UIUC Dept. of Mathematics Seminar Calendar

     October 2009          November 2009          December 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    1  2  3  4  5  6  7          1  2  3  4  5
  4  5  6  7  8  9 10    8  9 10 11 12 13 14    6  7  8  9 10 11 12
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 25 26 27 28 29 30 31   29 30                  27 28 29 30 31

Wednesday, January 28, 2009

Algebra, Geometry and Combinatorics Seminar
3:00 pm in 445 Altgeld Hall, Wednesday, January 28, 2009

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Submitted by ayong.

Andrew Berget (University of Minnesota)
Representations Generated by Decomposable Tensors
Abstract: We form the smallest general linear or symmetric group representation containing a decomposable tensor. The vectors forming this tensor give rise to a linear matroid which, to some extent, controls the irreducible decomposition of these representations. The multiplicities of hook shaped irreducibles are determined by the characteristic polynomial of the matroid, but the multiplicities of other shapes remain a mystery,in general. After some examples and discussion of multiplicities, some interesting open problems will be presented. No familiarity with matroids will be assumed.

Wednesday, February 4, 2009

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

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Elizabeth Townsend Beazley (University of Chicago)
Non-emptiness of Affine Deligne-Lusztig Varieties
Abstract: Deligne-Lusztig varieties, which can be thought of as Frobenius-twisted Schubert varieties, were invented for studying the representation theory of finite Chevalley groups. We introduce several affine variants, one lying inside the affine Grassmannian and another in the affine flag variety, and recall the history of progress made in studying these affine versions. In particular, we demonstrate two methods for proving that certain affine Deligne-Lusztig varieties are non-empty as sets. Both methods are combinatorial in nature, the first of which uses the combinatorics of a certain set of Newton polygons, and the second of which involves the combinatorics of lengths of elements in finite Coxeter groups.

Wednesday, February 11, 2009

Algebra, Geometry and Combinatorics Seminar
3:00 pm in 445 Altgeld Hall, Wednesday, February 11, 2009

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Stephen Griffeth (University of Minnesota)
Three problems in the representation theory of rational Cherednik algebras
Abstract: The rational Cherednik algebra is an algebra attached to a complex reflection group W and depending on a set of central parameters indexed by the conjugacy classes of reflections in W. Its representation theory is roughly analogous to that of a simple Lie algebra, but many important questions remain unresolved.

The eponymous problems are to classify the unitary irreducible representations, classify the finite dimensional irreducible representations, and to construct canonical bases for the W-invariant subspace in a given irreducible module.

For the symmetric group, Berest-Etingof-Ginzburg classified the finite dimensional irreducible modules, and Etingof-Stoica made a conjectural classification of the unitary irreducibles. As it turns out, their conjecture follows from work of Suzuki, and a similar technique ought to classify the unitary irreducibles for all the groups in the infinite family G(r,p,n). Though we have many examples of finite dimensional representations for the rational Cherednik algebra of type G(r,p,n), there is not even a conjectural classification available in general.

The last problem, giving canonical bases for the W-invariant subspace, is closely related the theory of non-crossing partitions.

Wednesday, February 18, 2009

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

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Leon Chekov (Steklov Mathematical Institute/Michigan State U.)
Graph description of Teichmuller spaces and algebras of geodesic functions
Abstract: Since pioneering works of Penner, the fat graph (combinatorial) description of moduli spaces of Riemann surfaces with punctures advanced far enough to incorporate cases of Riemann surfaces with holes (V.V.Fock) and recently Riemamm surfaces with orbifold points (Fock+Goncharov, L.Ch.). Probably the most interesting object are algebras of geodesic functions (governed mainly by the Goldman bracket) that appear in this approach. I present, first, a short excursion into the graph description of moduli spaces, introduce general algebras of geodesic functions (classical and quantum) and then specify these algebras to two important cases related to A_n and D_n algebras and to algebras of Stokes parameters (Dubrovin, Ugaglia) and to algebras of the groupoid of upper triangular matrices (Bondal).

Wednesday, February 25, 2009

Algebra, Geometry and Combinatorics Seminar
3:00 pm in 445 Altgeld Hall, Wednesday, February 25, 2009

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Tom Nevins (UIUC Math)
Mirabolic Langlands duality
Abstract: The geometric Langlands program aims at a "spectral decomposition" of certain derived categories, in analogy with the spectral decomposition of function spaces provided by the Fourier transform. I'll explain such a geometrically-defined spectral decomposition of categories for a particular geometry that arises naturally in connection with integrable systems (more precisely, the quantum Calogero-Moser system) and representation theory (of Cherednik algebras). The category in this case comes from the moduli space of vector bundles on a curve equipped with a choice of ``mirabolic'' structure at a point. The spectral decomposition in this setting may be understood as a case of ``tamely ramified geometric Langlands.'' In the talk, I won't assume any prior familiarity with the geometric Langlands program, integrable systems, or Cherednik algebras.

