Algebra Seminars
Seminars are held in 358 Parker Hall on Tuesdays at 2:30pm.
Upcoming Algebra Seminars
Past Algebra Seminars
DMS Algebra Seminar
Sep 24, 2024 02:30 PM
358 Parker Hall
DMS Algebra Seminar
Sep 17, 2024 02:30 PM
358 Parker Hall
DMS Algebra Seminar
Sep 10, 2024 02:30 PM
358 Parker Hall
DMS Algebra Seminar
Sep 03, 2024 02:30 PM
358 Parker Hall
DMS Algebra Seminar
Aug 27, 2024 02:30 PM
358 Parker Hall
DMS Algebra Seminar
Aug 20, 2024 02:30 PM
ZOOM
http://webhome.auburn.edu/~mkb0096/Fall2024Seminar.html
DMS Algebra Seminar
Apr 23, 2024 01:00 PM
358 Parker Hall and ZOOM
DMS Algebra Seminar
Mar 26, 2024 02:30 PM
250 Parker Hall
DMS Algebra Seminar
Mar 19, 2024 02:30 PM
358 Parker Hall
DMS Algebra Seminar
Mar 12, 2024 02:30 PM
358 Parker Hall
Sep 24, 2024 02:30 PM
358 Parker Hall
Speaker: Michael Brown (Auburn)
Title: Noncommutative sheaf cohomology
Abstract: I will give a brief introduction to noncommutative projective geometry, with a view toward a pair of projects concerning noncommutative sheaf cohomology (one joint with Daniel Erman and Greg Smith, and another joint with Prashanth Sridhar).
DMS Algebra Seminar
Sep 17, 2024 02:30 PM
358 Parker Hall
Speaker: Colin Crowley (University of Oregon)
Title: Orlik-Terao algebras and internal zonotopal algebras
Abstract: In 2017 Moseley, Proudfoot, and Young conjectured that the reduced Orlik-Terao algebra of the braid matroid was isomorphic as a symmetric group representation to the cohomology of a certain configuration space. This was proved by Pagaria in 2022. We generalize Pagaria's result from the braid arrangement to arbitrary hyperplane arrangements and recover a new proof in the case of the braid arrangement. Along the way, we give formulas for several other invariants of a hyperplane arrangement.
Joint with Nick Proudfoot.
DMS Algebra Seminar
Sep 10, 2024 02:30 PM
358 Parker Hall
Speaker: Sean Grate (Auburn University)
Title: Betti numbers of connected sums of graded Artinian Gorenstein algebras
Abstract: Considered as an algebraic analog for the connected sum construction from topology, the connected sum construction introduced by Ananthnarayan, Avramov, and Moore is a method to produce Gorenstein rings. Joint with Nasrin Altafi, Roberta Di Gennaro, Federico Galetto, Rosa M. Miró-Roig, Uwe Nagel, Alexandra Seceleanu, and Junzo Watanabe, we determine the graded Betti numbers for connected sums and fiber products of Artinian Gorenstein algebras, where the fiber product in the local setting was obtained by Geller. We also show that the connected sum of doublings is the doubling of a fiber product ring. I will discuss these results through some examples and Macaulay2 code.
DMS Algebra Seminar
Sep 03, 2024 02:30 PM
358 Parker Hall
Speaker: Ayah Almousa (University of South Carolina)
Title: Stirling numbers and Koszul algebras with symmetry
Abstract: Stirling numbers \(c(n,k)\) and \(S(n,k)\) of the first and second kind are the answers to two counting problems: how many permutations of n letters have \(k\) cycles, and how many set partitions of \([n]\) have \(k\) blocks? The \(c(n,k)\) also give the Hilbert function for certain well-studied Koszul algebras with symmetry: the cohomology of configurations of \(n\) distinct labeled points in \(d\)-space, also known as the Orlik-Solomon algebras and graded Varchenko-Gelfand algebras for type A reflection hyperplane arrangements. We discuss how the \(S(n,k)\) give the Hilbert series for their less-studied Koszul dual algebras. This includes relating the symmetric group action on the original algebras and on their Koszul duals, representation stability in the sense of Church and Farb, and branching rules that lift Stirling number recursions.
This is joint work with Vic Reiner and Sheila Sundaram.
DMS Algebra Seminar
Aug 27, 2024 02:30 PM
358 Parker Hall
Speaker: Austin Conner (Harvard University).
