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Have a good summer!Past Events
September 18
Keefe San Agustin
Applications of the lower algebraic K-groups
I'll define the first three groups in algebraic K-theory and then present some applications to classical number theory, topology and algebra; algebraic geometry and C*-algebras fit in here, too, but those will be ignored; and at risk of offending just about everyone, Bott Periodicity will never be mentioned. Specifics that will be touched upon, however, include:
Number Theory: Minkowski's theorem on the finiteness of the class number of a number field, Dirichlet's theorem on the structure of the ring of integers of a number field, and the Quillen-Lichtenbaum Conjecture relating the K-theory of the integers to odd zeta values;
Topology: Wall finiteness obstruction, Whitehead torsion, and, if we are brave, the Hatcher-Wagoner and Igusa Stability theorems; Algebra: Mercurjev and Suslin's K_2 result on the Brauer group (via Matsumoto and Wedderburn).
September 25
Picnic!
In lieu of a seminar this week, there will be a Welcome Back cookout this Thursday for math graduate students and faculty. We'll begin cooking by 11:30, food should be available by noon (hopefully sooner for those teaching/in class/office hour). Right now the forecast looks good, if it stays that way the cookout will be held in front of the 1st floor entrance. (If it rains, we'll be outside the second floor entrance.) The theme this year is "The Burger", as we will have beef burgers, turkey burgers, (chicken burgers?), and eggplant burgers. Featuring Andrew's homemade macaroni salad, and additional *sides* and dessert (TBD).
October 2
John Bergdall
The Kähler differentials
Let A be a commutative ring and B an A-algebra. We will generalize the notion of a differential form from differential geometry to introduce the module of differentials for B/A via a universal property. We will continue from there to build our structure: the de Rham complex of B relative to A, a discussion about connections and covariant differentiation and how to define curvature in this very general setting. This all corresponds to topics in differential geometry but that knowledge is not a prerequisite.
October 16
No Seminar. Colloquium!
October 23
CJ Wang
On Dyck Paths
October 30
Tathagata Sengupta
Moduli space of curves
I will try and explain the moduli problem for smooth curves, talk about the difficulties one runs into, and the cures and compromises one has to make to save the situation, and hopefully illustrate a few examples. A basic understanding of classical algebraic geometry should be useful, but I will try to keep it simple and less technical.
November 6
Keith Merrill
Pontryagin Duality
Pontryagin duality is a fundamental theorem about locally compact abelian groups. In particular, it lays the foundation for the generalization of the theory of Fourier series to a more general setting. In the talk, we will define and discuss the theorem and its proof, as well as its more substantial corollaries. As a precursor, I will also discuss related topics like Haar measure and Banach algebras, topics which every student should see. The talk will assume no prerequisites and should be accessible to everyone.
November 13
No Seminar. Colloquium!
November 20
Dipramit Majumdar
The Support Problem
Quote: Everything should be made as simple as possible, but not simpler. In 1988 Paul Erdos proposed the the following problem: Let x and y be positive integers with the property that for all positive integers n, the set of prime dividing (x^n) - 1 is equal to the set of primes dividing (y^n)-1. Is it true x=y?
In 1995 Rogriganez and Schoof proved it in affirmative and generalized it to number fields and elliptic curves. I will present their proof for the number fields and tell major problems in case of the elliptic curves and how those were resolved. I will try to keep the material understandable to everyone, but basic knowledge of algebraic number theory will be helpful.
December 4
No Seminar. Colloquium!
February 12
Jonah
Alternating Sign Matrices
March 5
No Seminar. Colloquium!
March 12
Matt Graham
Topos Theory
Last year I gave two talks that tried to answer a few questions: What kind of category looks most like Set? Why would one want to construct a category theory that acted like Set? It turns out that a topos is the category that one gets by building a category with all of the universal properties that are lurking in the background of set theory. In these talks I tried to use what you already knew about mathematics to introduce some of the basic constructions of category theory. I will let you be the judge of how well I illuminated category theory, by opening up the attached pdf of my talk. I would suggest giving it a once over, paying particular attention to pullbacks, subobjects, and the subobject classifier, as they will be used heavily and only introduced briefly in the talk on Thursday. I think there are only a couple of typos mainly transposing an f with a g (or vice versa) on a couple of slides. Link to Last Years Talk
So what the heck am I talking about this week?:
I will construct the logical operations negation, conjunction, disjunction, implication, complements, intesections, and unions categorically. Why? So glad you asked... Well it turns out that applying these categorical objects to a topos allows for a rigorous generalization of logic itself. In particular, one can create a logic that coincides with intuitionism. Intuitionist logic is exactly the same as classical logic except the statement (not a or a) is not a tautology, and hence proofs by contradiction fail. This has some interesting ramifications in theoretical physics, which someday I hope to understand. By allowing for a different logical system it might be possible to make logical sense of the physical observations in quantum mechanics, since you wouldn't be forced to say that a particle is either here or not here.
An aside on Intuitionism: Some great mathematicians have been labeled as intuitionists, Poincare, Brouwer, and Leopold Kronecker to name a few. Kronecker has been quoted as saying, "God made the integers, all the rest is the work of man." He rejected the notions of infinite set and irrational number as mystical not mathematical. He maintained that the logical correctness of a theory does not imply the existence of the entities that it purported to describe. They are devoid of any significance unless they can actually be produced. Not all intuitionists were/are as extreme as Kronecker.
March 19
Keefe San Agustin
Operads, A-infinity Algebras, Trees and Infinite Loop Spaces
I'll give an anti-historical talk on the connections between the items of the title. This is just the preface to a very modern book, the end chapters of which have yet to be written. The stable mapping class group, Artin braids, some homological algebra, and the ubiquitous pictures of "little disks" and their homology will all make an appearance, but really this talk could alternatively be titled simply ``The Associahedron" and we'll stick to the underlying simplicity that the definition of an operad brings to the picture. The primary thrust will be in presenting simple examples that act as advertisements for the further study of the viewpoint presented here. But for the number theorists in the crowd, I'll try to briefly explain at the very end the relevance of this set-up to the study, due to Grothendieck and, later, Kontsevich, of the absolute Galois group over the rationals via its "homotopical action" on a certain moduli space whose definition naturally involves an operad over an A-infinity algebra.
March 26
Stephen Hermes
Derived and Triangulated Categories and Their Connexions to Auslander-Reiten Theory
The basic technique of homological algebra is to approximate a module by a suitable resolution (ex., free, or injective, or flat, or...) and extrapolate information about the module via the resolution. The derived category D(mod-R) gives us a way to identify a module and all of its resolutions together as a single object, at the cost of some of the original category's structure. However, we will see that there remains some useful structure; namely, the derived category is a triangulated category. This will turn out to be enough to use the techniques of Auslander-Reiten Theory to study the derived category D(mod-A) when A is a path algebra.
April 2
No Seminar. Colloquium!