Mathematics

The cyclotomic polynomial, topologically

The cyclotomic polynomial is the minimal polynomial over the rational numbers Q satisfied by a primitive n-th root of unity. Its coefficients turn out to be integers, whose interpretation and signs are somewhat mysterious. We will explain how to interpret these coefficients topologically, as computing the homology of certain simplicial complexes. The interpretation builds upon previous work (joint with Jeremy Martin) uncovering a strange connection between Q-bases for a cyclotomic extension of Q, and the higher dimensional spanning trees introduced by Gil Kalai and studied further by Ron Adin. The talk is based on joint work with Gregg Musiker; see arxiv.org/abs/1012.1844.

Homological rigidity of Schubert varieties

Schubert varieties are distinguished by the property that their homology classes form an additive basis of the integral homology of a rational homogeneous variety G/P. The work discussed in this talk is motivated by the following question: in addition to the Schubert variety S, what are the varieties Y representing the Schubert class [S]? More generally, one might ask: what are the varieties representing an integer multiple of [S]?

Remarkably, in the case that G/P is cominuscule (e.g. X is a Grassmannian), the varieties Y representing an integer multiple of [S] are characterized by a system of PDE, known as the Schur system. We say the Schubert class is Schur rigid when the Schubert varieties are the only irreducible integrals (solutions) of the Schur system.

The main result is a complete list of the Schur rigid Schubert classes. One (of two) sine qua non of the work is a new description of the Schubert varieties by an integer and a marking of the Dynkin diagram of G. (This generalizes the well-known partition description in the case that G/P is a Grassmannian.) The characterization also yields a uniform description of the (irreducible components of the) singular locus Sing(S) across all cominuscule G/P.

This work is joint with Dennis The.

Symmetric Chain Decompositions: How to find them?

Many posets of fundamental interest have rank sizes that are symmetric around the middle and are unimodal, but the question of whether they partition into chains that are symmetric around the middle and don't skip ranks remains elusive. We will discuss the search for such symmetric chain decompositions, considering in particular the poset L(m,n) . The elements of L(m,n) are the integer lists (a_1,...,a_m) such that 0<=a_1<=...<=a_m <=n , ordered by a<=b if and only if a_i<=b_i for 1<=i<=m.

K-orbits on G/B, Richardson varieties, and a positive rule for (p,q)-Schubert constants

For G a complex, reductive algebraic group, the fixed point subgroup of an involution of G is typically denoted K, and is referred to as a symmetric subgroup. K acts on the flag variety G/B (by left translations) with finitely many orbits. The geometry of such orbits and their closures is important in the infinite-dimensional representation theory of real forms of G.

One interesting example of a symmetric pair is (G,K) =(GL(p+q), GL(p) x GL(q)). Restricting attention to this example, I will discuss a recent result which establishes that a number of the K-orbit closures in this case coincide with certain Richardson varieties. When combined with a theorem of M. Brion on expressing the class of such an orbit closure in the Schubert basis, this observation implies a positive (in fact, multiplicity-free) rule for certain Schubert structure constants c_{u,v}^w --- those for which u,v form what I refer to as a "(p,q)-pair".

Program

Kevin Purbhoo (Waterloo) 9:30-10:30
Michael Larsen (IU) 11:00-12:00
Claudia Pollini (ND) 2:00-3:00
Wenbo Niu (Purdue) 3:00-4:00

Linearly dependent powers of quadratic forms

Given a positive integer d, let Phi(d) denote the smallest r so that there exist r pairwise non-proportional complex quadratic forms q_i(x,y) = a_i x^2 + 2b_i x y + c_i y^2 with the property that {q_i^d} is linearly dependent. We are interested in computing Phi(d) and characterizing the minimal sets. For example, when d=2, the essentially unique minimal set comes from the Pythagorean parameterization: {x^2 - y^2, x y , x^2 + y^2}. Using a classical map of Felix Klein, the three pairs of linear factors of these quadratics are associated with the three pairs of antipodal vertices of the regular octahedron.

Pivot distributions and Weierstrass nongaps

We show how various properties of a linear code are captured by the collection of its dimension-length profiles. The dimension-length profile of a matrix describes the column positions of the pivots after the matrix is brought into row echelon form. The collection of dimension-length profiles describes the distribution of the pivots considered over all possible column permutations of the matrix. A code is MDS if and only if for every permutation of the columns in the generator matrix the pivot columns are the leading columns. The zeta function of a code that we introduced in previous work will be used to describe the deviation from this distribution when the code is not MDS. We will point out how the zeta function is the exact analogue of the zeta function for curves over a finite field and how for curves it describes the distribution of Weierstrass nongaps.