Wednesday, March 11, 2009

Algebra, Geometry and Combinatorics Seminar
4:10 pm in 345 Altgeld Hall, Wednesday, March 11, 2009

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Bruce Reznick (UIUC Math)
The obvious inner product for homogeneous polynomials
Abstract: There is an obvious, and frequently rediscovered, inner product for the vector space of homogeneous polynomials in n variables and fixed degree. The inner product has its roots in 19th century mathematics. Analysts called it the Fisher inner product. Algebraic geometers used it to define apolarity. It arises in many applications ranging from combinatorics and number theory to moment problems and numerical analysis. I will discuss some of the basic properties and present a menu of opportunities for future seminars.

Wednesday, March 18, 2009

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

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Mihai Ciucu (U. Indiana/Georgia Tech)
Monomer correlations on the square lattice
Abstract: In 1963 Fisher and Stephenson conjectured that the correlation function of two oppositely colored monomers in a sea of dimers on the square lattice is rotationally invariant in the scaling limit. More precisely, the conjecture states that if one of the monomers is fixed and the other recedes to infinity along a fixed ray, the correlation function is asymptotically $C d^(-1/2)$, where $d$ is the Euclidean distance between the monomers and $C$ is a constant independent of the slope of the ray. In 1966 Hartwig rigorously determined $C$ when the ray is in a diagonal direction, and this remains the only direction settled in the literature. We generalize Hartwig's result to any finite collection of monomers along a diagonal direction. This can be regarded as a counterpart of a result of Zuber and Itzykson on n-spin correlations in the Ising model. A special case proves that two same-color monomers interact as predicted by physicists Moessner and Sondhi in 2002.

Wednesday, April 1, 2009

Algebra, Geometry and Combinatorics Seminar
3:00 pm in 445 Altgeld Hall, Wednesday, April 1, 2009

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Kyungyong Lee (Purdue U.)
The Hilbert schemes of points
Abstract: The famous n! conjecture can be stated in an elementary language. In fact it asserts that the dimension of the vector space spanned by all derivatives of a certain bivariate analogue of the Vandermonde determinant is equal to n!. Earlier results of Haiman and Garsia had shown that the n! conjecture implied the Macdonald positivity conjecture. Later Haiman proved the n! conjecture, and the proof is closely related to the algebraic and geometric properties of isospectral Hilbert schemes of points on the plane. I'll discuss how some of the results in the plane case can or cannot be generalized to the higher dimensional case.

Wednesday, April 8, 2009

Algebra, Geometry and Combinatorics Seminar
3:00 pm in 445 Altgeld Hall, Wednesday, April 8, 2009

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Iwan Duursma (UIUC Math)
Geometry of secret sharing
Abstract: The Shamir secret sharing scheme uses univariate interpolation to recover an unknown value of a polynomial (the secret) from known values (the shares). The scheme generalizes in various directions. We discuss the scheme in the setting of graphs and coding theory, and in the setting of multivariate polynomials and algebraic functions. Schemes defined using curves are particularly important, both for their asymptotic performance and for their application to secure multi-party computation. We show that properties of such schemes are often much better than expected and we give best known bounds for their parameters (joint work with Seungkook Park and Radoslav Kirov; see also http://agtables.appspot.com).

Wednesday, April 15, 2009

Wednesday, April 22, 2009

Wednesday, April 29, 2009

Algebra, Geometry and Combinatorics Seminar
3:00 pm in 445 Altgeld Hall, Wednesday, April 29, 2009

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Cristian Lenart (SUNY Albany)
On Combinatorial Formulas for Macdonald Polynomials
Abstract: Macdonald polynomials are generalizations of the irreducible characters of semisimple Lie algebras depending on two parameters; they appear in statistical physics, as eigenfunctions of a certain family of commuting differential operators. Haglund, Haiman and Loehr exhibited a combinatorial formula for the type A Macdonald polynomials in terms of a pair of statistics on fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary Lie type in terms of the corresponding affine Weyl group. In this talk, I relate the above developments, by explaining how the Ram-Yip formula compresses to a new formula, which is similar to the Haglund-Haiman-Loehr one but contains considerably fewer terms; in this context, the statistics on Young diagrams mentioned above follow naturally from more general concepts. I also explain how this work extends to types B and C, where no analog of the Haglund-Haiman-Loehr formula exists. The talk is largely self-contained.

Friday, September 4, 2009

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

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Submitted by llpku.

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.

Friday, October 2, 2009

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

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Submitted by llpku.

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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Submitted by llpku.

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.