UIUC Dept. of Mathematics Seminar Calendar

       May 2012              June 2012              July 2012      
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
        1  2  3  4  5                   1  2    1  2  3  4  5  6  7
  6  7  8  9 10 11 12    3  4  5  6  7  8  9    8  9 10 11 12 13 14
 13 14 15 16 17 18 19   10 11 12 13 14 15 16   15 16 17 18 19 20 21
 20 21 22 23 24 25 26   17 18 19 20 21 22 23   22 23 24 25 26 27 28
 27 28 29 30 31         24 25 26 27 28 29 30   29 30 31

Tuesday, January 24, 2012

Algebra, Geometry and Combinatoric
2:00 pm in 345 Altgeld Hall, Tuesday, January 24, 2012

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Submitted by darayon2.

Dave Anderson (University of Washington)
Arc spaces and equivariant cohomology
Abstract: When an algebraic group acts on a smooth complex variety X, it also acts on the arc space of X, an infinite-dimensional space parametrizing germs of curves in X. In joint work with Alan Stapledon, we develop a new perspective on the equivariant cohomology of X, by replacing X with its arc space. Under certain hypotheses, these infinite-dimensional varieties allow us to obtain a geometric basis (over the integers!) for equivariant cohomology, as well as geometric representatives for cup products as intersections. I'll explain how this leads to a new invariant of singularities, and illustrate our approach with examples from toric varieties and flag varieties.

Tuesday, January 31, 2012

Algebra, Geometry and Combinatoric
2:00 pm in 345 Altgeld Hall, Tuesday, January 31, 2012

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Submitted by darayon2.

Jenna Rajchgot (Cornell)
Compatibly split subvarieties of the Hilbert scheme of points in the plane
Abstract: Consider the Hilbert scheme of n points in the affine plane and the divisor "at least one point is on a coordinate axis". One can intersect the components of this divisor, decompose the intersection, intersect the new components, and so on to stratify the Hilbert scheme by a collection of reduced (indeed, "compatibly Frobenius split") subvarieties. This may prompt one to ask, "What are these subvarieties?" or, better, "What are all of the compatibly split subvarieties?" I'll begin by providing the answer for some small values of n. Following this, I'll restrict to a specific affine patch (now for arbitrary n) and describe a degeneration of the compatibly split subvarieties to Stanley-Reisner schemes.

Tuesday, April 3, 2012

Algebra, Geometry and Combinatoric
2:00 pm in 345 Altgeld Hall, Tuesday, April 3, 2012

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Submitted by darayon2.

Bridget Tenner (DePaul University)
Repetitions and patterns
Abstract: A permutation $w$ can be written as a product of adjacent transpositions, and such a product of shortest length, $\ell(w)$, is called a reduced decomposition of $w$. The difference between $\ell(w)$ and the number of distinct letters appearing in a (any) reduced decomposition of $w$ is $\text{rep}(w)$; that is, this statistic describes the amount of repetition in a reduced decomposition of $w$. In this talk, we will explore this statistic $\text{rep}(w)$, and find that it is always bounded above by the number of 321- and 3412-patterns in $w$. Additionally, these two quantities are equal if and only if $w$ avoids the ten patterns 4321, 34512, 45123, 35412, 43512, 45132, 45213, 53412, 45312, and 45231.

Tuesday, April 10, 2012

Friday, April 27, 2012

Algebra, Geometry and Combinatoric
4:00 pm in 341 Altgeld Hall, Friday, April 27, 2012

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Submitted by darayon2.

Allen Knutson (Cornell)
Manifold atlases consisting of Bruhat cells
Abstract: A Bruhat cell is a finite-dimensional affine space, but comes with many additional structures: a stratification, a torus action, a Poisson structure, a Frobenius splitting, a totally nonnegative part... When we have a manifold with these structures, we can ask whether it can be given an atlas of charts consisting of Bruhat cells.
I'll give a construction of a Coxeter diagram D (sometimes) associated to a manifold M with a stratification Y. If the Bruhat atlas program can be carried out for M, it gives a poset antiisomorphism between Y and an order ideal in the Bruhat order W_D. We have checked this combinatorics ex post facto for partial flag manifolds and wonderful compactifications of groups. The full program is complete for the Grassmannian, where the diagram is the affine Dynkin diagram, by work of [K-Lam-Speyer] and [Snider]. For most other examples, D is neither finite nor affine.
This work is joint with Xuhua He and Jiang-Hua Lu.

Friday, August 10, 2012

Algebra, Geometry and Combinatoric
2:00 pm in 143 Altgeld Hall, Friday, August 10, 2012

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Submitted by ayong.

