Kalina Mincheva

Algebra and Combinatorics Seminar

We meet weekly on Wednesday at 3:00 PM (Central time) over Zoom.

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

Schedule: Spring 2021

• Sankhaneel Bisui, Tulane University,
  Stable Harbourne-Huneke containment and Chudnovsky's Conjecture. abstract.
  

• Thai Nguyen, Tulane University,
  The Least Generating Degree of Symbolic Powers and Ideal Containment Problem. abstract.
  What is the smallest degree of a homogeneous polynomial that vanishes to order $m$ on a given finite set of points, or more generally on some algebraic variety in projective space? A classical result of Zariski and Nagata tells us the set of such polynomials is the $m$-th symbolic power of the defining ideal $I$ of the variety. To bound the generating degree of the symbolic powers of $I$, we can study containment between symbolic powers and ordinary powers of $I$. Conversely, knowing bounds for generating degree can help us study containment. My talk will be an introduction to this subject. I will also present some results from our joint work with Sankhaneel Bisui, Eloísa Grifo and Tài Huy Hà.

• Joseph Skelton, Tulane University,
  Invariants and properties of symbolic powers of edge and cover ideals. abstract.
  In this talk I will address several questions about symbolic powers of edge and cover ideals. The containment between ordinary and symbolic powers of edge ideals has been an active area of research for decades. As a result the resurgence number and Waldschmidt constant are of particular interest. The regularity of symbolic powers of edge ideals has been motivated by a conjecture of N.C. Minh which states that $reg I(G)^{(s)} = reg I(G)^s$ for any $s\in \mathbb{N}$. For cover ideals we are motivated by the results of Villlarreal showing that whiskering a graph results in a Cohen-Macaulay graph which, in turn, implies the cover ideal of the whiskered graph has linear resolution. Necessary and sufficient conditions on $S\subset V(G)$ cover ideal of the graph whiskered at $S$, $J(G\cup W(S))$ is Cohen-Macaulay. While symbolic powers of the cover ideal do not necessarily have linear resolution I will show necessary conditions on $S$ such that symbolic powers of $J(G\cup W(S))$ have componentwise linearity.

• Mike Joyce, Tulane University,
  A Survey of Hopf Algebras and Quantum Groups. abstract.
  We will introduce Hopf algebras and quantum groups through some of their key properties and some of the simplest examples. We will discuss R-matrices and the Yang-Baxter equation and then survey some of their manifestations in other areas of mathematics.

• Christopher Manon, University of Kentucky,
  When is a (projectivized) toric vector bundle a Mori dream space? abstract.
  Like toric varieties, toric vector bundles are a rich class of varieties which admit a combinatorial description. Following the classification due to Klyachko, a toric vector bundle is captured by a subspace arrangement decorated by toric data. This makes toric vector bundles an accessible test-bed for concepts from algebraic geometry. Along these lines, Hering, Payne, and Mustata asked if the projectivization of a toric vector bundle is always a Mori dream space. Su\ss and Hausen, and Gonzales showed that the answer is "yes" for tangent bundles of smooth, projective toric varieties, and rank 2 vector bundles, respectively. Then Hering, Payne, Gonzales, and Su\ss showed the answer in general must be "no" by constructing an elegant relationship between toric vector bundles and various blow-ups of projective space, in particular the blow-ups of general arrangements of points studied by Castravet, Tevelev and Mukai. In this talk I'll review some of these results, and then show a new description of toric vector bundles by tropical information. This description allows us to characterize the Mori dream space property in terms of tropical and algebraic data. I'll describe new examples and non-examples, and pose some questions. This is joint work with Kiumars Kaveh.