Title: Geometry and the complexity of matrix multiplication
Abstract: In 1968, Strassen discovered a faster way to multiply matrices than the standard row-column multiplication. Since then a fundamental question in theoretical computer science has been to determine just how fast matrices may be multiplied. Strassen's algorithm is at core a certain method to multiply 2 by 2 matrices using 7 multiplications. The data describing this method is equivalently an expression to write the structure tensor of the 2 by 2 matrix algebra as a sum of 7 decomposable tensors. Any such decomposition of an n by n matrix algebra yields a Strassen type algorithm, and Strassen showed that one essentially cannot do better than algorithms coming from such decompositions. Bini later showed all of the above remains true when we allow the decomposition to depend on a parameter and take limits. I discuss a technique for lower bounds for this decomposition problem, border apolarity. Two key ideas to this technique are (i) to not just look at the sequence of decompositions, but the sequence of ideals of the point sets determining the decompositions and (ii) to exploit the symmetry of the tensor of interest to insist that the limiting ideal has an extremely restrictive structure. I discuss its applications to the matrix multiplication tensor and other tensors potentially useful for obtaining upper bounds via Strassen's laser method.
DMS Algebra Seminar
Aug 20, 2024 02:30 PM
ZOOM
Speaker: Ritvik Ramkumar (Cornell University)
Title: Hilbert scheme of points on threefolds
Abstract: The Hilbert scheme of \(d\) points on a smooth variety \(X\), denoted by Hilb\(^d(X)\), is an important moduli space with connections to various fields, including combinatorics, enumerative geometry, and complexity theory, to name a few. In this talk, I will introduce this object and review some well-known results when \(X\) is a curve or a surface. The main focus of this talk will be on the case when \(X\) is a threefold. Specifically, I will present some tantalizing open questions and delve into describing the structure of the smooth points on Hilb\(^d(X)\). If time permits, I will also discuss the mildly singular points of this Hilbert scheme.
This is all joint work with Joachim Jelisiejew and Alessio Sammartano.
Seminar schedule:
DMS Algebra Seminar
Apr 23, 2024 01:00 PM
358 Parker Hall and ZOOM
Please note unusual time
Speaker: Matthew Speck
Title: Determinants of Sums of Normal Matrices
Abstract: Recent efforts in matrix theory have been concerned with describing invariants of matrices with "nice" properties. In this talk, we address a conjecture on the determinant of the sum of a pair of normal matrices. Reducing this conjecture to the problem of providing non-negative solutions for a system of linear equations without full rank, we use tools from representation theory to describe the modifications required to provide such solutions.
This is my doctoral defense and joint work with my committee chair, Luke Oeding.
DMS Algebra Seminar
Mar 26, 2024 02:30 PM
250 Parker Hall
Parker Hall 250. (Note the unusual room!)
Speaker: Sean Grate (Auburn)
Title: Problems in computational algebraic geometry: Lefschetz properties and toric varieties
Abstract: Starting with Lefschetz properties, moving on to toric varieties and Castelnuovo-Mumford regularity, and finishing with other miscellaneous projects, I will give an overview of the research I have conducted while at Auburn University. A common theme among all these projects is the strong presence (and necessity) of computation. As such, there will be many examples written in Macaulay2 and Python to help understand where these projects came from and how they were completed.
DMS Algebra Seminar
Mar 19, 2024 02:30 PM
358 Parker Hall
Speaker: Hal Schenck (Auburn)
Title: Free curves, eigenschemes, and pencils of curves
Abstract: Let \(R=K[x,y,z]\). A reduced plane curve \(C = V(f)\) in \(P^2\) is free if its associated module of tangent derivations Der\((f)\) is a free \(R\)-module, or equivalently if the corresponding sheaf \(T_{P^2}(−log C)\) of vector fields tangent to \(C\) splits as a direct sum of line bundles on \(P^2\). In general, free curves are difficult to find, and in this note, we describe a new method for constructing free curves in \(P^2\). The key tools in our approach are eigenschemes and pencils of curves, combined with an interpretation of Saito's criterion in this context. Previous constructions typically applied only to curves with quasihomogeneous singularities, which is not necessary in our approach. We illustrate our method by constructing large families of free curves.
This is joint work with Roberta Di Gennaro, Giovanna Ilardi, Rosa Maria Mirò-Roig, Hal Schenck, and Jean Vallès.
DMS Algebra Seminar
Mar 12, 2024 02:30 PM
358 Parker Hall
Speaker: Ian Tan (Auburn)
Title: Tensor factorizations and orbit classification
Abstract: We introduce a general framework for obtaining tensor factorizations with respect to various group actions. Our motivation comes from orbit classification problems that arise in quantum information. For example, the singular value decomposition gives real diagonal normal forms under left and right multiplication by unitary matrices. Similarly, the higher order singular value decomposition of De Lathauwer et al. was used by B. Kraus to find normal forms for generic n-qubit tensors under the action of the local unitary group. Our perspective explains these tensor factorizations and gives new ones for the "local" special orthogonal group and for the group of Stochastic Local Operations with Classical Communication (SLOCC). Consequently, we obtain SLOCC normal forms for generic n-qubit tensors.
This is joint work with Luke Oeding.