Algebra and Number Theory Seminar
Organized by Lilit Martirosyan and Michael Freeze
Department of Mathematics & Statistics, UNCW
Organized by Lilit Martirosyan and Michael Freeze
Department of Mathematics & Statistics, UNCW
The UNCW Algebra & Number Theory Seminar brings together researchers and students interested in algebra, number theory, representation theory, quantum algebra, and related areas. The seminar provides a forum for research talks, mathematical exchange, and interaction among faculty, students, and visiting scholars.
The seminar has been founded and organized by Lilit Martirosyan since 2018. And, beginning in Fall 2026, is co-organized by Lilit Martirosyan and Michael Freeze.
Abstract: TBD
Cosponsored by Interdisciplinary Math-Physics Seminar
Abstract: TBD
Cosponsored by Department Colloquium
Abstract: TBD
Cosponsored by Joint Department Colloquium and Interdisciplinary Math-Physics Seminar
Abstract: TBD
Cosponsored by Interdisciplinary Math-Physics Seminar
Spring 2026 Semester
Abstract: If p is a prime written in base 10 as digits dᵣ, dᵣ₋₁, ..., d₀, then the polynomial f(x) = dᵣxʳ + ... + d₁x + d₀ is irreducible over the integers. This result, due to Arthur Cohn, has been generalized in a number of directions. In particular, we investigate what happens when the coefficients are not restricted to digits and the base is arbitrary. For example, if f(x) is as above with r ≤ 31, the coefficients dⱼ are simply nonnegative, and the value f(10) is prime, then f(x) is irreducible. We will survey some recent work on these generalizations, carried out with several of my former students: Morgan Cole, Scott Dunn, Joseph Foster, Sam Gross, Jacob Juillerat, and Jeremiah Southwick.
Cosponsored by Department Colloquium and Algebra & Number Theory Seminar
Abstract: Tensor categories provide a powerful framework for understanding symmetry beyond classical group theory, with important applications in representation theory, quantum groups, and mathematical physics. In this talk, we begin with a brief conceptual introduction to tensor categories and motivating examples. We then turn to the problem of classifying tensor categories with prescribed structural properties, focusing on the exceptional Lie type G₂. We discuss recent results in both the non-symmetric and symmetric ribbon settings, and explain how structural constraints and braid-theoretic methods lead to rigidity and reconstruction phenomena. In particular, these results show that such categories are highly constrained and, in the non-symmetric case, arise from standard quantum group constructions.
Abstract: Tambara-Yamagami (TY) categories form one of the simplest families of non-pointed fusion categories. Despite their elementary description, these categories play a central role in mathematical physics. Studying extensions of TY fusion rings can provide a controlled framework to explore the new phenomena that arise when one moves beyond the classical TY setting. In this talk, I will focus on two extensions of the TY fusion ring, as studied by Jordan-Larson and Galindo-Lentner-Moller. One of the most useful tools to study fusion rings is the classification of their non-negative integer matrix (NIM-) representations. NIM-reps can be used as an effective method to detect algebra objects in the fusion category underlying the fusion ring. In this talk, we will compute and classify the irreducible NIM-reps of both proposed extensions as well as detect the algebra objects associated to these NIM-reps. This work is joint with Agustina Czenky, Emily McGovern, Monique Muller, and Ana Ros Camacho.
Fall 2025 Semester
Abstract: The so-called von Neumann algebras are well-known concepts in the area of mathematical physics. They can be seen non-commutative versions of measure spaces due to the fact the abelian von Neumann algebras can be represented as a space of functions on a measure space. Also their classification is a big achievement in this research area. More recently, there has also been work on non-commutative Lp spaces that are constructed via von Neumann algebras. We will discuss those constructions and will show that they also give rise to rich structures even though there is not a notion of points but only of functions.
Abstract: Correlation functions for modules of vertex operator algebras (VOAs) have long been known to satisfy nice modular transformation properties. A famous example of this is given by the Moonshine module, a VOA whose graded dimension is a modular function. Zhu proved that, more generally, correlation functions for modules of C_2-cofinite rational VOAs satisfy nice modular transformation properties. In this talk, we will discuss modular transformation properties of correlation functions with zero modes inserted. No prior knowledge of VOAs will be assumed. This talk is based on joint work with Christoph A. Keller.
Abstract: Motivated by the special case of the Monstrous Moonshine example G= M, V=V^♮we define Hecke-Adams operators on suitable pairs of a group G and graded module V ∈ R(G)[q] with modular Thompson series characters. These operators form a Hecke algebra. This is a work in progress.
Abstract: A fiber functor is an embedding of a tensor category into vector spaces. For representation categories of groups or Hopf algebras, the canonical forgetful functors are the main examples, but truly exotic examples are hard to come by. In this talk we will introduce a family of exotic fiber functors on the category of representations of SLn in positive characteristic, arising from combinatorial objects called triangle presentations associated to Bruhat-Tits buildings.
Abstract: Vertex operator algebras (VOAs) are central objects in modern mathematics and mathematical physics, with connections to number theory, representation theory, and conformal field theory. In this lecture, I will introduce VOAs through Isaac Newton’s method of forward differences, showing how this elementary tool provides a natural doorway into the formal calculus underlying vertex algebras. Along the way, I will highlight how Newton’s perspective unexpectedly echoes modern algebraic structures, and how these ideas touch on themes that appear in current research. The talk will be accessible to graduate students and advanced undergraduates, with no prior background in VOAs assumed.
