Seminars Sorted by Series

Vertex decomposition, introduced by Provan and Billera in 1980, is an inductive strategy for breaking down and understanding simplicial complexes. A simplicial complex that is vertex decomposable is shellable, hence Cohen--Macaulay. Through the...

Influential work of Hodge from the 1940s led the way in using Gröbner bases to combinatorially study the Grassmannian. We follow Hodge's approach to investigate certain subvarieties of the Grassmannian, called positroid varieties. Positroid...

We heard last week in Daoji's talk about positroid varieties, which are subvarieties in the Grassmannian defined by cyclic rank conditions, and which are related to Schubert varieties. In this talk, we will provide a criterion for whether positroid...

Equivariant cohomology was introduced in the 1960s by Borel, and has been studied by many mathematicians since that time.  The talks will be an introduction to some of this work.  We will focus on torus-equivariant cohomology (as well as Borel-Moore...

In 2008, looking to bound the face vectors of tropical linear spaces, Speyer introduced the g-invariant of a matroid, defined in terms of exterior powers of tautological bundles on Grassmannians. He proved its coefficients nonnegative for matroids...

We will present recent applications of enumerative algebra to the study of stationary states in physics. Our point of departure are classical Newtonian differential equations with nonlinear potential. It turns out that the study of their stationary...

We consider the space of configurations of n points in the three-sphere $S^3$, some of which may coincide and some of which may not, up to the free and transitive action of $SU(2)$ on $S^3$. We prove that the cohomology ring with rational...

In 1999, Pitman and Stanley introduced the polytope bearing their name along with a study of its faces, lattice points, and volume. This polytope is well-studied due to its connections to parking functions, lattice path matroids, generalized...

K-theory arose in the 1950s from Grothendieck’s formulation of the Riemann-Roch theorem – that is, from attempts to calculate spaces of sections of vector bundles on a variety X via intersection theory on X.  An equivariant version was introduced...

A remarkable result of Brändén and Huh tells us that volume polynomials of projective varieties are Lorentzian polynomials. The dual notion of covolume polynomials was introduced by Aluffi by considering the cohomology classes of subvarieties of a...

Postnikov's divided symmetrization, introduced in the context of volume polynomials of permutahedra, possesses a host of remarkable ``positivity'' properties. These turn out to be best understood using a family of operators we call quasisymmetric...

We show that various classical theorems of linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases of a general result that involves a tiling of a closed oriented surface by...

We will discuss combinatorial and algebraic aspects of regular Hessenberg varieties, a large class of subvarieties of the flag variety G/B. For the special case of Peterson varieties, we show their equivariant structure constants are non-negative...

The Kadomtsev-Petviashvili (KP) Equation has deep connections to algebraic curves, with solutions constructed from Riemann theta functions in the style of Krichever. As a curve undergoes tropical degeneration, its theta function simplifies to a...

In this talk I will discuss various connections between the dynamics of integrable Hamiltonian flows, gradient flows, and combinatorial geometry. A key system is the Toda lattice  which describes the dynamics of interacting particles on the line. I...

We describe three constructions of spherical roots of spherical varieties via embddings, Borel orbits, and harmonic analysis, and give a hint for why they yield the same results. Then we describe more recent extensions of this theory to arbitrary G...

I will discuss various applications of a combinatorial model for the (torus equivariant) quantum K-theory of flag manifolds G/B, called the quantum alcove model. This is a uniform model for all Lie types, based on Weyl group combinatorics. It first...

For a complex elliptic curve $E$ and a point $p$ of order $n$ on it, the images of the points $p_k=kp$ under the Weierstrass embedding of $E$ into $CP^2$ are collinear if and only if the sum of indices is divisible by $n$. We prove that for $n$ at...

In their recent paper, Giansiracusa and Manaker introduced a notion of tropical subrepresentations of linear representations by considering linear actions on tropical linear spaces. In particular, this framework naturally brings matroids into the...

In this talk, I will describe a new definition, joint with Bivas Khan, for a tropical toric vector bundle on a tropical toric variety. This builds on the tropicalizations of toric vector bundles, and can be used to define tropicalizations of vector...

The moduli space of pointed rational curves has a natural action of the symmetric group permuting the marked points.  In this talk, we will present a combinatorial formula for the induced representation on the cohomology of the moduli space, along...

Multi-variate residues on Grassmannians $G(k,n)$ and moduli spaces $M_{0,n}$ are ubiquitous in the study of scattering amplitudes; they provide a powerful and essential tool. Amenable theories include the biadjoint scalar, NLSM, Yang-Mills, gravity...

For an embedded stable curve over the real numbers we introduce a hyperplane arrangement in the tangent space of the Hilbert scheme. The connected components of its complement are labeled by embeddings of the graph of the stable curve to a compact...

The study of the topology of hyperplane arrangement complements has long been a central part of combinatorial algebraic geometry. I will talk about intersection pairings on the twisted (co)homology for a hyperplane arrangement complement, first...

The second lecture features the nuts and bolts of the invariants from first lecture, which we call foundations. We explain the structure theorem for foundations of ternary matroids, which is rooted in Tutte's homotopy theorem. We show how this...