• Kiumars Kaveh, University of Pittsburgh,
  On combinatorics of Arthur's trace formula, convex polytopes, and toric varieties. abstract.
  I start by discussing two beautiful well-known theorems about decomposing a convex polytope into an signed sum of cones, namely the classical Brianchon-Gram theorem and Lawrence-Varchenko theorem. I will then explain a generalization of the Brianchon-Gram which can be summerized as "truncating a function on the Euclidean space with respect to a polytope". This is an extraction of the combinatorial ingredients of Arthur's ''convergence'' and ''polynomiality'' results in his famous trace formula. Arthur's trace formula concerns the trace of left action of a reductive group $G$ on the space $L^2(G/\Gamma)$ where $\Gamma$ is a discrete (arithmetic) subgroup. The combinatorics involved is closely related to compactifications of ''locally summetric spaces'' (which btw are hyperbolic manifolds). Our ''combinatorial truncation'' can be thought of as an analogue of Arthur's truncation over a toric variety (in place of a compactification of a locally symmetric space). This is joint work in progress with Mahdi Asgari (Oklahoma State).

• Mike Joyce, Tulane University,
  Quantum Groups, R-Matrices, Quantum Yang-Baxter Equation and Solvable Lattice Models. abstract.
  In this talk, we will briefly define quantum groups and focus on one specific example, quantum $\mathfrak{sl}_2$. Quantum group modules yield interesting R-matrices that arise from the almost cocommutativity of the quantum group. These R-matrices satisfy the Quantum Yang-Baxter Equation (QYBE). We'll connect this to an area of intense current research, solvable lattice models, which have applications in algebraic combinatorics, number theory, and probability. We'll see how quantum $\mathfrak{sl}_2$ explains properties of the six vertex model, the simplest and most well-understood solvable lattice model.

• Robert Walker, University of Wisconsin-Madison,
  Uniform Asymptotic Growth of Symbolic Powers of Ideals. abstract.
  Algebraic geometry (AG) is a major generalization of linear algebra which is fairly influential in mathematics. Since the 1980's with the development of computer algebra systems like Mathematica, AG has been leveraged in areas of STEM as diverse as statistics, robotic kinematics, computer science/geometric modeling, and mirror symmetry. Part one of my talk will be a brief introduction to AG, to two notions of taking powers of ideals (regular vs symbolic) in Noetherian commutative rings, and to the ideal containment problem that I study in my thesis. Part two of my talk will focus on stating the main results of my thesis in a user-ready form, giving a "comical" example or two of how to use them. At the risk of sounding like Paul Rudd in Ant-Man, I hope this talk will be awesome.

• Taylor Brysiewicz, MPI Leipzig,
  Nodes on Quintic Spectrahedra. abstract.
  A spectrahedron in $\mathbb{R}^3$ is the intersection of a 3-dimensional affine linear subspace of $d\times d$ real matrices with the cone of positive-semidefinite matrices. Its algebraic boundary is a surface of degree $d$ in $\mathbb{C}^3$ called a symmetroid. Generically, symmetroids have $(d^3-d)/6$ nodes over $\mathbb{C}$ and the real singularities are partitioned into those which lie on the spectrahedron and those which do not. This data serves as a coarse combinatorial description of the spectrahedron. For $d=3$ and $4$, the possible partitions are known. In this talk, I will explain how we determined which partitions are possible for $d=5$. In particular, I will explain how we used numerical algebraic geometry and an enriched hill-climbing algorithm to find explicit examples of spectrahedra witnessing each partition.

• Abu Thomas, Tulane University,
  Steiner Configurations Ideals: Containment and Colorability. abstract.
  We show that Stable Harbourne Conjecture and Stable Harbourne--Huneke Conjecture hold for the defining ideal of a Complement of a Steiner configuration of points in $\mathbb{P}^n_k$. We study the relation between a particular notion of colorability of hypergraphs associated to Steiner configurations of points. We also find results on the containment problem for the cover ideal associated to these special hypergraphs. We can also show that Chudnovsky's Conjecture and Demailly's Conjecture are satisfied by the ideal defining Complement of Steiner configuration of points. This is a joint work with E. Ballico, G. Favacchio and E. Guardo. We dedicate the paper to L. Millazzo who passed away in 2019.