Junior Algebraic Geometry 2003-2004

Teru Thomas 3/31

Jin Park 4/7

Mike Broshi 4/14-4/21 Representability of the Hilbert scheme and some work on the Picard scheme

Chris Bremer 4/28 fourier transforms on algebraic groups

10/08 Haris Skiadas, "It's a harmonic world we live in."
In this first talk I am going to give you a brief introduction/exposition of the basic ideas in Complex Algebraic Geometry, focusing on harmonic forms, the Hodge theorem, and the Lefschetz decomposition, including the hard Lefschetz theorem. This is in some pretty precise sense all the structure that the cohomology of a projective smooth algebraic variety has. I will probably assume some familiarity with the complex numbers, manifolds, vector bundles, and a little bit of sheaves.

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10/15 Haris Skiadas, "Grothendieck's dream, and some of his nightmares."
I will try to describe Grothendieck's construction of the category of pure motives, along with the series of conjectures that tie the whole thing up and maybe some of their consequences. This talk is independent of the previous one, although the first talk will in some sense serve as motivation. Some familiarity with basic intersection theory is preferable but not absolutely necessary.

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10/22 Haris Skiadas, "The (other) man who beat Grothendieck."
I will tie up loose ends from the previous talk and describe Uwe Janssen's theorem which in essence characterizes the relation of numerical equivalence and the corresponding `numerical equivalence motives.' Knowledge of the material in the previous talk is essential. Some familiarity with category theory will probably also be essential.

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10/29 David Ogilvie, "Geometric Class Field Theory"
I will explain the baby model for Geometric Langlands in the unramified abelian case. It is well known that the Jacobian of a complex algebraic curve X can be thought of as an avatar for H_1(x), a.k.a. the abelianized fundamental group of the curve. After reviewing this situation, I will explain how to translate these facts into a more geometric formulation, which compares local systems on X with "automorphic sheaves" on the Jacobian. Finally, I will explain what happens when one considers a curve X over a finite field, rather than the complex numbers. I will try to keep scary language to a minimum.

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11/5 Ben Wieland, "Mod p methods in characteristic 0."
I will present the Deligne-Illusie proof of the degeneration of the Hodge-de Rham spectral sequence. The original proof is due to Hodge using elliptic operators and for decades it was the only method. It gives rise to Hodge structures on the cohomology. The result is not true in characteristic p, but a limited version is true, is easy to prove, using Frobenius, and suffices to prove the characteristic 0 result.

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11/12 Chris Bremer, "An Introduction to Local Systems with Irregular Singularities."
I will review the theory of local systems on complex quasi-projective varieties, focusing on the correspondence between local systems and connections with regular singularities. I will then try and motivate the study of irregular singularities as a failure of the Grothendieck-Deligne comparison theorem.

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11/19 Chris Bremer, "Irregular Singularities and the Failure of the Grothendieck-Deligne Theorem"
In this talk I will discuss the theory of algebraic connections on quasi-projective varieties. There is a natural equivalence of categories between analytic connections and local systems on a variety; however, the analogy does not extend to algebraic connections. Somehow, algebraic connections have a bit of extra data called "irregularity." I will discuss what the irregularity of a connection is, and hopefully give some examples of irregular connections.

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12/3 Jinhyun Park, "A hopefully very accessible introduction to higher algebraic K-theory."
Abstract: To celebrate the "declaration of the year of motives" at the Geometric Langlands Seminars, and to redeem and console lost souls who attended and were terrified, we prepared this talk. Briefly, I'd like to talk about "a" history of the development of K-theories, and give some definitions. After that, following Quillen's K-theory, I'd like to state some important and essential results about them. No proof will be provided, unless it is very straightforward or obvious. If time permits, I'll probably talk about a relationship between K-theories and Chow groups, a.k.a. Bloch's formula using BGQ-resolution, or some more stuffs of my interest, if we still have time after that. Prerequisites: basic knowledge on some terminologies in algebraic geometry and homological algebra. should be accessible to lots of first year students as well. Notes for this talk are available here.

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1/21 Chris Bremer, "An Introduction to Hypergeometric Functions".
I will be speaking on the definition and applications of hypergeometric functions after Gelfand and Kapranov. For those of you who have a distaste for transcendental functions, fear not: Gelfand's theory of hypergeometric functions comes from a representation theoretic point of view. Furthermore, hypergeometric functions may also be used to study the homology groups of local systems on a family of hypersurface complements. I hope to give a survey of how hypergeometric functions allow us to understand each of these things.

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1/28-1/4 Haris Skiadas, "27 lines in a cubic surface"
This will be a series of two talks aimed at showing that any cubic surface can be obtained by blowing-up P^2 at 6 points in general position, and because of this construction it contains exactly 27 lines. In the first talk we will discuss some general stuff about surfaces, and the structure of blow-ups, and then show that by constructing an appropriate linear system on P^2 blown-up at six points we get an embedding of it into P^3 as a cubic. Then we will see how the 27 lines arise naturally from the geometry of blow-ups, and show that there are no more lines. Next week we'll talk about the Castelnuovo-Enriques criterion on when a divisor can be blown-down, and use that to show that any cubic arises in this way. I will try to keep the prerequisites to a bare minimum, and I believe that any grad student will be able to follow most of the talks.

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2/11 Dave Balduzzi, "Coherent Sheaves on P^n."
Following Beilinson, I'll construct a resolution of the diagonal of P^n and use it to find generators for K(P) and D(P) and an equivalence of categories between D(P) and D(modules over an assoc. alg.) Time permitting I'll talk about generalizations. Prerequisites: familiarity with sheaves and complexes.

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2/18 Roman Fedorov, "Algebraically Integrable Systems."
I shall try to explain some connection between integrable systems and algebraic geometry. I start with the definition of Algebraically Integrable Systems. Then I give a famous example of the Hitchin System (this will occupy most of the talk). It is very natural from the point of view of algebraic geometry. Many classically known systems "embed," in some sense, into Hitchin's one. Time permits, I shall give some examples. Finally (if we have time), I shall tell something about the infinite dimensional case.
Prerequisites: basic course on Riemann surfaces, some familiarity with sheaves and cohomology.