UIUC Dept. of Mathematics Seminar Calendar

     January 2010          February 2010            March 2010     
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Friday, February 26, 2010

Special seminar on derived categories and applications
1:00 pm in 147 Altgeld Hall, Friday, February 26, 2010

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Tom Nevins (UIUC Math)
Recovery of algebraic varieties from their derived categories
Abstract: The (coherent or quasicoherent) derived category of an algebraic variety is a powerful but subtle invariant of the variety. In general, one cannot hope to recover the variety from its derived category just as a triangulated or DG category without additional structure (such as a choice of t-structure or monoidal structure). However, I'll explain and sketch the proof of a theorem of Bondal-Orlov that if the variety has ample canonical or anticanonical bundle, then the derived category is sufficient to determine the variety. In spite of the fancy words I'm using in this abstract, I'll try to make the lecture as low-tech as it can be, given that it's a talk about derived categories of algebraic varieties!

Women in Mathematics Seminar
2:00 pm in 141 Altgeld Hall, Friday, February 26, 2010

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Mee Seong Im [email] (UIUC Math)
Basic Intersection Theory for Non-Algebraic Geometers
Abstract: We will first compare the intersection multiplicity of a finite number of planar curves at a point in the Euclidean plane versus the ones in the projective plane. Now what happens when two or more polynomials in two variables have a nonconstant factor in common, or how should we understand self-intersection? How do we then count the intersection multiplicity at a point? Does the notion of intersection multiplicity at a point still make sense or should we extend this notion to the intersection multiplicity at a planar curve? What about the intersection multiplicity of a finite number of polynomials in several variables? What conditions should we impose in order to obtain finite nonnegative integer as the intersection number? Is it possible to obtain negative integers as the intersection number? Background in algebraic geometry is not necessary to understand and investigate these questions.

Algebra, Geometry and Combinatorics Seminar
3:00 pm in 445 Altgeld Hall, Friday, February 26, 2010

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Bruce Reznick (University of Illinois)
The cones of real convex forms
Abstract: For each (n,m), let K_&ob;n,m&cb; denote the set of real forms p(x_1,...,x_n) of degree m which are convex as functions. Then K_&ob;n,m&cb; is a closed convex cone. It is been known to some (not many) that a positive homogeneous polynomial p of degree m is convex if and only if the dehomogenization p(1,x_2,...,x_n)^(1/m) is convex, and under that interpretation, K_&ob;2,4&cb; and K_&ob;2,6&cb; were independently described by V. I. Dmitriev and the speaker. An example of an extremal form in K_&ob;2,6&cb; is x^6 + 45 x^2y^4 + 18 y^6. Under the natural inner product, the dual cone is generated by all products L_1^2*L_2^&ob;m-2&cb;, where the L_i's are real linear forms. If p is a positive definite form, then there exists N so that (\sum x_j^2)^N*p is convex. This talk was motivated by Blekherman's recent probabilistic proof that there exist convex forms which are not a sum of squares of polynomials. Neither he, nor, alas, the speaker, have found any particular examples yet. http://www.math.uiuc.edu/~reznick/banff.html

Sheaves of Dimension 1 and Gromov-Witten Theory
4:00 pm in 243 Altgeld Hall, Friday, February 26, 2010

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Artan Sheshmani [email] (UIUC Math.)
Donaldson Thomas theory of A_n \times P^1, I
Abstract: Let A_n denote the minimal resolution of the quotient C^2/Z_n+1. Consider the threefold X=A_n\times P^1. The Donaldson-Thomas invariants of X are obtained by integration against the virtual fundamental cycle of the moduli scheme of ideal shaves of X of dimension at most equal to 1. Despite the fact that X is noncompact the torus-fixed component of the moduli scheme is compact. By considering the distinct points of P^1 and the fibers of X over these points we obtain torus equivariant divisors inside X, hence we can construct the moduli scheme of ideal sheaves of X which intersect the equivariant divisors transversely. We discuss the strategy to compute the DT partition function of relative ideal sheaves in this setup.