Department of

Mathematics


Seminar Calendar
for Algebra, Geometry and Combinatorics events the year of Thursday, October 12, 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.
    September 2017          October 2017          November 2017    
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Thursday, January 19, 2017

3:00 pm in 347 Altgeld Hall,Thursday, January 19, 2017

Pizzas and Toric surfaces with Kazhdan-Lusztig atlases

Balazs Elek (Cornell University)

Abstract: A Bruhat atlas, introduced by He, Knutson and Lu, on a stratified variety is a way of modeling the stratification locally on the stratification of Schubert cells by opposite Schubert varieties. They described Bruhat atlases on many interesting varieties, including partial flag varieties and on wonderful compactifications of groups. We will discuss some results toward a classification of varieties with Bruhat atlases, focusing on the 2-dimensional toric case. In this case, the answer may be stated in terms of the moment polygon of the toric surface, which one should first slice up, then put toppings on, much like one would do while preparing a pizza.

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.

Friday, September 1, 2017

3:00 pm in 341 Altgeld Hall,Friday, September 1, 2017

Prism Tableaux and Alternating Sign Matrices

Anna Weigandt (UIUC)

Abstract: A prism tableau is an overlay of semistandard tableaux. In joint work with A. Yong, prism tableaux were used to provide a formula for Schubert polynomials. This expression directly generalizes the tableau rule for Schur polynomials. We study fillings of more general prism shapes. The resulting polynomials are multiplicity free sums of Schubert polynomials. Each prism shape determines an alternating sign matrix. This allows us to give a prism formula for the multidegree of an alternating sign matrix variety.

Wednesday, September 20, 2017

3:00 pm in 343 Altgeld Hall,Wednesday, September 20, 2017

Symmetric group representations and Z

Anshul Adve (UIUC)

Abstract: We discuss implications of the following statement about the representation theory of symmetric groups: every integer appears infinitely often as an irreducible character evaluation, and every nonnegative integer appears infinitely often as a Littlewood-Richardson coefficient and as a Kronecker coefficient. This is joint work with A. Yong.

Friday, September 29, 2017

3:00 pm in 341 Altgeld Hall,Friday, September 29, 2017

Bott-Samelson varieties and combinatorics

Laura Escobar (UIUC)

Abstract: Schubert varieties parametrize families of linear spaces intersecting certain hyperplanes in C^n in a predetermined way. In the 1970’s Hansen and Demazure independently constructed resolutions of singularities for Schubert varieties: the Bott-Samelson varieties. In this talk I will describe their relation with associahedra. I will also discuss joint work with Pechenick-Tenner-Yong linking Magyar’s construction of these varieties as configuration spaces with Elnitsky’s rhombic tilings. Finally, based on joint work with Wyser-Yong, I will give a parallel for the Barbasch-Evens desingularizations of certain families of linear spaces which are constructed using symmetric subgroups of the general linear group.

Tuesday, October 3, 2017

2:00 pm in 243 Altgeld Hall,Tuesday, October 3, 2017

Singularities of semisimple Hessenberg varieties.

Erik Insko (Florida Gulf Coast University)

Abstract: Semisimple Hessenberg varieties are subvarieties of the flag variety with important connections to representation theory, algebraic geometry, and combinatorics. Like Schubert varieties, the structure of semisimple Hessenberg varieties can be studied using the combinatorics of the symmetric group. In this talk, we will define these varieties and give a combinatorial criterion for identifying singular points in certain semisimple Hessenberg varieties. This is based on joint work with Martha Precup. At 3-4pm in Algeld 245 there will be an IGL/ICLUE seminar by Insko

Friday, October 13, 2017

3:00 pm in 341 Altgeld Hall,Friday, October 13, 2017

Depth in classical Coxeter groups

Alexander Woo (University of Idaho)

Abstract: Given a list of n objects, one would like to sort them - meaning rearrange them so that they are in order - as "cheaply" as possible. One naturally wants to know how cheaply a given list can be sorted and what the cheapest method for a given list actually is. The answer depends on the notion of "cheaply" one uses. One notion of "cost", called depth, was introduced by Petersen and Tenner, not just for permutations as in the original problem, but for elements of arbitrary Coxeter groups. They gave a combinatorial formula, with a constructive proof, for the depth of a permutation. We find a similar formula for symmetries of the n-cube. This is joint work with Eli Bagno, Riccardo Biagioli, and Moti Novick. This talk is intended to be ICLUE friendly

