Kalina Mincheva

Algebra and Combinatorics Seminar

We meet weekly on Wednesday at 3:00 PM (Central time), Gibson 126

Organizers: Daniel Bernstein, Mahir Can, Tài Huy Hà, Kalina Mincheva.

Spring 2024

• Souvik Dey, Charles University (Czech Republic).
  On partial trace ideals of one-dimensional local rings. abstract.
  In this talk, based on joint work with S. Kumashiro, we define and study a slight generalization of the notion of partial trace ideals and h-invariant of S. Maitra. We show that for one-dimensional local rings, h-invariant of a module is finite if and only if the co-length of its trace is so. For ideals in nice enough local domains of dimension one, we give an explicit tangible formula for the h-invariant. We also discuss some characterizations of rings, including three-generated numerical semigroup rings, whose canonical ideal have low h-invariant, and how the h-invariant of the canonical module changes with respect to forming fiber products and gluing of numerical semigroup rings.

• Taylor Brysiewicz, University of Western Ontario (Canada).
  Algebraic Matroids, Monodromy, and the Heron Variety. abstract.
  Heron's formula gives the area of a triangle in terms of the lengths of its sides. More generally, the volume of any simplex is determined by its edge-lengths via a Cayley-Menger determinant. In this talk, I will discuss which sets of volumes of faces of an $n$-simplex determine other volumes. The answer to this question is encoded in the algebraic matroid of the Heron variety. Whether this determination is in terms of a formula in terms of radicals is controlled by the monodromy groups of certain branched covers. We answer these questions for $n\leq 6$ by combining techniques in computational group theory, computer algebra, field theory, and numerical algebraic geometry. Of particular focus is recovering the 10 edge lengths of a $4$-simplex from its $10$ triangular face areas, a problem motivated by applications in theoretical physics.

• Takayuki Hibi, Osaka University (Japan).
  Pick's formula and Castelnuovo polytopes abstract.
  Let $P \subset \mathbb{R}^d$ be a lattice polytope of dimension $d$. Let $b(P)$ denote the number of lattice points belonging to the boundary of $P$ and $c(P)$ that to the interior of $P$. It follows from the lower bound theorem of Ehrhart polynomials that, when $c(P) > 0$, $vol(P) ≥ (d \cdot c((1) P) + (d − 1) \cdot b(P) − d^2 + 2)/d!   (1)$, where $vol(P)$ is the (Lebesgue) volume of $P$. Pick’s formula guarantees that, when $d = 2$, the inequality (1) is an equality. One calls $P$ Castelnuovo if $c(P) > 0$ and if the equal sign holds in (1). A quick introduction to Ehrhart theory of lattice polytopes will be presented. Furthermore, a historical background on polarized toric varieties to explain the reason why one calls Castelnuovo will be briefly reviewed.

• Janet Vassilev, University of New Mexico.
  Differential operators: simplicity and combinatorial properties of affine semigroup rings. abstract.
  We will discuss the ring of differential operators of an affine semigroup ring $R$ and how combinatorial properties of the affine semigroup translate into both the simplicity of the ring of differential operators, $D(R)$, and the simplicity of the ring as a $D(R)$-module. This is joint work with Berkesch, Chan, Matusevich, Page and Traves.

• Julia Lindberg, UT Austin.
  Invariants of SDP Exactness in Quadratic Programming. abstract.
  In this talk I will consider the Shor relaxation of quadratic programs by fixing a feasible set and considering the space of objective functions for which the Shor relaxation is exact. I first give conditions under which this region is invariant under the choice of generators defining the feasible set. I then will describe this region when the feasible set is invariant under the action of a subgroup of the general linear group. If time permits, I will conclude by applying these results to quadratic binary programs by giving an explicit description of objective functions where the Shor relaxation is exact and discuss algorithmic implications of this insight.

• Jane Coons, University of Oxford.
  Algebraic geometry of likelihood inference in rational partition models. abstract.
  In this talk, we investigate the geometry of parameter inference in statistical models that are contained in a toric variety. The classical iterative proportional scaling algorithm, or IPS, numerically computes the maximum likelihood estimate of a given vector of counts for such a toric model. We study the conditions under which IPS produces the exact maximum likelihood estimate, or MLE, in finitely many steps. Since IPS produces a rational function at each step, a necessary condition is that the model must have rational maximum likelihood estimator. However, the convergence is highly parametrization-dependent; indeed, one monomial parametrization of a model may exhibit exact convergence in finitely many steps while another does not. We introduce the generalized running intersection property, which guarantees exact convergence of IPS. As the name suggests, this strictly generalizes the well-known running intersection property for hierarchical models. This generalized running intersection property can be understood in terms of the geometry of the toric model, and models that satisfy this property can be obtained by performing repeated toric fiber products of linear ideals. We also draw connections between models that satisfy the generalized running intersection property and balanced, stratified staged trees.

• Gill Grindstaff, University of Oxford.
  Expanding statistics in phylogenetic tree space. abstract.
  For a fixed set of n leaves, the moduli space of weighted phylogenetic trees is a fan in the n-pointed metric cone. As introduced in 2001 by Billera, Holmes, and Vogtmann, the BHV space of phylogenetic trees endows this moduli space with a piecewise Euclidean, CAT(0), geodesic metric. This has be used to define a growing number of statistics on point clouds of phylogenetic trees, including those obtained from different data sets, different gene sequence alignments, or different inference methods. However, the combinatorial complexity of BHV space, which can be most easily represented as a highly singular cube complex, impedes traditional optimization and Euclidean statistics: the number of cubes grows exponentially in the number of leaves. Accordingly, many important geometric objects in this space are also difficult to compute, as they are similarly large and combinatorially complex. In this talk, I’ll discuss specialized regions of tree space and their subspace embeddings, including affine hyperplanes, partial leaf sets, and balls of fixed radius in BHV tree space. Characterizing and computing these spaces can allow us to extend geometric statistics to areas such as supertree contruction, compatibility testing, and phylosymbiosis.

• William Bernardoni, Case Western Reserve University.
  Space, Spectra, and Semiring Systems of Equations. abstract.
  In this talk we will give two motivations for building theory and methodologies around systems of equations over idempotent semirings. We will show how a theory of equations over idempotent semirings could be used in both real world applications, such as creating a solar system wide internet, as well as to create new mathematical tools in areas such as commutative algebra. We will first briefly discuss how the computational problem of routing in a deep space satellite network can be reduced to solving a matrix equation over specific idempotent semirings and how this model allows one to solve secondary problems such as determining storage requirements in a network. We will then see how idempotent semirings can be used as a tool to study commutative algebra. Through the Giansiracusa's generalized valuation theory one can study the spectrum and structure of commutative rings through valuations into idempotent semirings and the maps between them. We will conclude by examining what it means to "solve a system of equations" and how these problems can be modelled categorically.

• Tài Huy Hà, Tulane University.
  Algebraic Perspective of Polynomial Interpolations in Several Variables. abstract.
  The polynomial interpolation problem considers the construction and the number of polynomials of a fixed degree that vanishes at a given set of point with prescribed multiplicities. In this talk, we will examine a few open problems for polynomial interpolations in several variables. We will discuss algebraic approaches to these problems, particularly, one that investigates containment between powers of ideals.

Past seminars.