Auburn Algebra Seminar

Spring 2023

Seminars will be held in Parker 358 on Tuesdays from 2:30 to 3:20.

Schedule:

Janauary 17: Michael Brown (local)

Title: N_p conditions for curves in weighted projective space

Abstract: This is joint work with Daniel Erman. Say we have a closed embedding of a smooth curve C into projective space, and let R be its homogeneous coordinate ring. A famous theorem of Green gives a tight relationship between a certain geometric property of the embedding (namely the degree of the line bundle inducing it) and a certain homological property of R (the linearity of its free resolution over the coordinate ring of projective space). The goal of this project is to generalize this result to curves in weighted projective space.


January 24: Matthew Speck (local)

Title: Determinants of Sums of Normal Matrices

Abstract: Marcus (1972) and de Oliveira (1982) conjectured bounds on the determinantal range of the sum of a pair of normal matrices with prescribed eigenvalues. We show that this determinantal range is a flattened solid twisted permutahedron, which is, in turn, a finite union of flattened solid twisted hypercubes with prescribed vertices. This complete geometric description, in particular, proves the conjecture. Our techniques are based on classical Lie theory, geometry, and combinatorics. This is joint work with Luke Oeding.


January 31: Doug Leonard (local)

Title: Induced Structure

Abstract: I care about integral domains with structure, and how that structure should induce structure on their integral closures and resolutions of same. I'll use examples defined by y^3+yx+x^5=0 and z^6+z^3x+y^3x^2=0 to investigate how the computer algebra systems Macaulay2 and Singular view integral closures and resolutions while ignoring such induced structure.


February 7: Qingyuan Jiang (University of Edinburgh)

Title: Derived Grassmannians, derived Schur functors, and their applications.

Abstract: In this talk, we will revisit Grothendieck's theory of Grassmannians and flag schemes and the theory of Schur and Weyl module functors studied in $GL_n$-representation theory from the perspective of derived algebraic geometry (DAG). We will explain how to use the DAG framework to extend these theories from modules to complexes, and the numerous theoretical benefits of doing so. Next, we will show how these two new theories are connected by a derived generalization of the Borel--Bott--Weil theorem. Finally, we will discuss how this framework broadens the application range of classical theories and sheds new light on many classical problems, including the study of derived categories of singular schemes, and of Hilbert schemes and compactified Jacobians of integral curves, as well as their applications to a variety of recent research topics including Hecke correspondences for surfaces and two-dimensional categories. The talk will be based on papers arXiv:2202.11636 and arXiv:2212.10488.


February 14: Luke Oeding (local)

Title: A hyperdeterminant on Fermionic Fock Space

Abstract: Twenty years ago Cayley's hyperdeterminant, the degree four invariant of a certain polynomial ring over the complex numbers under an action of a special linear group, was popularized in modern physics as it separates genuine entanglement classes in the three qubit Hilbert space and is connected to entropy formulas for special solutions of black holes. In this note we compute the analogous invariant on the fermionic Fock space for N=8, i.e. spin particles with four different locations, and show how this invariant projects to other well-known invariants in quantum information. We also give combinatorial interpretations of these formulas.


February 21: Prashanth Sridhar (local)

Title: Periodicity of Fitting ideals in free resolutions

Abstract: We examine the periodicity of ideals of minors in minimal free resolutions of finitely generated modules over local rings with a focus on their asymptotic behavior. The heuristics rely on the general principle that the initial part of the resolution is influenced by the module’s relations, while the singularities of the ring have a telling effect on the asymptotic behavior. For example, we show that for any finite module over a local complete intersection, the Fitting ideals of any size in its minimal free resolution are eventually 2-periodic (although the resolution is not). Moreover, if the stable value of a given size of minors is non-zero, it contains some power of the Jacobian ideal. Results of this type hold true in other settings as well (ex: Golod rings or residue field over any local ring), but do not hold in general. This is joint work with Michael K. Brown and Hailong Dao.