Alexander Woo (Univ. of Idaho)
Local complete intersection Schubert varieties
Abstract: I will talk about a characterization of the Schubert varieties which are local complete intersections (lci) using pattern avoidance. One direction of the proof is by constructing an explicit minimal set of equations cutting out neighborhoods of Schubert varieties at the identity. This leads to some interesting combinatorics involving Fulton's essential set and the Schubert varieties defined by inclusions, which have a mysterious link to certain hyperplane arrangements called inversion arrangements. The other direction requires a generalization of pattern avoidance known as mesh pattern avoidance. This is joint work with Henning Ulfarsson (Reykjavik U.).

Tuesday, September 18, 2012

Algebra, Geometry and Combinatorics
2:00 pm in 345 Altgeld Hall, Tuesday, September 18, 2012

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Submitted by bwyser.

Ben Wyser [email] (UIUC Math)
Geometry and Combinatorics of K-Orbits on the Flag Manifold
Abstract: The orbits of a symmetric subgroup on a flag manifold ("K-orbits") are of importance in the representation theory of real Lie groups, and have been studied extensively from this perspective. The closures of such orbits are generalizations of Schubert varieties, and any geometric and/or combinatorial question one has about Schubert varieties can equally well be posed about these more general orbit closures. However, while the geometry and combinatorics of Schubert varieties have been studied exhaustively, even apart from their role in representation theory, K-orbits and their closures have received far less attention from these perspectives. I will discuss the K-orbit analogue of a story which is well understood in the case of Schubert varieties. Namely, I will describe how one can compute representatives for the torus-equivariant cohomology classes of K-orbit closures, and how these formulas can be interpreted as Chern class formulas for certain types of degeneracy loci. This is in parallel with the well-known story, due to Lascoux-Schutzenberger, Fulton, Pragacz, Graham, et. al., of double Schubert polynomials as representatives for the torus-equivariant classes of Schubert varieties, and their interpretation as Chern class formulas for the classes of degeneracy loci associated to flagged vector bundles.

Tuesday, September 25, 2012

Algebra, Geometry and Combinatorics
2:00 pm in 345 Altgeld Hall, Tuesday, September 25, 2012

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Submitted by bwyser.

Michael Joyce [email] (Tulane University)
Type A Symmetric Varieties and Schubert Polynomials
Abstract: Symmetric varieties, such as the variety which parameterizes smooth quadric hypersurfaces in a projective space, admit decompositions into orbits of a Borel subgroup, analogous to the Schubert cell decomposition of Grassmannians and flag varieties. Studying the analogue of the weak order for these varieties leads to interesting poset structures on the sets of involutions, fixed point free involutions, certain signed involutions, and some further variants of these objects. We explicitly describe the maximal chains in these posets and use the calculations to provide interesting factorizations of certain multiplicity-free sums of Schubert polynomials. This is joint work with Mahir Can and Ben Wyser.

Tuesday, October 2, 2012

Tuesday, October 9, 2012

Monday, November 26, 2012

Tuesday, November 27, 2012

Algebra, Geometry and Combinatorics
2:00 pm in 345 Altgeld Hall, Tuesday, November 27, 2012

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Submitted by bwyser.

Jayadev Athreya [email] (UIUC Math)
Counting special trajectories for right-angled billiards and pillowcase covers, II.
Abstract: In joint work with A. Eskin and A. Zorich, we compute volumes of moduli spaces of meromorphic quadratic differentials on CP, via enumerating pillowcase covers. One motivation comes from understanding the (weak) quadratic asymptotics for counting special trajectories for billiards in polygons whose angles are integer multiples of 90 degrees. This talk is a continuation of, but will be independent from, my Ergodic Theory seminar on Monday. In particular, you do not need to have attended the Ergodic Theory talk to understand this talk.

Tuesday, December 11, 2012

Algebra, Geometry and Combinatorics
2:00 pm in 345 Altgeld Hall, Tuesday, December 11, 2012

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Submitted by bwyser.

Daniele Rosso [email] (University of Chicago)
Mirabolic Hecke Algebras
Abstract: The Hecke Algebra of the symmetric group can be defined as the convolution algebra of GL(V) invariant functions on the variety of pairs of complete flags in a vector space V of dimension d over the finite field with q elements. In the ‘mirabolic’ setting, we consider the variety of triples of two complete flags and a vector in V. The convolution algebra of GL(V) –invariant functions on this variety is very interesting and it was first described by Solomon. For generic q it is a semisimple algebra and its irreducible representations can be parametrized by the partitions of all integers from 0 to d. I will describe some analogues of classical algebraic and combinatorial objects that are found in this new setting. For example, we’ll see Jucys-Murphy elements and their action on Gelfand-Zeitlin bases for irreducible representations. This will give an analogue of the Fock space structure for the category of representations of the Hecke algebra, and lead to a proposed definition of an affine version of the algebra of Solomon (or, equivalently, a mirabolic version of the Affine Hecke algebra). This is work in progress, some of it joint with Jonathan Sun. In addition to discussing some current results, I will point out some directions in which the research is going.