematical maturity, and, most importantly, enthusiasm for the subject. I will try to
make as many comments as possible about the relation of the subject of each talk to other areas of
mathematics, but a full understanding of those comments is not required for following the main storyline.

Certain parts of algebra will come up more frequently than others; some notes on these topics can be found here.

Meetings of the seminar

Friday, January 14. An introduction to the orbit method

Abstract: In a nutshell, the orbit method is a large collection of theorems and techniques
in representation theory, which grew out of the principle that the irreducible representations
of a group G are classified by the orbits of the coadjoint action of G on the dual space to its
Lie algebra. The interpretations of the words 'irreducible', 'representation', 'group', 'orbit',
'coadjoint action', 'dual space' and 'Lie algebra' may all change (no kidding!) depending on
the context in which one works. In my talk I will give an overview of some (though far
from all) areas of representation theory in which the orbit method plays a prominent role.

I will start by recalling the theorems obtained by Alexandre Kirillov in the 1950s-1960s
for nilpotent Lie groups. There, the orbit method gives a classification of all irreducible
unitary representations as well as explicit formulas for their characters. I will then move
on to representations of finite nilpotent groups and nilpotent p-adic groups. In each case
all the necessary notions will be recalled and the main results will be carefully formulated,
although no proofs will be given. The future lectures will be devoted to a more detailed
exposition of the orbit method for finite and p-adic nilpotent groups.
Friday, January 21. The orbit method for finite and p-adic nilpotent groups I

Abstract: I will explain the ideas behind the proofs of the main results of the orbit
method in two settings: for finite nilpotent groups of "small" nilpotence class and
for nilpotent groups over the field of p-adic rational numbers.
Friday, January 28. The orbit method for finite and p-adic nilpotent groups II

Abstract: I will finish the explanation of "how the orbit method works" for p-adic
unipotent groups. In particular, I will discuss the notion of polarizations for a Lie
algebra and the way in which this notion is used to give an explicit construction of
an irreducible representation corresponding to a chosen coadjoint orbit.

Notes for the orbit method lectures (courtesy of Hieu Ngo) in PDF
Friday, February 4. No meeting
Friday, February 11. No meeting
Friday, February 18. An introduction to local fields

Abstract: We will begin a series of lectures, which may last until the end of the seminar.
The main goal of these lectures is to understand the statements (if not the proofs) of two
approaches to local class field theory: the one via Tate duality and the Lubin-Tate theory.
Depending on the pace at which we will be able to go, one thing I will try to explain with
detailed proofs is the calculation of the Brauer group of a local field (and why it is a key
ingredient in local class field theory). In general, my approach will be the opposite of
being self-contained, since much of the beauty of class field theory (and number theory
in general) is lost when one sets the goal of presenting proofs that require the minimal
amount of general machinery. On the other hand, all the necessary definitions will be
stated carefully during the lectures.

In the first lecture I will recall Hensel's lemma and its applications to the study of
Galois extensions of local fields. For instance, if k is an algebraically closed field of
characteristic zero and K=k((t)) is the field of formal Laurent series in one variable
over k, we will obtain an explicit description of the absolute Galois group of K (it is
non-canonically isomorphic to the profinite completion of Z). This result can be viewed
as a relatively trivial special case of local class field theory.

For other local fields, such as Q_p or k((t)) when k is a field of characteristic p>0, the
absolute Galois group is much more complicated, although using Hensel's lemma and
related tools we will still be able to say something interesting about it. In particular, we
will discuss the "tame inertia group" and the related notions of tame and wild ramification
for extensions of local fields.

No prior knowledge of number theory (apart from the definition and some elementary
properties of p-adic numbers) will be assumed. On the other hand, I will expect some
familiarity with basic field and Galois theory (including the definition of algebraic closure,
normal and separable extensions, and the Galois groups of infinite Galois extensions).
Friday, February 25. Local Class Field Theory, part I

Abstract: In the first part of the talk we will finish the discussion of general facts about
local fields and their extensions. In particular, I will explain what the maximal tamely
ramified extension of a local field looks like, and what its Galois group is. We will also
talk about the Galois group of an arbitrary finite extension L/K of local fields. In the
case where the residue field of K has characteristic p>0 (which is the interesting case),
we will see that the inertia group (which consists of all elements of the Galois group that
act trivially on the residue field of L) is solvable and has a unique p-Sylow subgroup.
In particular, when the residue field of K is finite, Gal(L/K) is itself solvable.

