Welcome to the Algebra and Geometry Seminar at Iowa State University, organized by Jonas Hartwig, Jason McCullough, and Tathagata Basak.

During Fall 2020, the seminar runs virtually on **Tuesdays at 2:10pm–3:00pm** over Zoom/WebEx. To attend you must be on the seminar mailing list, where invitation links to the talks will be posted. If you wish to be added or removed from the mailing list, please contact one of the organizers.

Topics include:

- associative algebras and commutative rings,
- representation theory and Lie theory,
- connections to combinatorics, geometry and physics.

April 20, 2021

Reduction algebras and \(\mathfrak{osp}(1|2)\)

Dwight A. Williams II (Iowa State University)

Reduction algebras associated to a pair of Lie algebras \(\mathfrak{G},\mathfrak{g}\) have been shown to act irreducibly on the space of primitive vectors of certain \(\mathfrak{G}\)-modules. The role of reduction algebras extends to the super case: Here we consider the diagonal reduction algebra of the pair of Lie superalgebras \((\mathfrak{osp}(1|2)\times\mathfrak{osp}(1|2), \mathfrak{osp}(1|2))\) as a double coset space having an associative diamond product and give generators with relations.

April 13, 2021

Constructions in Combinatorial Commutative Algebra, Old and New

Justin Lyle (University of Arkansas)

We discuss some new constructions in combinatorial commutative algebra and tie them to classical topics considered by Stanley, Munkres, Hibi and others. In particular, we discuss the recently introduced higher nerve complexes of a simplicial complex, which extend the classical nerve complex, and relate them to certain balanced simplicial complexes. Taking advantage of the balanced structure, we prove that reduced homologies of these higher nerves capture a wealth of algebraic information about the associated Stanley-Reisner ring, including it's depth, it's h-vector, and information about Serre conditions. We discuss connections between these topics and related open problems, particularly in local algebra. This is primarily based on joint work with Brent Holmes.

March 23, 2021

Singularities of Rees-like algebras

Jason McCullough (ISU)

Given their importance in constructing counterexamples to the Eisenbud-Goto Conjecture, it is reasonable to study the algebra and geometry of Rees-like algebras further. Given a graded ideal \(I\) of a polynomial ring \(S\), its Rees-like algebra is \(S[It, t^2]\), where \(t\) is a new variable. Unlike the Rees algebra, whose defining equations are difficult to compute in general, the Rees-like algebra has a concrete minimal generating set in terms of the generators and first syzygies of \(I\). Moreover, the free resolution of this ideal is well understood. While it is clear that the Rees-like algebra of an ideal is never normal and only Cohen-Macaulay if the ideal is principle, I will explain that it is often seminormal, weakly normal, or F-pure. I will also discuss the computation of the singular locus, how the singular locus is affected by homogenization, and the structure of the canonical module, class group, and Picard group. This talk is joint work with Paolo Mantero and Lance E. Miller.

March 16, 2021

F-singularities and homological invariants of blowup algebras

Jonathan Montaño (New Mexico State University)

In this work we introduce the notion of symbolic F-purity which is satisfied by important classes of ideals such as determinantal and squarefree monomials. Symbolic F-purity implies that the corresponding symbolic blowup algebras have good F-singularities and implies the convergence of certain homological invariants. In this talk we discuss these and other consequences of this new notion, and further generalizations to filtrations of ideals. This is joint work with Alessandro De Stefani and Luis Núñez-Betancourt.

February 16, 2021

Uniform Asymptotic Growth of Symbolic Powers of Ideals

Robert Walker (UW Madison)

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.

A first course in AG would be helpful but I review what I need for my thesis problem.

A first course in AG would be helpful but I review what I need for my thesis problem.

November 10, 2020

Some quantum symmetries of path algebras

Amrei Oswald (University of Iowa)

Much like group actions formalize the notion of symmetry, Hopf actions of quantum groups formalize the notion of quantum symmetry. We investigate an example of quantum symmetry by studying Hopf actions of \(U_q(\mathfrak{b})\) and \(U_q(\mathfrak{sl}_2)\) on path algebras. First, we parametrize these actions using linear algebraic data. Then, we attempt to describe the "building blocks" of these actions by viewing them as tensor algebras in the tensor categories \(\mathsf{rep}(U_q(\mathfrak{b}))\) and \(\mathsf{rep}(U_q(\mathfrak{sl}_2))\). We construct an equivalence between categories of bimodules in \(\mathsf{rep}(U_q(\mathfrak{b}))\) (or \(\mathsf{rep}(U_q(\mathfrak{sl}_2))\)) and a subcategory of certain finite-dimensional representations of associative algebras, explicitly given in terms of quivers with relations.

November 3, 2020

Hochschild cohomology of general twisted tensor products

Pablo S. Ocal (Texas A&M)

The Hochschild cohomology is a tool for studying associative algebras that has a lot of structure: it is a Gerstenhaber algebra. This structure is useful because of its applications in deformation and representation theory, and recently in quantum symmetries. Unfortunately, computing it remains a notoriously difficult task. In this talk we will present techniques that give explicit formulas of the Gerstenhaber algebra structure for general twisted tensor product algebras. This will include an unpretentious introduction to this cohomology and to our objects of interest, as well as the unexpected generality of the techniques. This is joint work with Tekin Karadag, Dustin McPhate, Tolulope Oke, and Sarah Witherspoon.

