Department of

Mathematics


Seminar Calendar
for Algebra, Geometry and Combinatorics Seminar events the year of Friday, April 21, 2017.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
      March 2017             April 2017              May 2017      
 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  6
  5  6  7  8  9 10 11    2  3  4  5  6  7  8    7  8  9 10 11 12 13
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                        30                                         

Thursday, February 16, 2017

3:00 pm in 347 Altgeld Hall,Thursday, February 16, 2017

Set-Valued Skylines

Cara Monical (UIUC)

Abstract: Set-valued tableaux play an important role in combinatorial $K$-theory. Separately, semistandard skyline fillings are a combinatorial model for Demazure atoms and key polynomials. We unify these two concepts by defining a set-valued extension of semistandard skyline fillings and then give analogues of results of J. Haglund, K. Luoto, S. Mason, and S. van Willigenberg.

Thursday, February 23, 2017

3:00 pm in 347 Altgeld Hall,Thursday, February 23, 2017

A non-partitionable Cohen-Macaulay complex

Bennet Goeckner (The University of Kansas)

Abstract: Stanley conjectured in 1979 that all Cohen-Macaulay complexes were partitionable. We will construct an explicit counterexample to this conjecture, which also disproves a related conjecture about the Stanley depth of monomial ideals. This talk is based on joint work with Art Duval, Caroline Klivans, and Jeremy Martin. No prerequisite knowledge of simplicial complexes or commutative algebra will be assumed.

Friday, March 3, 2017

1:00 pm in 347 Altgeld Hall,Friday, March 3, 2017

Embedded resolution of singularities in dimension two

Bernd Schober (University of Toronto)

Abstract: When studying a singular variety one aims to find a variety that shares many properties with the original one, but that is easier to handle. One way to obtain this is via resolution of singularities. In contrast to the quite well understood situation over fields of characteristic zero, only little is known in positive or mixed characteristic and resolution of singularities remains still an important open problem. One of the key ideas over fields of characteristic zero is the notion of maximal contact. After briefly explaining its power, I will point out problems that arise in positive characteristic. Then I will focus on the known two-dimensional case and will discuss the resolution algorithm constructed by Cossart, Jannsen and Saito. Finally, I will explain how polyhedra can be used to detect the improvement of the singularity along the process. This is joint work with Vincent Cossart.

Thursday, March 9, 2017

3:00 pm in 347 Altgeld Hall,Thursday, March 9, 2017

A Combinatorial approach to Supersymmetry

Yan Zhang (San Jose State University)

Abstract: Adinkras are combinatorial tools created to study representations in supersymmetry. Besides having inherent interest for physicists, adinkras offer many easy-to-state and accessible open problems for mathematicians from many diverse subfields including Clifford algebras, posets, coding theory, and algebraic topology. I will discuss some results and problems, but mostly focusing on sharing some very pretty combinatorial objects with you.

Thursday, March 16, 2017

3:00 pm in 347 Altgeld Hall,Thursday, March 16, 2017

Levi subgroup actions on Schubert varieties in the Grassmannian

Reuven Hodges (Northeastern University)

Abstract: Let L be the Levi part of the stabilizer in GL_N(C) (for left multiplication) of a Schubert variety X(w) in the Grassmannian. For the induced action of L on C[X(w)], the homogeneous coordinate ring of X(w) (for the Plucker embedding), I will give a combinatorial description of the decomposition of C[X(w)] into irreducible L-modules. Using this combinatorial description, I give a classification of all Schubert varieties X(w) in the Grassmannian for which C[X(w)] has a decomposition into irreducible L-modules that is multiplicity free. This classification is then used to show that certain classes of Schubert varieties are spherical L-varieties. Also, I will describe interesting related results on the singular locus of X(w) and multiplicities at points in X(w).