Friday, October 27, 2017

3:00 pm in 341 Altgeld Hall,Friday, October 27, 2017

Affine Growth Diagrams

Tair Akhmejanov (Cornell University)

Abstract: We introduce a new type of growth diagram, arising from the geometry of the affine Grassmannian for $GL_m$. These affine growth diagrams are in bijection with the $c_{\vec\lambda}$ many components of the polygon space Poly($\vec\lambda$) for $\vec\lambda$ a sequence of minuscule weights and $c_{\vec\lambda}$ the Littlewood--Richardson coefficient. Unlike Fomin growth diagrams, they are infinite periodic on a staircase shape, and each vertex is labeled by a dominant weight of $GL_m$. Letting $m$ go to infinity, a dominant weight can be viewed as a pair of partitions, and we recover the RSK correspondence and Fomin growth diagrams within affine growth diagrams. The main combinatorial tool used in the proofs is the $n$-hive of Knutson--Tao--Woodward. The local growth rule satisfied by the diagrams previously appeared in van Leeuwen's work on Littelmann paths, so our results can be viewed as a geometric interpretation of this combinatorial rule.

Friday, November 3, 2017

3:00 pm in 341 Altgeld Hall,Friday, November 3, 2017

On the Gorensteinization of Schubert varieties via boundary divisors

Sergio Da Silva (Cornell University)

Abstract: A variety being Gorenstein can be a useful property to have when considering questions in birational geometry. Although Schubert varieties are Cohen-Macaulay, they are not Gorenstein in general. I will describe a convenient way to find a "Gorensteinization" for a Schubert variety by considering only one blow-up along its boundary divisor. We start by reducing to the local question, one involving Kazhdan-Lusztig varieties. These affine varieties can be degenerated to a toric variety defined using the Stanley-Reisner ideal of a subword complex. The blow-up of this variety along its boundary is now Gorenstein. Carefully choosing a degeneration of the blow-up allows us to extend this result to Schubert varieties.

Thursday, November 9, 2017

3:00 pm in 147 Altgeld Hall,Thursday, November 9, 2017

On the three matrix analog of Gerstenhaber's theorem

Jenna Rajchgot (Jenna Rajchgot)

Abstract: In 1960, Gerstenhaber proved that the algebra generated by two commuting dxd matrices has vector space dimension at most d. The analog of this statement for four or more commuting matrices is easily seen to be false. The three matrix version remains open. After providing some history and context, I'll translate this three commuting matrix statement into an equivalent statement about certain maps between modules, and prove that this commutative-algebraic reformulation is true in special cases. I'll end with some combinatorial questions about three dimensional analogs of Young diagrams, which, if answered, would settle the three matrix analog of Gerstenhaber's theorem for other infinite families of examples. This is joint work with Matthew Satriano.

Wednesday, November 15, 2017

3:00 pm in 345 Altgeld Hall,Wednesday, November 15, 2017

Vanishing of Littlewood-Richardson polynomials is in P

Colleen Robichaux (UIUC)

Abstract: J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni independently proved that determining the vanishing of Littlewood-Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood-Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas-A. Yong. Our proof then combines a saturation theorem of D. Anderson-E. Richmond-A. Yong, a reading order independence property, and E. Tardos' algorithm for combinatorial linear programming. This is joint work with A. Adve and A. Yong.

Friday, December 1, 2017

3:00 pm in 343 Altgeld Hall,Friday, December 1, 2017

Equations of Kalman Varieties

Amy Huang (University of Wisconsin)

Abstract: Given a subspace L of a vector space V, the Kalman variety consists of all matrices of V that have a nonzero eigenvector in L. I will discuss how to apply Kempf Vanishing technique with some more explicit constructions to get a long exact sequence involving coordinate ring of Kalman variety, its normalization and some other related varieties in characteristic zero. Time permitting I will also discuss how to extract more information from the long exact sequence including the minimal defining equations for Kalman varieties.

Tuesday, December 19, 2017

2:00 pm in 241 Altgeld Hall,Tuesday, December 19, 2017

Nash blowups of Grassmannian Schubert varieties

Alexander Woo (University of Idaho)

Abstract: The Nash blowup is a natural construction of a partial desingularization of a variety. We identify the normalized Nash blowup of a Schubert variety in the Grassmannian with a Schubert variety in a partial flag manifold and derive some consequences. This is joint work with Ed Richmond and William Slofstra.