In the previous lecture I briefly mentioned one of the main results of local class field
theory: a description of the Galois group of the maximal abelian extension of a local
field. In the second part of the talk I will give a more precise formulation of the result.
If time permits, I will also state the corresponding theorem in global class field theory
and explain the compatibility between the two.
Friday, March 18. Introduction to Lubin-Tate theory

Abstract: Let K be a locally compact non-archimedean field, that is, either
F_q((t)) or a finite extension of Q_p. Some of the main problems studied in
local class field theory are as follows.

(1) Compute the Galois group over K of the maximal abelian extension K^{ab}
of K, in terms that only depend on K itself and not on any finite extensions of K.

(2) Explicitly describe a collection of elements in an algebraic closure of K such
that K^{ab} can be obtained from K by adjoining these elements.

For example, the Galois group Gal(K^{ab}/K) can be naturally identified with
the profinite completion of the group K^* of nonzero elements of K (the latter is
equipped with the standard topology), and if K=Q_p, one can obtain K^{ab} by
adjoining all roots of unity to K (this is the "local Kronecker-Weber theorem").

The goal of this talk (and perhaps the next one) will be to give detailed statements
of the main results of Lubin-Tate theory, which generalizes the local Kronecker-Weber
theorem and provides one explicit construction of an isomorphism between
Gal(K^{ab}/K) and the profinite completion of K^*.
Friday, March 25. The main results of Lubin-Tate theory

Abstract: Let K be a locally compact non-archimedean field, that is, either F_q((t)) or
a finite extension of Q_p. We will discuss the notion of a Lubin-Tate formal group over
the ring of integers of K, which allows one to explicitly describe K^{ab} (the maximal
abelian extension of K) as a subfield of a given algebraic closure of K. It also yields a
construction of an isomorphism between Gal(K^{ab}/K) and the profinite completion
of the multiplicative group of K.
Friday, April 1. The main results of Lubin-Tate theory, part II

Abstract: Last time I stated the theorem about the existence and uniqueness of the
local Artin reciprocity isomorphism, proved some basic results about Lubin-Tate series,
and started discussing formal groups and formal modules. This week I will explain the
construction of the Lubin-Tate formal module over the ring of integers of a local field
(associated to a given Lubin-Tate power series over that ring), and the way one can use
this formal module to give a construction of the local Artin map.

Notes for the local class field theory lectures (courtesy of Hieu Ngo) in PDF

Archive of the older seminar webpages:

Winter 2010 meetings of the seminar

Friday, January 8. Structure of finite fields (Mitya Boyarchenko)
Notes in PDF (appearing here by courtesy of Dylan Moreland)

Abstract: A good understanding of finite fields and their extensions is essential in many areas
of pure mathematics (e.g., number theory, algebraic geometry, representation theory) as well
as applied mathematics (e.g., coding theory). Fortunately, the basic results on finite fields are
easy to state and prove.
In the first part of the lecture I will explain the classification of finite fields and of their
extensions, and I will explain what the corresponding Galois groups are. (There will be no need
to know anything about Galois theory before the lecture.) Complete proofs will be presented.
In the second part of the lecture I will talk about various situations in which finite fields arise
and describe some of their applications. I will also explain what the "absolute Galois group" of a
finite field looks like.

Prerequisites: For the first half of the lecture the prerequisites are fairly modest. If all the terms
in the statement "the quotient of a commutative ring by a maximal ideal is a field" are familiar
to you, you will have no trouble following the first part. However, even if you only know what
a field is, you will most likely get something useful out of the lecture.
The second half will consist of several topics, requiring different kinds of background. I'd
rather not list all of them here, and say instead that the main prerequisite is enthusiasm for
learning something new about the connections between various parts of algebra.
Friday, January 29. Representations of finite groups (Mitya Boyarchenko)
Notes in PDF

Abstract: This lecture will be devoted to complex representations of finite groups and their
characters. The goal is to present the basic theory, with emphasis on giving the simplest and
most conceptual proofs of the main results. In particular, we will see that (finite dimensional)
representations of a given finite group are classified up to isomorphism by their characters;
we will explore the connection between representations of a given finite group and modules
over its group algebra; and we will prove the orthogonality relations for the irreducible
characters of a given finite group. If time permits, we will discuss the notion of an induced
representation (which is the most important tool in all of representation theory).

