There are deep and subtle relations between the numbers of vertices, edges, triangles, and higher dimensional faces of a polytope. The so-called g-theorem makes these precise by giving necessary and sufficient conditions for an integer vector to be counting the faces of a polytope. In this introductory talk, we’ll discuss Stanley’s ingenious proof of the g-theorem, which invokes the topology of algebraic varieties. This will take us into the realm of toric varieties, where combinatorics and algebraic geometry intertwine. This bird’s-eye view will not include hard algebraic geometry—interesting details will be the subjects of future meetings of the learning seminar.

Arising in fields ranging from algebraic geometry to theoretical physics, toric varieties provide a concrete link between combinatorial geometry and algebraic geometry. In this introductory talk, we’ll focus our attention on one of the fundamental types of toric varieties, toric varieties constructed from fans.

This talk will begin building up a dictionary relating geometric properties of toric varieties to combinatorial properties of fans. In particular, we will analyze singularities and compactness of toric varieties. As a by-product of our dictionary, we will see how to form a fan from a normal toric variety.

Continuing from last week, we will continue discussing properness and projectivity of toric varieties. Along the way, we will study the divisors and line bundles on toric varieties with some explicit examples.

In this talk I will will be looking at toric geometry from a symplectic point of view. We will begin by looking at symplectic manifolds. There are natural actions of Lie groups on such manifolds and we will define certain maps called moment maps. In the case of torus actions, we will discuss results by Atiyah-Guillemin-Sternberg and Delzant, relating convex polytopes and toric manifolds, via moment maps.

We will review the basic theory of Chow groups and study its intersection product for simplicial toric varieties. We will show that the Chow group endowed with the intersection product is isomorphic to the cohomology ring, thus showing various corollaries such as relating Betti numbers and the number of cones in the associated fan. Finally, we will show a description of the cohomology ring in terms of generators and an ideal of relations. We will also prove analogous but stronger statements for nonsingular toric varieties.

We will show that for projective simplicial toric varieties, the Chow group is isomorphic to its cohomology, thus showing various corollaries such as relating Betti numbers and the number of cones in the associated fan. Finally, we will show a description of the cohomology ring in terms of generators and an ideal of relations.

In this learning seminar, we have been converting questions about face counts of polytopes into questions about the cohomology of toric varieties. In the case of toric varieties associated to a (nice enough) fan, the cohomology is isomorphic to a combinatorial Chow ring. As their name suggests, the generators and relations of these rings can be described directly from the combinatorics of the fan. We will describe these rings, and give several detailed examples.

I will state the g-theorem characterizing the face numbers of simplicial polytopes, and discuss some related notions in commutative algebra: face rings, M-vectors, Cohen-Macaulay rings, etc. Then, time-permitting, I’ll briefly explain the relation to toric varieties that we’ve been discussing this semester.

Given a plane curve singularity C, one can define an algebraic variety called the compactified Jacobian of C. We introduce a class of “generic” curves, and describe the homology of the corresponding compactified Jacobians in terms of combinatorics of non-coprime rational q,t-Catalan numbers. All notions will be introduced in the talk, this is a joint work with Mikhail Mazin and Alexei Oblomkov.

In this talk, we describe a connection between full dimensional lattice polytopes and torus invariant Cartier Divisors on toric varieties. We will introduce a result connecting the volume of a polytope with the self-intersection number of its corresponding divisor. This result allows us to compute volumes of polytopes using intersection theory, prove properties of mixed volumes using multilinearality of intersection pairings, and even deduce the isoperimetric inequality from Hodge Index Theorem. Finally, we will introduce the Bernstein-Khovanskii- Kushnirenko theorem, which roughly says the solutions to a system of Laurent polynomial equations in a certain toric variety is generically counted by the mixed volume of polytopes associated with the equations. The talk will focus on illustrating the theorems with examples, rather than going through the proofs.

The correspondence between lattice polytopes and divisors on a toric variety is a basic incarnation of a connection that extends far beyond toric varieties. In this talk, we will give a gentle introduction to Newton-Okounkov bodies, which provide a correspondence between convex geometry and divisors on a general variety. While Newton-Okounkov bodies are in generally very far from being polyhedral, we will highlight an important situation in which the Newton-Okounkov body is polyhedral and what consequences this has for the geometry of the associated variety.

An introductory talk on polytopes and canoncial forms to give an overview of the subject.