The Spring 2024 Algebra & Number Theory Seminar was organized and led by Lilit Martirosyan as a sequence of interconnected, accessible lectures for students and faculty. With no dedicated funding for visiting speakers that semester, the series focused on building background in algebra, representation theory, and number theory through lectures ranging from classical results to modern ideas. Talks typically drew approximately 10–25 participants, and lecture notes were circulated to attendees following the seminars.
Abstract: An introduction to cyclotomic polynomials and several classical results concerning polynomial irreducibility. The talk discusses foundational ideas and results associated with Gauss, Kronecker, Schönemann, Eisenstein, Dedekind, Landau, and Schur, illustrating the historical development of techniques that remain important in algebra and number theory.
Prerequisites: Basic linear algebra; some familiarity with abstract algebra is helpful.
Abstract: A historical and mathematical introduction to the development of representation theory, beginning with Dedekind’s work and continuing through Frobenius’ theorem, the theory of characters of finite-group representations, Frobenius reciprocity, and the emergence of Young diagrams. The talk explains how these classical ideas became foundational tools in modern representation theory.
Prerequisites: Basic linear algebra; some familiarity with abstract algebra is helpful.
Abstract: A continuation of the discussion of the development of representation theory from Dedekind’s work to Frobenius’ theory of finite-group representations. Topics include Frobenius’ theorem, characters of finite-group representations, Frobenius reciprocity, and the emergence of Young diagrams. The talk highlights how these classical ideas became fundamental tools in modern representation theory.
Prerequisites: Basic linear algebra; some familiarity with abstract algebra is helpful.
Abstract: An introduction to characters of representations of symmetric groups, induced and restricted modules, Frobenius reciprocity, and Young diagrams. These ideas provide some of the basic tools for understanding the representation theory of symmetric groups and prepare the way for the combinatorial classification of their irreducible representations.
Prerequisites: Basic linear algebra; some familiarity with abstract algebra is helpful.
Abstract: An introduction to Young diagrams, tableaux, and Young symmetrizers, and how these combinatorial constructions are used to describe the irreducible representations of symmetric groups. The talk develops one of the fundamental classification results in the representation theory of finite groups.
Prerequisites: Basic linear algebra; some familiarity with abstract algebra is helpful.
Abstract: A continuation of the study of Young diagrams and representations of symmetric groups. Topics include the proof of the main classification theorem, hook formulas, the Frobenius character formula, and an introduction to Schur–Weyl duality connecting symmetric-group representations with matrix groups and Lie algebras.
Prerequisites: Basic linear algebra; some familiarity with abstract algebra is helpful
Abstract: An introduction to Schur–Weyl duality, the remarkable connection between representations of symmetric groups and representations of matrix groups and Lie algebras. The talk also introduces the Schur functor through examples and discusses how these ideas lead to several directions in modern representation theory.
Prerequisites: Basic linear algebra; some familiarity with abstract algebra is helpful.
Abstract: An introduction to the basic ideas of category theory, an important language and framework used throughout modern mathematics. The talk introduces fundamental concepts and illustrates how categorical ideas provide a unified way to describe mathematical structures and the relationships between them.
2018-2023 - Earlier talks
These talks predate the current Algebra & Number Theory Seminar. During this period, we primarily supported departmental colloquium talks; the dedicated Algebra & Number Theory Seminar developed later under its current name and format.
Abstract: In this expository talk, I will first show a glimpse on quantum field theory (QFT) and its mathematical formalism. I will then have a closer look at the simplest type of QFT, namely 0-dimensional scalar QFT (aka combinatorial QFT). In the last part of the talk, I will exhibit how these general ideas generalize to the case of random matrix and tensor models (seen as 0-dimensional QFT models).
Abstract: A convenient way to construct infinite dimensional algebras is via paths. This is also useful to describe combinatorial information such as dimensions and multiplicities of representations of groups. More recently, they have also been found useful to study generalizations of group representations, called tensor categories.
Abstract: The Euler constant is one of the basic constants in mathematics. In my talk I will focus on analogues that arise on studying the finer asymptotic behavior of certain counting functions. One such analogue can for example be used to disprove a conjecture Ramanujan made in his very first letter to Hardy on the counting function of the number of integers <=x that can be written as a sum of two squares when x becomes large. I will also discuss my disproof (joint work with Kevin Ford and Florian Luca) of a conjecture of Yasutaka Ihara on the positivity of the Euler constant for cyclotomic number fields.
Abstract: Through the examples of edge pairings in a polygon (higher Catalan numbers), branched coverings of the Riemann sphere (Hurwitz theory), and Mirzakhani-type identities in hyperbolic geometry of surfaces, I will show that implementing the idea of cutting/gluing can lead to an inductive solution for several problems of enumeration of surfaces. This induction decreases the “complexity” of the surface (measured by its Euler characteristic), and always takes the same form, with only difference lying in the initial conditions. I will also try to give a brief overview of the history and the recent applications of this universal structure, which are closely related to physics (random matrices, 2d quantum gravity, and topological field theories). We will introduce along the way the few notions needed from geometry so that no prior knowledge is assumed.
Abstract: Let G act via matrices on a vector space V. It is a classical problem to determine how the action of G on the n-
fold tensor product of V decomposes into irreducible representations. This is encoded by its centralizer algebra, i.e. the algebra of linear operators commuting with the group action. In more recent times, this study was extended to centralizer algebras of q-deformations of Lie algebras, called quantum groups, motivated by new results in topology, mathematical physics and tensor categories. This provided more information even for the Lie group cases. We will first review the classical results for general linear and orthogonal groups. We then talk about more recent results involving spinor representations and exceptional Lie algebras.