No previous knowledge of any part of representation theory will be assumed.

Prerequisites: I will assume that you are comfortable with basic linear algebra, although no
nontrivial theorems from this subject will be used in my talk. I will also assume that you know
the definition of a group, but I don't think it is important to remember any group theory apart
from this definition. (Of course, if you only know the definition of a group, it may be hard for
you to get interested in studying group representations.) Some familiarity with rings/algebras
and modules over them will be helpful, but not indispensable.
Friday, February 12. The main theorem of Galois theory (Mitya Boyarchenko)
Notes in PDF

Abstract: A finite field extension L/K is said to be Galois if K is equal to the fixed field in L
of the group of automorphisms G of L over K. In that case G is denoted by Gal(L/K) and
called the Galois group of L over K. The main theorem of Galois theory provides a natural
bijection between the set of intermediate fields F in a finite Galois extension L/K and the
set of subgroups of the Galois group Gal(L/K).

In my talk I will give a full proof of this result, which to the best of my knowledge differs
from any of the proofs that can be found in the literature. The novel feature of the approach
I will explain is that Wedderburn's structure theory for simple Artinian rings is used to give
a short and direct proof of the main theorem of Galois theory, while the notions of a normal
field extension and of a separable field extension play no role whatsoever.

(Of course, those notions are indispensable for a deeper understanding of Galois theory and
its applications. However, it is psychologically satisfying to know that the main theorem
itself can be proved directly from the definition of a Galois extension mentioned above,
as long as one is willing to accept some of the key results of Wedderburn's theory.)

Prerequisites: I will assume that you are comfortable with the notions of a ring, a module
over a ring, and of a finite extension of fields. It will not be necessary to remember any
nontrivial theorems having to do with the above notions. In particular, I will fully state the
main result of Wedderburn's theory, and everything else will be proved "from scratch."
Friday, March 12. Introduction to Morita Theory (Mitya Boyarchenko)
Notes in PDF (courtesy of Dylan Moreland)

Abstract: If A is an (associative, unital) ring, the category A-mod of (left) A-modules
is of great interest in ring theory and homological algebra. Morita's theory (a term that
I am using very loosely here) studies the following two (closely related) questions.
First, to what extent can we recover the ring A from the category A-mod? Second,
what properties or invariants of A can be directly extracted from the category A-mod?

I will present some of the most elementary results that are known along these lines,
introducing important and widely used notions of category theory along the way. For
example, I will explain that if Mat_n(A) is the ring of n-by-n matrices with entries in
a given ring A, then the categories A-mod and Mat_n(A)-mod are equivalent. I will
also show that if A and B are rings such that A-mod and B-mod are equivalent, then
A and B have isomorphic centers, using the notion of "Bernstein center" of a category.

I will also explain how the ring A itself can be recovered from the category A-mod
equipped with some additional data. In a later part of the talk I will discuss similar
questions in the context of group representations. In particular, I will describe the
additional data one needs to recover a group G from its category of representations.

Prerequisites: I will assume knowledge of some basic notions and results of category
theory. For instance, you should be comfortable with the content of, and all the terms
appearing in, the following statement: A fully faithful and essentially surjective functor
is an equivalence of categories. Prior exposure to Yoneda's lemma will be helpful, but
not absolutely essential. I will also assume familiarity with basic techniques one uses
to study modules over a ring, such as the notions of a direct sum of modules, of a
finitely generated module, and of a projective module. For the last third or so of the
lecture you will need to know what the tensor product of vector spaces over a field is.
Friday, March 19. Induced Representations (Mitya Boyarchenko)
Notes in PDF

Abstract: The construction of induced representations is perhaps the single most
important tool in all of representation theory. I will present this construction and
explore some of its basic properties and applications, mostly in the setting of finite
groups. In particular, I will prove Mackey's criterion for the irreducibility of an
induced representation; I will show that for a finite nilpotent group every complex
irreducible representation is induced from a 1-dimensional representation of a
subgroup; and I will use induced representations to explicitly construct some
interesting irreducible representations of finite groups such as the general linear
group or the special linear group over a finite field. If time permits, I will mention
some other contexts in which induced representations also play a prominent role
(notably, the representation theory of Lie groups and Lie algebras).