October 27, 2020

Interpolation problems

Paolo Mantero (University of Arkansas)

Homogeneous polynomial interpolation problems are challenging problems in Algebraic Geometry. They ask for information regarding hypersurfaces of given degrees passing with given multiplicity through a given set X of points in a projective space.

A theorem by Zariski and Nagata allows one to translate the problem into a question about the Hilbert function of symbolic powers of the ideal I_X defining the set of points, so the problem can be tackled with Commutative Algebra tools as well.

In the present talk we will review the history of some interpolation problems, state classical and recent results and discuss open problems and conjectures.

A theorem by Zariski and Nagata allows one to translate the problem into a question about the Hilbert function of symbolic powers of the ideal I_X defining the set of points, so the problem can be tackled with Commutative Algebra tools as well.

In the present talk we will review the history of some interpolation problems, state classical and recent results and discuss open problems and conjectures.

Links: Video of lecture, Slides

October 20, 2020

The structure of Koszul algebras defined by four quadrics

Matt Mastroeni (Oklahoma State University)

If I is a graded ideal in a standard graded polynomial ring S over a field such that R = S/I is a Koszul algebra, a question of Avramov, Conca, and Iyengar asks whether the Betti numbers of R over S can be bounded above by binomial coefficients on the minimal number of generators of I. Building on previous affirmative answers for Koszul algebras defined by three quadrics and Koszul almost complete intersections with any number of generators, we give a strong affirmative answer to the above question in the case of four quadrics. In the process, we prove structure theorems allowing for a complete description of the possible ideals of four quadrics defining Koszul algebras over an algebraically closed field. We will attempt to highlight the main ingredients that go into this classification. This work is joint with Paolo Mantero.

Links: Video of lecture, Slides

October 13, 2020

From Lie algebras to Lie superalgebras II

Dwight A. Williams II (ISU)

(Continuation from last week.)

Links: Slides

October 6, 2020

From Lie algebras to Lie superalgebras I

Dwight A. Williams II (ISU)

A Lie superalgebra is said to be a generalization of a Lie algebra. But how exactly do Lie superalgebras extend algebraic, geometric, and categorical examples of Lie algebras? This talk is the first of two discussions on the similarities and differences between Lie algebras and Lie superalgebras, beginning with the underlying vector spaces or super vector spaces. The goal in part one is to develop a familiarity with the role of parity and a non-trivial braiding in the category of super vector spaces in order to support the computations featured in the sequel---developments in the infinite-dimensional representation theory of a particular basic classical Lie superalgebra.

Links: Slides

September 29, 2020

Exact Factorizations of Finite Tensor Categories

Issac Odegard (ISU)

Exact factorizations of fusion categories, which gives a categorical generalization the notion of exact factorizations of finite groups, were originally studied by Shlomo Gelaki. In this talk, we explore the generalization of exact factorizations of fusion categories to finite tensor categories.

Links: Slides

August 27 and September 3, 2020 at 2:10pm - 3:00pm

An Introduction to Koszul Algebras

Jason McCullough (ISU)

Koszul Algebras are abundant in algebraic geometry and combinatorics. They are necessarily defined by quadratic (and linear) relations but not all quadratic algebras are Koszul. The precise definition concerns the (Castelnuovo-Mumford) regularity of the coefficient field. A famous theorem of Avramov-Eisenbud-Peeva says that a graded (commutative) K-algebra A is Koszul if and only if every finitely generated module has finite regularity over A. Koszul algebras enjoy a beautiful duality that expands the duality between exterior and symmetric algebras.

This will be a series of 2 talks. Talk 1 will be an introduction to Koszul Algebras with lots of examples and non-examples. Talk 2 will focus on the commutative case and techniques for showing an algebra is or is not Koszul. Everyone is welcome but some experience with resolutions or homology is useful.

This will be a series of 2 talks. Talk 1 will be an introduction to Koszul Algebras with lots of examples and non-examples. Talk 2 will focus on the commutative case and techniques for showing an algebra is or is not Koszul. Everyone is welcome but some experience with resolutions or homology is useful.

August 20, 2020 at 2:10pm - 3:00pm

Resurgence via Asymptotic Resurgence

Michael DiPasquale - Colorado State University

If an ideal defines a projective variety, then its sth symbolic power consists of polynomials which vanish to order s along this variety. Thus symbolic powers are an important geometric analogue of taking regular powers. There is significant interest in the containment problem; that is studying which pairs (r,s) satisfy that the symbolic sth power is contained in the regular rth power. A celebrated result of Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede states that the n*r symbolic power is contained in the rth regular power if I is an ideal in a polynomial ring with n variables. In an effort to quantify these containment results more precisely, the notions of resurgence and asymptotic resurgence of an ideal were introduced by Bocci and Harbourne and Guardo, Harbourne, and Van Tuyl. We discuss some aspects of resurgence which can be studied via asymptotic resurgence. This is joint work with Ben Drabkin. No knowledge of symbolic powers will be assumed.

Links: slides

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