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Algebra

Fall 2022

 

Seminars are held in 358 Parker Hall on Tuesdays at 2:30pm.
DMS Algebra Seminar
Mar 28, 2023 02:30 PM



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Speaker: Emanuele Ventura, Politecnico di Torino

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DMS Algebra Seminar
Mar 23, 2023 02:30 PM
358 Parker Hall


PLEASE NOTE SEMINAR IS ON THURSDAY
 
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Speaker: 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.
DMS Algebra Seminar
Mar 21, 2023 02:30 PM



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Speaker: 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 underlie 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.

DMS Algebra Seminar
Feb 21, 2023 02:30 PM
358 Parker Hall


Speaker: Prashanth Sridhar, Auburn, 

 

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.

DMS Algebra Seminar
Feb 14, 2023 02:30 PM
358 Parker Hall


Speaker: Luke Oeding (Auburn)
 
 
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.
DMS Algebra Seminar
Feb 07, 2023 02:30 PM
ZOOM


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Speaker: 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.

DMS Algebra Seminar Pretalk
Feb 02, 2023 09:00 AM
ZOOM


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Speaker: Hang Huang

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

DMS Algebra Seminar
Jan 31, 2023 02:30 PM
358 Parker Hall


Speaker: Doug Leonard (Auburn University)

 
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.
DMS Algebra Seminar
Jan 24, 2023 02:30 PM
358 Parker Hall


Speaker: Matthew Speck, Auburn University

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.
DMS Algebra Seminar
Jan 17, 2023 02:30 PM
358 Parker Hall


Speaker: Michael Brown

Title: N_p conditions for curves in weighted projective space
 
Abstract: 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.
 
This is joint work with Daniel Erman.
DMS Algebra Seminar
Nov 29, 2022 02:30 PM
ZOOM


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Speaker: Mark Walker (University of Nebraska)

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Last Updated: 09/20/2022