The theme of the third lecture is the notion of points over F1, the field with one element. Several heuristic computations led to certain expectations on the set of F1-points: for example the Euler characteristic of a smooth projective complex...

Schubert Calculus studies cohomology rings in (generalized) flag varieties, equipped with a distinguished basis - the fundamental classes of Schubert varieties - with structure constants satisfying many desirable properties. Cotangent Schubert...

A valuation is a finitely additive measure on the class of all convex compact subsets of $R^n$. Over the past two decades, a number of structures has been discovered on the space of translation invariant smooth valuations. Recently, these findings...

In a landmark paper in 1992, Stanley developed the foundations of what is now known as the Kazhdan--Lusztig--Stanley (KLS) theory. To each kernel in a graded poset, he associates special functions called KLS polynomials. This unifies and puts a...

The conjecture in combinatorics that has received perhaps the most attention over the last 50 years is McMullen's g-conjecture. It provides a complete characterisation of the number of $i$-dimensional faces in a triangulation of an $(d - 1)$...

The construction by Mikhalkin of a non-planar tropical cubic curve in R^3 of genus 1 marked a significant breakthrough in the study of combinatorial tropical varieties. It was the first known example of a non-realizable tropical variety, with the...

Equivariant cohomology was introduced in the 1960s by Borel, and has been studied by many mathematicians since that time.  The talks will be an introduction to some of this work.  We will focus on torus-equivariant cohomology (as well as Borel-Moore...

Tropical ideals are combinatorial objects that abstract the behavior of the collections of subsets of lattice points that arise as the supports of all polynomials in an ideal. Their structure is governed by a sequence of ‘compatible’ matroids and...

In 2008, looking to bound the face vectors of tropical linear spaces, Speyer introduced the g-invariant of a matroid, defined in terms of exterior powers of tautological bundles on Grassmannians. He proved its coefficients nonnegative for matroids...

The theory of stable polynomials features a key notion called proper position, which generalizes interlacing of real roots to higher dimensions. I will show how a Lorentzian analog of proper position connects the structure of spaces of Lorentzian...

Algebraic statistics employs techniques in algebraic geometry, commutative algebra and combinatorics, to address problems in statistics and its applications. The philosophy of algebraic statistics is that statistical models are algebraic varieties...

I will start by a gentle introduction to operadic structures by drawing a parallel with classical associative structures. Then we will see how those structures can be applied to matroid theory via three examples: Chow rings, Orlik--Solomon algebras...

K-theory arose in the 1950s from Grothendieck’s formulation of the Riemann-Roch theorem – that is, from attempts to calculate spaces of sections of vector bundles on a variety X via intersection theory on X.  An equivariant version was introduced...

The ring of symmetric functions has a linear basis of Schur functions $s_{\lambda}$ indexed by partitions $\lambda = (\lambda_1 \geq \lambda_2 \geq \ldots \geq 0 )$. Littlewood-Richardson coefficients $c^{\nu}_{\lambda, \mu}$ are the structure...

Abstract: In this talk I will construct a “quasisymmetric flag variety”, a subvariety of the complete type A  flag variety built by adapting the BGG geometric construction of divided differences to the newly introduced “quasisymmetric divided...

We present a combinatorial approach to the Deligne-Mumford-Knudsen compactification of the moduli space of n distinct points on the projective line $P^1$. The idea is to choose a totally symmetric embedding of the orbits of generic points into a...

A real plane algebraic curve C is called expressive if its defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of C. We give a necessary and sufficient criterion for expressivity (subject...

Given a triple (X,π,s) consisting of a homogeneous space X=G/P, a dominant weight π giving a projective embedding of X, and a reduced expression s for the minimal coset representative of w_0 in the parabolic quotient W/W_P, we construct a polytope...

Suppose given a class of finite combinatorial structures, such as graphs or total orders. Nate Harman and I recently introduced a notion of measure in this context: this is a rule assigning a number to each structure such that some axioms are...

Configuration spaces of points in graphs are nonsmooth analogs of braid arrangements, appearing in robotics applications and in theory of moduli spaces of tropical curves. While their cohomology is extremely difficult to understand, and depends on...

In part 1, I will survey the history of total positivity, beginning in the 1930's with the introduction of totally positive matrices, which turn out to have surprising linear-algebraic and combinatorial properties. I will discuss some modern...

I will show that any Schubert or Richardson variety R in a flag manifold G/P is equivariantly rigid and convex. Equivariantly rigid means that R is uniquely determined by its equivariant cohomology class, and convex means that R contains any torus...

The long-standing F-Conjecture asserts that there is a very simple description for the closed cone of effective curves on the moduli space M_{g,n}\bar of stable n-pointed curves of genus g as being determined by a finite collection of so-called F...

The Khovanskii-Teissier inequality provides the fundamental log-concavity property of intersection numbers of divisors of algebraic varieties, extending the Alexandrov-Fenchel inequality of convex geometry. In this talk I will explain, and attempt...

I will present a (cohomological) Fourier-Mukai transform for tropical Abelian varieties and give some applications, including a (generalized) Poincaré formula (for non-degenerate line bundles on tropical Abelian varieties).

Based on joint work with...