UIUC Dept. of Mathematics Seminar Calendar

      July 2009             August 2009           September 2009   
 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                      1          1  2  3  4  5
  5  6  7  8  9 10 11    2  3  4  5  6  7  8    6  7  8  9 10 11 12
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                        30 31

Friday, September 4, 2009

Algebra, Geometry and Combinatorics Seminar
3:00 pm in 445 Altgeld Hall, Friday, September 4, 2009

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Zach Teitler (Texas A&M University)
Ranks of polynomials
Abstract: The Waring rank of a polynomial of degree d is the least number of terms in an expression for the polynomial as a sum of dth powers. The problem of finding the rank of a given polynomial and studying rank in general has been a central problem of classical algebraic geometry, related to secant varieties; in addition, there are applications to signal processing and computational complexity. Other than a well- known lower bound for rank in terms of catalecticant matrices, there has been relatively little progress on the problem of determining or bounding rank for a given polynomial (although related questions have proved very fruitful). I will describe new upper and lower bounds, with especially nice results for some examples including monomials and cubic polynomials. This is joint work with J.M. Landsberg.

Friday, September 11, 2009

Friday, September 18, 2009

Algebra, Geometry and Combinatorics Seminar
3:00 pm in 445 Altgeld Hall, Friday, September 18, 2009

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Luis Serrano (University of Michigan, Ann Arbor)
The shifted plactic monoid
Abstract: We introduce a shifted analog of the plactic monoid of Lascoux and Schtzenberger, the shifted plactic monoid. It can be defined in two different ways: via the shifted Knuth relations, or using Haimans mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood-Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur P-function; a shifted counterpart of the Lascoux-Schtzenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.

Saturday, September 26, 2009

Friday, October 2, 2009

Algebra, Geometry and Combinatorics Seminar
3:00 pm in 445 Altgeld Hall, Friday, October 2, 2009

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Eugene Mukhin (Indiana University-Purdue University Indianapolis)
Algebraic Bethe Ansatz
Abstract: The method of algebraic Bethe Ansatz can be used to connect the Representation Theory (over complex numbers) to many areas of mathematics, including Integrable Systems (the KDV and KP hierarchies, the Calogero-Moser system), Special Functions (orthogonal and multiple orthogonal polynomials, hypergeometric solutions of the KZ equations), Algebraic Geometry (Schubert Calculus, the B. and M. Shapiro conjecture), Combinatorics (Kostka polynomials, Capelli identities, crystalls). These relations produce a number of difficult and important theorems. I will survey some recent results in this area. (Based on a joint project with V. Tarasov and A. Varchenko.)

Friday, October 9, 2009

Algebra, Geometry and Combinatorics Seminar
3:00 pm in 445 Altgeld Hall, Friday, October 9, 2009

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Patricia Hersh (North Carolina State University)
Subword complexes, 0-Hecke algebras and a map to Bruhat order
Abstract: Anders Bjorner characterized which finite, graded partially ordered sets are closure posets of regular CW complexes, and he also observed that a regular CW complex is homeomorphic to the order complex of its closure poset. One might therefore hope to use combinatorics to determine topological structure for stratified spaces such as real Schubert varieties; however, it is possible for two different CW complexes with very different topological structure to have the same closure poset. I will discuss a stratified space from total positivity theory which Sergey Fomin and Michael Shapiro conjectured to be a regular CW complex homeomorphic to a ball. I proved this conjecture using a new regularity criterion together with a map to this space from a simplex. This talk will also highlight more recent joint work with Drew Armstrong using an induced poset map from a Boolean algebra to Bruhat order to give a new proof that the proper part of Bruhat order is homotopy equivalent to a sphere.

Friday, October 16, 2009

Algebra, Geometry and Combinatorics Seminar
3:00 pm in 445 Altgeld Hall, Friday, October 16, 2009

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Aaron Lauve (Texas A&M University)
Skew Littlewood Richardson rules from Hopf algebras
Abstract: We use the rational action of a Hopf algebra on its dual to study products of skew Schur functions in the ring of symmetric functions. The result is a version of the Littlewood-Richardson rule for skew Schur functions that simplifies, and proves, a conjecture of Assaf and McNamara (recent preprint). We also establish similar skew Littlewood-Richardson formulas for Schur P- and Q-functions, and the noncommutative ribbon Schur functions of Gelfand, Krob, et al. (1991). (This is joint work with Thomas Lam and Frank Sottile.)

