UIUC Dept. of Mathematics Seminar Calendar

     January 2010          February 2010            March 2010     
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Tuesday, January 26, 2010

Graduate Student Algebraic Geometry Working Seminar
1:00 pm in 347 Altgeld Hall, Tuesday, January 26, 2010

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Stephen Maguire [email] (UIUC Math)
Understanding First-Order Deformations (Part 1)
Abstract: We will concentrate on deformation theory and its applications to moduli theory, Mori Program, and birational geometry this semester. We are using Robin Hartshorne's book titled Deformation Theory. I will give the definition of deformation, talk about why we should care about such a thing, and give a proof that H⁰(Y, N_Y/X) parametrizes the deformation of a closed subscheme Y in X.

Mathematics Colloquium - Special Lecture 2009-10
4:00 pm in 245 Altgeld Hall, Tuesday, January 26, 2010

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Sam Payne (Stanford University and Clay Mathematics Institute)
Nonarchimedean algebraic geometry
Abstract: The usual norm on the complex numbers and its associated analytic geometry (holomorphic functions and differential forms) have been fundamental tools for understanding the geometry and topology of complex algebraic varieties since the beginnings of the subject. Nonarchimedean norms, such as the p-adic norm on the rational numbers, also have an associated analytic geometry which has been used in number theory, but is just beginning to be applied in other areas of mathematics, such as algebraic geometry and dynamical systems. This talk will be an introduction to nonarchimedean geometry with an explanation of its combinatorial manifestation in tropical geometry and relations to intersection theory.

Please join us for refreshments at 3:30 p.m. in the Common Room, 321 Altgeld Hall.

Tuesday, February 2, 2010

Graduate Student Algebraic Geometry Working Seminar
1:00 pm in 347 Altgeld Hall, Tuesday, February 2, 2010

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Mee Seong Im [email] (UIUC Math)
Understanding First-Order Deformations (Part 2)
Abstract: This is the second of a series of lectures from Robin Hartshorne's Deformation Theory. I will discuss structures such as a (sub)scheme flat over the dual numbers (this is thought of as an infinitesimal thickening a scheme evenly spread out over the base), some applications of the Tⁱ functors (e.g., given a k-algebra B where k is a field, deformations of B over the dual numbers are classified by T¹(B/k,B) and its obstructions lie in T²(B/k,B)), the infinitesimal lifting property, and deformation of rings (this has to do with finding various extensions of a ring B by a B-module). I will also talk about when trivial deformations and trivial extensions are the only possibility for a scheme. Since this is a seminar for graduate students, I will attempt to provide as many examples as possible during the lecture.

Monday, February 8, 2010

Tuesday, February 9, 2010

Graduate Student Algebraic Geometry Working Seminar
1:00 pm in 347 Altgeld Hall, Tuesday, February 9, 2010

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Steve Maguire [email] (UIUC Math, 1-3PM)
Higher Order Deformations and Obstruction Theory (Part 3)
Abstract: I will talk about higher order deformations and obstruction theory applied to subschemes, invertible sheaves, vector bundles and coherent sheaves. Jimmy Shan and I will prove when a locally free sheaf over a scheme X can be deformed when X is deformed, the necessity and the sufficiency of the vanishing of its obstruction for the existence of global deformation, and which group the automorphism of the deformed sheaf is isomorphic to. The first hour will be devoted to lecture while the second hour (2-3PM) will be devoted to working on problems from Hartshorne's book.

Algebraic Geometry Seminar
3:00 pm in 243 Altgeld Hall, Tuesday, February 9, 2010

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Kevin Tucker (University of Michigan)
On the Number of Log Canonical Centers and Compatibly F-Split Subvarieties
Abstract: In this talk, I will explain joint work with Karl Schwede which gives identical enumerative bounds for two very different mathematical objects. The first, log canonical centers, are certain loci of singularities arising in the study of higher dimensional complex algebraic geometry. They are particularly important for inductive arguments and adjunction theorems in the minimal model program. The second arise in positive characteristic and are those subvarieties which inherit a fixed splitting of Frobenius. These compatibly F-split subvarieties have been very useful in both representation theory and commutative algebra. In the process of describing our result and giving an idea of the proof, I hope to also impart the mathematical philosophy that (mysterious) connections between complex algebraic geometry and positive characteristic commutative algebra can lead to new and interesting results in both of these areas.

Tuesday, February 16, 2010

Tuesday, February 23, 2010

Tuesday, March 30, 2010

Tuesday, April 6, 2010

Tuesday, April 13, 2010