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.

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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