Friday, October 23, 2009

Algebra, Geometry and Combinatorics Seminar
3:00 pm in 445 Altgeld Hall, Friday, October 23, 2009

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Mahir Bilen Can (Tulane University)
Cross section lattices
Abstract: The cross section lattice of an algebraic monoid (Zariski closure of an algebraic group) is defined and investigated by Mohan Putcha in 80's. It is a finite lattice of idempotents parametrizing the two sided orbits of the group of invertible elements acting on the monoid.
In this talk after reviewing some of the background we are going to look into the combinatorial properties of the cross section lattices. In particular we are going to determine which cross section lattices are supersolvable, and we are going to study their quasi-symmetric functions.
In a recent paper Lex Renner shows that if the algebraic monoid comes from an irreducible representation of a semisimple group then the minimal element of the cross section lattice provides information about the singularity of the associated affine toric variety (Zariski closure of a maximal torus in the monoid). If time permits, we are going to talk about our conjecture on the shape of a cross section lattice when the toric variety is rationally smooth. (No prior knowledge of algebraic monoids is required.)

Wednesday, November 4, 2009

Friday, November 6, 2009

Friday, November 13, 2009

Wednesday, November 18, 2009

Algebra, Geometry and Combinatorics Seminar
12:00 pm in 445 Altgeld Hall, Wednesday, November 18, 2009

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Jonah Blasiak (University of Chicago)
Cyclage, catabolism, and the affine Hecke algebra
Abstract: It is classically known that the ring of coinvariants C[y_1,...,y_n]/(e_1,...,e_n), thought of as an S_n-module with S_n acting by permuting the variables, is a graded version of the regular representation of S_n. However, how a decomposition of the coinvariants into irreducibles is compatible with its ring structure remains a mystery. In particular, there are difficult combinatorial conjectures for the graded characters of certain subquotients of this ring. We describe a promising approach to understanding such subquotients using the canonical basis of the extended affine Hecke algebra. We show that a subalgebra of this Hecke algebra has a cellular subquotient which is a q-analog of the ring of coinvariants and, further, that this subquotient has cellular quotients which are q-analogs of the Garsia-Procesi modules. This cellular picture gives a clear explanation of the appearance of cyclage and catabolism in the combinatorial description of these modules.

Friday, December 4, 2009

Algebra, Geometry and Combinatorics Seminar
3:00 pm in 445 Altgeld Hall, Friday, December 4, 2009

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Anders Skovsted Buch (Rutgers University)
Pieri rules for the K-theory of cominuscule Grassmannians
Abstract: The K-theoretic Schubert structure constants of a homogeneous space G/P are known to have signs that alternate with codimension by a result of Brion. For Grassmannians of type A, these constants are computed by a generalization of the classical Littlewood-Richardson rule that counts set-valued tableaux. The K-theory ring of any Grassmann variety is generated by special Schubert classes that correspond to partitions with a single row. I will present positive combinatorial formulas for the structure constants in products involving special Schubert classes on any cominuscule Grassmannian. For maximal orthogonal Grassmannians this confirms the Pieri case of a conjecture of Thomas and Yong. This is joint work with Vijay Ravikumar.

Friday, February 12, 2010

Algebra, Geometry and Combinatorics Seminar
3:00 pm in 445 Altgeld Hall, Friday, February 12, 2010

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Martha Yip ( University of Wisconsin)
A Littlewood-Richardson rule for Macdonald polynomials
Abstract: Macdonald polynomials are orthogonal polynomials associated to root systems and depend on parameters q and t. The double affine Hecke algebra H is a fundamental tool for studying Macdonald polynomials, which can be constructed by applying intertwining operators on the polynomial representation of H. Using objects known as alcove walks, we give a combinatorial description for the coefficients in the expansion of a product of two Macdonald polynomials. At q=0, the formula specializes to the formula of Schwer for Macdonald spherical functions in terms of positively folded walks, and at q=t, this formula specializes to the formula of Littelmann for Weyl characters in terms of the Littelmann path model.

Friday, February 26, 2010

Friday, March 5, 2010

Friday, April 2, 2010

Friday, April 16, 2010

Friday, April 23, 2010

Saturday, May 1, 2010