• Chelsea Walton - A Century of Frobenius Algebras, Room TBA

• 2026, August 17-21 (4-5pm)

• Abstract. Born over 100 years ago, Frobenius algebras have undergone a radical transformation from their classical representation-theoretic origins into the backbone of modern Topological Quantum Field Theory. In this lecture series, we will take a journey from F.G. Frobenius’s 19th-century foundations to the category-theoretic breakthroughs of today. The outline is as follows.

1.  Frobenius algebras over a field. 

• We will cover the history, numerous characterizations, examples of Frobenius algebras over a field, as well as self-injective algebras and other generalizations.

• Lecture Note

2. Preliminaries on monoidal categories.

• This is the setting for modern applications of Frobenius algebras, and the speaker will only assume basic knowledge of categories and functors for this lecture.

• Lecture Note

3. Algebraic structures in monoidal categories.

• We will introduce algebras, coalgebras, and Frobenius algebras in the settings from Lecture 2.

• Lecture Note

4. Ties to 2-dimensional Topological Quantum Field Theories (2-d TQFTs).

• We will highlight how (variants of) Frobenius algebras in monoidal categories are used to construct oriented, unoriented, and extended 2-d TQFTs.

• Lecture Note

5. Representations and functorial constructions. 

• Now that we appreciate Frobenius algebras, we will show how to build more and how they are represented.

• Lecture Note

This is based on the speaker's book in preparation, "Symmetries of Algebras, Volume 2".

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• Gordana Todorov - Algebras and categories: Auslander, Cluster, Preprojective and Higher Version of Those, Room 403.

• 2026, May 18-22 (4-5pm), Room 403.

• Topics: 

1. Quiver representations, Gabriel, Dlab, Ringel — Dynkin diagrams, Auslander-Reiten theory.

2. Cluster algebras and cluster categories.

3. Higher Auslander algebras and cluster tilting subcategories.

4. Preprojective algebras and higher preprojective algebras.

5. Preprojective structure on subcategories.

Lecture Notes

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• Ivan Dimitrov -  Root systems, generalizations and applications, Room 403.

• 2026, May 19 and 21 (3-4pm),

• 2026, June 1, 3 and 5 (4-5pm).

• Abstract. Root systems were introduced by Wilhelm Killing in 1889 as a tool for classifying Lie algebras. Since then, they have become ubiquitous, appearing in various contexts in seemingly unrelated areas of Mathematics. In this lecture series, I will introduce and study root systems and will define a generalization of root systems. I will explain how this more general structure applies to various problems, e.g. hyperplane arrangements. Finally, I will introduce and study inversion sets of roots.

The exposition will be mostly self-contained, assuming Linear algebra and some Group theory background.

Key words. Root systems, Coxeter groups, Hyperplane arrangements, Inversion sets of roots

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• Quanshui Wu - Equivalence of Quotient Categories, with Applications to Noncommutative Resolutions, Room 578

• 2026, August 6 (4-5pm),

• 2026, August 7 (3-4pm),

• 2026, August 7 (4:10-5:10pm),

• Abstract: The lectures begin with the fundamentals of various constructions of localizations and quotient categories. We then introduce the notion of F-ampleness in an abelian category, where F is a Gabriel topology on a ring. Next, we present a structure theorem stating that an abelian category C is equivalent to mod-A/F if and only if C admits an F-ample generator under some mild condition, where mod-A/F denotes the quotient category of Mod-A with respect to F . We also revisit the Popescu–Gabriel theorem from the perspective of F-ample generators. We then apply F -ample generators to study equivalences of quotient categories and establish a Morita-type theory. Based on these equivalences, we propose a definition of noncommutative resolutions of AS-Gorenstein isolated singularities and prove that such noncommutative resolutions are generalized AS-regular algebras.

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