Affine Mirkovic-Vilonen polytopes.

Kashiwara developed combinatorial objects called crystals to study the representation theory of complex simple Lie groups and Lie algebras. The construction is quite involved, but one can often realize the same combinatorics by more elementary means. One useful realization is based on the Mirkovic-Vilonen polytopes of the title. I will describe what these polytopes are, and why they are interesting. I will then explain current work giving an analogous construction for symmetric affine Kac-Moody algebras. For affine sl(2) the construction is purely combinatorial. For other symmetric affine types the definitions are combinatorial, but we need some geometry (quiver varieties) to prove that everything works. I will explain these ideas mainly via examples and pictures. This is joint work with Pierre Baumann, Thomas Dunlap and Joel Kamnitzer.

Patch Ideals and Hessenberg varieties

Patch ideals encode neighbourhoods of a variety in GLn /B. For the Peterson varieties we determine generators for these ideals and show they are complete intersections, and thus Cohen-Macaulay and Gorenstein. Consequently, we combinatorially describe the singular locus of the Peterson variety; give an explicit equivariant K-theory localization formula; and extend some results of [B. Kostant ’96] and of D. Peterson to intersections of Peterson varieties with Schubert varieties. (This is joint work with Alexander Yong) I will also discuss my further work that explores generalizations to other Hessenberg varieties.

Higher pentagram maps, weighted directed networks, and cluster dynamics

Introduced by R. Schwartz about 20 years ago, the pentagram map acts on plane polygons, considered up to projective equivalence, by drawing the diagonals that connect second-nearest vertices and taking the new polygon formed by their intersections. The pentagram map is a discrete completely integrable system whose continuous limit is the Boussinesq equation, a completely integrable PDE of soliton type. In this talk, I shall describe a new family of discrete completely integrable dynamical systems, including the pentagram map, and explain their connection to the theory of cluster algebras, a new and rapidly growing area with numerous connections to diverse fields of mathematics.

Arc spaces and equivariant cohomology

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.

Compatibly split subvarieties of the Hilbert scheme of points in the plane

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.

Discrete Integrable Systems and Cluster Algebras

Recursive systems arising from integrable quantum spin chains, such as Q,T and Y-systems display remarkable combinatorial properties. These are actually part of a more general mathematical structure called Cluster Algebras, introduced by Fomin and Zelevinsky around 2000, and which has found a host of mathematical applications so far, ranging from the theory of total positivity, Teichm?ller space geometry, to the representation theory of quantum groups. A cluster algebra is a sort of dynamical system describing the mutation of a vector of data along the edges of an infinite tree, with rules guaranteeing that only Laurent polynomials of the initial data are generated. A longstanding conjecture of Fomin and Zelevinsky states that these have non-negative integer coefficients. In this talk, we will describe the very simple example of discrete integrable systems, and use their exact solutions in terms of paths on graphs or networks to explain this positive Laurent phenomenon. Non-commutative extensions will also be discussed.

Macdonald Polynomials and the Hilbert Series of the Quotient Ring of Diagonal Coinvariants

Macdonald polynomials are symmetric functions in a set of variables X which also depend on two parameters q,t. In this talk we describe how a formula of Haiman for the Hilbert series of the quotient ring of diagonal coinvariants in terms of Macdonald polynomials implies a much simpler expression for the Hilbert series involving matrices satisfying certain constraints..

Repetitions and patterns

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 $\textsf{rep}(w)$; that is, this statistic describes the amount of repetition in a reduced decomposition of $w$. In this talk, we will explore this statistic $\textsf{rep}(w)$, 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.

Finite dimensional representations of KLR algebras

Khovanov-Lauda-Rouqier algebras are a family of algebras that appear in categorifying quantum groups. I will talk about the category of finite-dimensional representations of these algebras - classifying the simple representations, giving some understanding of higher Ext groups, and the related combinatorial structures. No previous knowledge of KLR algebras will be assumed.

Manifold atlases consisting of Bruhat cells

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.

Program

Alexandra Seceleanu 9:30-10:30
Dan Erman 11:00-12:00
Allen Knutson 2:00-3:00
Ravi Vakil 3:00-4:00