**Fall 2023** (Thursdays from 12:00 p.m. to 12:50 p.m.)

i) **September 21**

**Speaker: Charlotte Ure (ISU)**

**Talk Title: Twisting of Algebras and Comodules**

Abstract: Deformations of algebra structures arise naturally in the study of quantum symmetries and have many applications in representation theory and noncommutative algebra. For example, one may twist the algebra structure by a graded isomorphism. In this talk, I will introduce and compare different methods of deformation of algebras such as twisting by an automorphism or by a 2-coycle. The talk is based on joint work with Hongdi Huang, Van C. Nguyen, Kent B. Vashaw, Padmini Veerapen, and Xingting Wang.

ii) **October 12** (10:00am-10:50am)

**Speaker: Samantha Kirk (Bradley University)**

**Talk Title: A Vertex Algebra Construction of Representations of Twisted Toroidal Lie Algebras **

Abstract:

Given a simple finite-dimensional Lie algebra and an automorphism of finite order, one can construct a twisted toroidal Lie algebra. Similar to twisted affine Lie algebras, which are well-studied in the literature, we can construct representations of twisted toroidal Lie algebras with the help of vertex algebras. In this talk, I will discuss twisted modules of vertex algebras, and I will show how representations of twisted toroidal Lie algebras can be constructed from such twisted modules.

*Joint work with Bojko Bakalov*

iii)** October 19**

**Speaker: Lucian Ionescu (ISU)**

**Talk Title: Real Numbers: A New (Quantum) Look**

Abstract: The continuum of Real Numbers is central in Math, but in conflict with modern Physics, Sciences, Alg. Geom., Number Theory etc. I will present a Number Theory oriented completion of the rational numbers, using sequential topology (no metric/distance used) based on continued fractions and modular group PSL(2;Z) (Mobius transformations of the rational circle).

The background material presented is interesting in itself: Continued Fractions and Modular Group (2D-congruence arithmetic), fractions and Farey sequence, providing a deeper understanding of Pythagorean triples, algebraic numbers etc.

Applications and implications to Mathematics in general (and Physics) will be briefly mentioned at the end; for example, Platonic solids are just “dessins d’enfant” on modular curves with Belyi maps, as previously presented in the seminar.

This is work in progress together with Anurag Kurumbail, just opening a new direction of research .

iv) **November 16**

**Speaker: Benton Duncan (ISU)**

**Title: What is an operator algebra?**

Abstract: I will discuss the definition of operator algebras as well as their abstract characterizations. Teh abstract characterization for C*-algebras is much cleaner, and I will survey work on connecting arbitrary operator algebras with associated C*-algebras.

v) **November 30**

**Speaker: Thuy (Christy) Bui **(Ph.D. Candidate in Operations Research at Rutgers Business School)

**Title:** **A game-theoretic approach to patrolling problems**

**Abstract:** This research studies the problems of finding optimal patrols to prevent adversarial attacks on a network, for example, addressing how to patrol a border to guard against infiltration or how to optimally patrol an airport or shopping mall to minimize the risk of terrorist attacks. We consider the problems (games) in continuous time and space where the attack could take place anywhere on the network at any time. The game is modeled as a zero-sum game between an Attacker and a Patroller. The Attacker will choose a time and a point to attack the network for a fixed amount of time. The Patroller picks a patrol of the network. The Patroller wins if he visits the attacked point while the attack is occurring; otherwise, the Attacker wins. We present (i) a solution for any arbitrary networks as long as the attack time is sufficiently short, (ii) a solution for all tree networks with any attack time, and (iii) a solution in some cases for complete networks**. **This is joint work with Steve Alpern, Thomas Lidbetter, and Katerina Papadaki.

vi) **December 7**

**Speaker: Matt Speck **(Ph.D. Candidate at Auburn University)** **

**Title:** **Progress on the Marcus-de Oliveira Conjecture**

**Abstract:** Marcus (1972) and de Oliveira (1982) independently conjectured bounds on the determinantal range of a sum of normal matrices given their respective eigenvalues. We will propose a proof method rooted in Lie theory and representation theory and discover some interesting combinatorics along the way. This talk will be accessible to anyone with mathematical curiosity, regardless of background. This is joint work with Luke Oeding.

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