Thursday, March 30, 2017

3:00 pm in 347 Altgeld Hall,Thursday, March 30, 2017

The m=1 amplituhedron and cyclic hyperplane arrangements

Steven Karp (UIUC)

Abstract: The m=1 amplituhedron and cyclic hyperplane arrangements The totally nonnegative part of the Grassmannian Gr(k,n) is the set of k-dimensional subspaces of R^n whose Plücker coordinates are all nonnegative. The amplituhedron is the image in Gr(k,k+m) of the totally nonnegative part of Gr(k,n), through a (k+m) x n matrix with positive maximal minors. It was introduced in 2013 by Arkani-Hamed and Trnka in their study of scattering amplitudes in N=4 supersymmetric Yang-Mills theory. Taking an orthogonal point of view, we give a description of the amplituhedron in terms of sign variation. We then use this perspective to study the case m=1, giving a cell decomposition of the m=1 amplituhedron and showing that we can identify it with the complex of bounded faces of a cyclic hyperplane arrangement. It follows that the m=1 amplituhedron is homeomorphic to a ball. This is joint work with Lauren Williams.

Thursday, April 13, 2017

3:00 pm in 347 Altgeld Hall,Thursday, April 13, 2017

A crystal structure on shifted tableaux, with applications to type B Schubert curves

Maria Gillespie (UC Davis)

Abstract: We present a new crystal-like structure on shifted (marked) semistandard skew tableaux. The raising and lowering operators commute with jeu de taquin slides, and detect the type B Littlewood-Richardson condition as the highest weight entries. Certain substructures satisfy the Kashiwara crystal axioms for the root system of $\mathrm{GL}_n$. If time permits, we will discuss applications of our new operators to Schubert curves in the orthogonal Grassmannian. This is joint work with Jake Levinson and Kevin Purbhoo.

Thursday, April 20, 2017

3:00 pm in 347 Altgeld Hall,Thursday, April 20, 2017

A Murnaghan-Nakayama rule for quantum cohomology of the flag manifold

Carolina Benedetti (Fields Institute and York University)

Abstract: Given k less than n and a hook \lambda inside a k,(n-k) box, Mészáros et. al. made use of right operators to provide a rule for the expansion of the quantum Schur polynomial s_{\lambda} in term of generators in the quantum Fomin-Kirillov algebra. In this talk, we will make use of Mészáros et. al. result to provide a different combinatorial interpretation of such expansion, using left operators. As a consequence, we will derive a combinatorial rule for the expansion of quantum power-sum polynomials. This is current work with N. Bergeron, L. Colmenarejo, F. Saliola, F. Sottile.

Friday, April 28, 2017

3:00 pm in 343 Altgeld Hall,Friday, April 28, 2017

On decomposition of the product of Demazure atom and Demazure characters

Anna Pun (Drexel University)

Abstract: t is an open problem to prove the Schubert positivity property combinatorially. Recently Haglund, Mason, Remmel, van Willigenburg et al. have studied the skyline fillings (a tableau-combinatorial object giving a combinatorial description to nonsymmetric MacDonald polynomials , proved by Haglund, Haiman and Loehr) specifically for the case of Demazure atoms (atoms) and key polynomials (keys). This suggests a new approach to a combinatorial proof of Schubert positivity property. In this talk, I will introduce Demazure atoms and key polynomials using skyline fillings called semi-standard augmented fillings (SSAFs) and define generalized Demazure atoms by some modifications on SSAF defining atoms and keys. I will illustrate the insertion algorithm on Demazure atoms proved by Mason and describe refinements of Littlewood-Richardson rule proved by Haglund, Mason and Willigenburg. Then I will describe an algorithm to prove the atom positivity property of the product of a monomial and a Demazure atom. The last result gives a positive support to the approach of the combinatorial proof of Schubert positivity property. If time allows, I will show some connection with polytopes and discuss some conjectures.

Tuesday, May 30, 2017

11:00 am in 243 Altgeld Hall,Tuesday, May 30, 2017

Equivariant Pieri Rules for Isotropic Grassmannians

Vijay Ravikumar (Chennai Mathematical Institute)

Abstract: We describe a manifestly positive Pieri rule for the torus-equivariant cohomology of Grassmannians of Lie types B, C, and D. To the best of our knowledge, this is the first such formula for sub-maximal Grassmannians. We also give a simple proof of the equivariant Pieri rule for the (type A) complex Grassmannian. Our method involves reducing equivariant Pieri coefficients to restrictions of special Schubert classes at torus fixed points in the equivariant cohomology ring of a different Grassmannian. This is joint work with Changzheng Li.