The amplituhedron is an object introduced by physicists in 2013 arising from their study of scattering amplitudes which has garnered much recent attention from physicists and mathematicians alike. Mathematically, it is a linear projection of a nonnegative Grassmannian to a smaller Grassmannian, via a map induced by a totally positive matrix. A Grassmann polytope, or Grasstope, is a generalization of the amplituhedron, defined to be such a projection by any matrix, removing one of the positivity conditions. In this talk, I will discuss joint work with Dmitrii Pavlov and Lizzie Pratt in which we study these objects, with hope that we may gain new insights by broadening our horizons and studying all Grasstopes.

I’ll explain the definition of a positive geometry and some motivation from the physics of scattering amplitudes.

We will explore the relationship between the moduli space M0n of n points on the projective line and scattering amplitudes for the simplest Quantum Field Theory, the cubic biadjoint scalar. We discuss a natural set of “u-variable” coordinates on M0n which satisfy a miraculous set of binary-type identities, and which are used to define a certain canonical form which has singularities on the boundary strata of a “curvy” associahedron in M0n. We ask the same questions for 6 points in the projective plane and produce a similarly beautiful curvy polytope and set of binary-type identities, and a differential form which has singularities on the boundary strata, connecting to the moduli space of del Pezzo surfaces. Time permitting, we may discuss some appearances of oriented matroids in physics. References: 1912.11764, 2306.13604 and 2212.11243.

The Grassmannian Gr(k,n) has a stratification into positroid varieties which can be indexed by many objects including move-equivalence classes of reduced plabic graphs and bounded affine permutations. Given a bounded affine permutation f, the positroid variety \(\Pi_f\) and the corresponding positroid cell \(\Pi_{f,\geq 0}\) in the TNN Grassmannian form a positive geometry \((\Pi_f, \Pi_{f,\geq 0})\). We will see how to compute the canonical form of a positroid variety using a reduced plabic graph corresponding to that variety.

Projectivizations of pointed polyhedral cones \(C\) are positive geometries with canonical forms that look like \(\Omega_C(x) = \frac{A(x)}{B(x)}dx\) with \(A, B\) polynomials. We are going to see that the numerator \(A(x)\) is given by the adjoint polynomial of the dual cone \(C^\vee\), which cancels unwanted poles outside the polytope.

No seminar

Throughout the semester, we have explored some aspects of polytopes and other spaces that exhibit a positive geometry. One space that we have seen is a positive geometry is the totally nonnegative Grassmannian, but it is not a polytope. However, like a polytope, the totally nonnegative Grassmannian is a regular CW complex homeomorphic to a closed ball, which was conjectured by Postnikov and proven by Galashin-Karp-Lam. In this talk, we will give a different, self-contained argument, also due to Galashin-Karp-Lam, that the totally nonnegative Grassmannian is homeomorphic to a closed ball. To do so, we will exhibit a homeomorphism by constructing a contractive flow on the totally nonnegative Grassmannian.

Positive geometries might be studied as generalizations of polytopes, and we’d like to import insights from combinatorial geometries associated with polytopes. A well-understood first case is that of hyperplane arrangements: there, the combinatorics of the arrangement controls much of the topology and arithmetic of the complement. We’ll survey relevant constructions and calculations for hyperplane arrangements, such as the characteristic polynomial, the cohomology ring, and point-counts over finite fields. Taking those as a model, one could later apply similar ideas to other positive geometries.

A del Pezzo surface can be obtained by blowing up the projective plane at points in general position. Such a surface contains lines that come from the point configuration in \(P^2\). We can also define a moduli space of del Pezzo surfaces as the parameter space of these point configurations. In this talk, we work over the real numbers and study real del Pezzo surfaces and the regions on them cut out by the lines; we also study the moduli space of real del Pezzo surfaces. In both cases, we find positive geometries. This is joint work with Nick Early, Alheydis Geiger, Marta Panizzut, and Bernd Sturmfels.

We construct the kinematic associahedron motivated by quantum field theory. We then construct a compactification for \(M_{0,n}^{+}\) motivated by string theory and realize it as a curvy associahedron (worldsheet associahedron) and a positive geometry. Finally, we introduce stringy integrals, which on some limit determines the kinematic associahedron and its canonical function, and on another limit gives a diffeomorphism between the interior of the kinematic associahedron and the worldsheet associahedron, thus relating their canonical functions and canonical forms.

The main goal of this talk is to introduce a method to compute the canonical form of a convex polytope by computing the volume of its polar dual polytope. We will first define the notion of a dual polytope and then compute the volume in a concrete example. Then we will give a proof that the dual volume actually gives the canonical form of the original convex polytope. Finally, we will introduce a more efficient way to compute the dual volume when the polytope has many vertices but few sides, via the Filliman Duality, which roughly says the volume of the polytope can be obtained from triangulating its dual.