February 28:

Title:

Abstract:


March 21: Edinah Gnang (Johns Hopkins University)

Title: Bounds on the Chow-rank of hypermatrices via Partial Differential Encoding of Boolean functions

Abstract: We introduce partial differential encodings of Boolean functions and their relaxations. These encodings enable us to determine the fraction of optimal encodings of some important Boolean functions. Our main result is a general method for deriving upper and lower bounds on the Chow-Rank of hypermatrices which underly polynomial transformed by some group or semi-group actions. Our method is a discrete analog of representation theory methods which devise bounds on the border rank via Lie group actions. The talk is based on joint work with Rongyu Xu.


March 23 (Thursday!): Fred Holweck (Université de Technologie de Belfort-Montbéliard)

Title: Spin representation and Fermionic Fock space

Abstract: In Quantum information, it is standard to study the classification of pure n-qubit states under the SLOCC (Stochastic Local Operations with Classical Communication) group. Mathematically this problem boils down to the study of orbits in $V=(\CC^2)^{\otimes n}$ under the semi-simple Lie group $G=SL_2(\CC)^{\times n}$. But what about other Lie groups G and their representations V ? In this talk after introducing a geometric framework to study classification of $n$-qubits quantum states in the language of auxiliary varieties of the highest weight orbit of the SLOCC group, I'll explain how similar constructions could be used for bosonic and fermionic states. In particular I will talk about the relation between the spin representation and quantum states described in Fermionic Fock space. This last identification was used by Luke Oeding and myself to provide a new invariant useful for quantum information.


March 28: Emanuele Ventura (Politecnico di Torino)

Title: Implicitisation and parameterisation in polynomial functors

Abstract: Closed subsets of polynomial functors are highly symmetrical varieties that live in infinite dimensions. Due to these large symmetries, they can be defined by finitely many polynomial equations. This is a recent result of Draisma. In this talk, I will discuss how to make this and more recent results of Bik-Draisma-Eggermont-Snowden algorithmic. Our algorithms use classical elimination in finite but ever-increasing dimension, but also separate routines, with no classical analogue, which search for a certificate that one has found enough equations.


April 4: Wanlin Li (Washington University in St. Louis)

Title: Algebraic and arithmetic properties of curves via Galois cohomology

Abstract: A lot of the algebraic and arithmetic information of a curve is contained in its interaction with the Galois group. This draws inspiration from topology, where given a family of curves over a base B, the fundamental group of B acts on the cohomology of the fiber. As an arithmetic analogue, given an algebraic curve C defined over a non-algebraically closed field K, the absolute Galois group of K acts on the etale cohomology of the geometric fiber and this action gives rise to various Galois cohomology classes. In this talk, we discuss how to use these classes to detect algebraic and arithmetic properties of the curve, such as the rational points (following Grothendieck's section conjecture) and whether the curve is hyperelliptic.


April 11: Cheng Meng (Purdue)

Title: Decompositions of local cohomology tables

Abstract: Let R be a standard graded polynomial ring over a field k, then the Boij-S\"oderberg theory describes the cone spanned by the Betti tables of finitely generated graded R-modules. This theory shows that the extremal rays of the cone are given by the pure Betti tables and every Betti table is a finite linear combination of pure Betti tables with positive rational coefficients. In this talk, we will focus on the cone spanned by the local cohomology tables of finitely generated graded R-modules of projective dimension at most 1. We will describe the extremal rays of the cone generated by local cohomology tables of such modules and show that not every local cohomology table of such a module is a finite linear combination of the tables on the extremal ray with positive rational coefficients.


April 18: Xianglong Ni (UC-Berkeley)

Title: Schubert varieties and the structure of codimension three perfect ideals

Abstract: In the orthogonal Grassmannian OG(n,2n), there are two (opposite) Schubert varieties of codimension three. Their restrictions to the big affine open cell are cut out by equations familiar to commutative algebraists. For example, one is defined by the submaximal Pfaffians of a generic skew matrix---thus, by the structure theorem of Buchsbaum and Eisenbud, it is the universal example of a Gorenstein ideal of codimension three on n generators. The other gives the universal example of an almost complete intersection of type n-3. I will explain how this example relates to Weyman's work on generic free resolutions of length three, and how it conjecturally extends to perfect ideals with other particular Betti numbers.


April 25:

Title:

Abstract: