Title: Linear Structural Equation Models

Abstract: Linear structural equation models (L-SEMs) are a class of multivariate statistical models which study possible causal dependencies among variables. These models are associated with a path diagram, a graph with a directed acyclic part and a bidirected part. When a model has been specified, it is of interest to determine whether the model parameters can be recovered from the covariance matrix which they define. In this talk we will introduce the topic of causal inference using L-SEMs and present recent progress on generic identifiability using algebraic methods.

Title: Initial degenerations of Grassmannians

Abstract: Let Gr_0(d,n) denote the open subvariety of the Grassmannian Gr(d,n) consisting of d-1 dimensional subspaces of P^(n-1) meeting the toric boundary transversely. We prove that Gr_0(3,7) is schoen in the sense that all of its initial degenerations are smooth. We use this to show that the Chow quotient of Gr(3,7) by the maximal torus in GL(7) is the log canonical compactification of the moduli space of 7 lines in P^2 in linear general position. This provides a positive answer to a conjecture of Hacking, Keel, and Tevelev from "Geometry of Chow quotients of Grassmannians."

Title: Khovanskii bases and rational, complexity $1$ $T$-varieties

 
Abstract: Khovanskii bases (introuduced by Kiumars Kaveh and Christopher Manon) are an important tool used to study $\mathbb{K}$-domains relative to a valuation, and in fact, are a vast generalization of the notion of a SAGBI basis. We begin by introducing Khovanskii bases and provide several examples. We are particularly interested in the case when a pair $(A, \mathfrak{v})$ has a finite Khovanskii basis. Next, we discuss $T$-varieties which are a natural generalization of toric varieties. We will see how affine $T$-varieties arise out of the quasi-combinatorial data of a polyhedral divisor via the construction of J"urgen Hausen and Klaus Altmann. Finally, we will see a result by Nathan Ilten and C. Manon: for any homogeneous valuation $\mathfrak{v}$ on the coordinate ring of a rational complexity $1$ $T$-variety, there is an embedding whose coordinates are a Khovanskii basis for $\mathfrak{v}$.   

Title: Asymptotic Algebra and FI-modules

Abstract: Symmetric ideals in increasingly larger polynomial rings that form an ascending chain arise in various contexts like algebraic statistics, commutative algebra, and representation theory. In this talk we discuss some recent results and open questions on the asymptotic behavior of algebraic/homological invariants of ideals in such chains. Our approach is based on FI-modules with varying coefficients and various related techniques. This talk is based on joint work with Dinh Van Le, Uwe Nagel, and Hop D. Nguyen.

Title: Motivic manifestations of matroids

Abstract: A matroid is a combinatorial object which abstracts the notion of (linear or algebraic) "independence. Recent work of Adiprasito-Huh-Katz demonstrates a surprising connection to algebraic geometry: every matroid possesses a cohomology ring'' which behaves like the Chow ring of a smooth projective variety. In this talk, I will define another "algebro-geometric'' matroid invariant, the motivic zeta function of a matroid. I will assume no prior knowledge of matroids or motivic zeta functions. This is work in progress with Jeremy Usatine.

Title: Balanced shellings on manifolds

Abstract: A classical result by Pachner states that any two PL homoeomorphic manifolds with boundary are related by a sequence of shellings and inverse shellings. We show that, for balanced manifolds, such a sequence can be chosen in such a way that in each step balancedness is preserved. This is joint work with Lorenzo Venturello.

Title: Primality tests, elliptic curves and field theory

Abstract: This talk would be of interest to anyone interested in algebraic varieties, field theory or number theory. We'll start by reviewing Pepin's primality test for Fermat numbers, which are numbers of the form F_n = 2^n+1. This test features a 'universal point' on an algebraic group (a multiplicative group) whose powers determine the primality of the given number. We will show how this is related to a field theory problem involving the splitting of rational primes for quadratic extensions. I'll also describe some directly analogous primality tests for other special types of numbers, which use elliptic curves. The field theory part of the problem is slightly elevated to the more general splitting of primes in abelian extensions. I will be working this summer on a project to develop a primality test that uses results about elliptic curves and non-abelian extensions, which would provide many more opportunities to develop tests for specific types of numbers.

Title: Subspace Polynomials and Cyclic Subspace Codes

Abstract: Subspace codes are collections of subspaces of the finite vector space $\mathbb{F}^n_q$ under the subspace metric. Of particular interest due to their efficient encoding/decoding algorithms are constant dimension cyclic codes where each codeword is a subspace of the same dimension and which are invariant under an action of $\mathbb{F}^{*}_{q^n}$.  We will see how to represent subspace codes using particular polynomials and deduce properties of the codes from the structure of these polynomials based on the work of Ben-Sasson et. al. in 2016. Using these results, we will give a construction of a constant dimension cyclic code containing multiple orbits under the $\mathbb{F}^{*}_{q^n}$ action with highest possible subspace distance.