Prerequisites: The main prerequisite is some familiarity with abstract groups and
their representations. Some background in other areas, such as Lie algebras, could
become useful at certain points during the talk. However, it will be irrelevant for
at least 80% of the lecture.
Friday, April 2. Fundamental groups in topology and algebraic geometry (Mitya Boyarchenko)
Notes in PDF (courtesy of Dylan Moreland)

Abstract: Classically the fundamental group of a topological space is defined as a
certain set of equivalence classes of continuous loops based at a given point (the
equivalence relation being homotopy of loops). This approach has the advantage of
being concrete and useful for some explicit calculations of fundamental groups. It
also has at least two disadvantages. First, even in the classical topological setting,
this approach is not the most convenient one (both from the technical viewpoint and
from the conceptual viewpoint). Second, it does not generalize to other contexts in
which fundamental groups play a prominent role (notably, algebraic geometry).

In the first half of the talk I will explain a different approach to the classical notion
of the fundamental group of a topological space. Namely, we will define it as
"something that classifies topological coverings" of the space (the precise meaning
of these words will be explained). The fact that this definition agrees with the more
standard one is a familiar theorem from algebraic topology. However, our definition
will not rely on the notion of a continuous path. In fact, the format in which it will
be given will provide good psychological preparation for understanding the notion
of the (étale) fundamental group in algebraic geometry.

In the second half of the talk I will introduce the (étale) fundamental group of an
algebraic variety and state some basic known facts about its behavior, focusing on
what is different from what one would expect by analogy with the classical case of
topological spaces (the difference becomes especially striking for algebraic varieties
over a field of characteristic p>0).

The talk will provide background that is absolutely indispensable for anyone wishing
to study geometric representation theory and the geometric Langlands program.

Prerequisites: The first part of the talk only requires knowledge of some basic
notions of point-set topology, such as those of a topological space, of a continuous
map, and of a homeomorphism. Familiarity with the standard definition of the
fundamental group will be very helpful for motivation, but is not formally necessary.

The second part will require two kinds of background. One is some basic knowledge
of algebraic geometry (of algebraic varieties over an algebraically closed field). The
other is the knowledge of elementary notions of category theory, including the notion
of a projective limit. For very elementary background on category theory (which
unfortunately does not include a discussion of limits), see:

http://www.math.lsa.umich.edu/seminars/kottwitz/categories.pdf
Friday, April 16. An introduction to D-modules (Mitya Boyarchenko)
Notes in PDF (courtesy of Dylan Moreland)

Abstract: The theory of D-modules is a very powerful and flexible tool that comes
in multiple flavors and has numerous applications. In particular, the algebraic theory
of D-modules plays a prominent role in many areas of representation theory and
algebraic geometry. In spite of the fact that this theory is by now highly developed,
it is still worth learning something about the historical origins of the subject.

In the talk I will discuss one of the origins, which goes back to something that may
seem completely unrelated, namely, a question asked by I.M. Gelfand about the
existence of meromorphic continuation (to the whole complex plane) of a certain
analytic function defined on the right half plane. J. Bernstein realized that this
question can be viewed as a special case of a theorem about D-modules on the
affine space, and thus he became one of the creators of the algebraic theory of
D-modules. I will explain the details of Bernstein's proof, introducing the relevant
language of D-module theory as we go along.

Prerequisites: Contrary to what the title might suggest, the prerequisites are in
fact quite modest. Since I will only be talking about D-modules on an affine space,
no knowledge of algebraic geometry will be required. The letter `D' in the term
"D-module" stands for the algebra of differential operators (in this case, on the
affine space), but we will define it formally during the lecture, so knowing anything
about differential operators will also not be important.

Prior exposure to very elementary complex analysis (namely, the notions of an
analytic function and of a meromorphic function) will be useful for understanding
the motivational part of the lecture, but not essential for everything that follows.

What the talk will need is some understanding of noncommutative algebras and
modules over them; notably, of graded algebras; of how to construct algebras by
generators and relations; and of how to work with short exact sequences.