Abstract: From any modular category (or braided fusion category, more generally) one obtains sequences of representations of the braid groups. We will explore the nature of these representations and describe several conjectures relating the complexity of these representations to a very basic quantity depending only on the tensor product structure of the category: the Frobenius-Perron dimension.

Abstract: For an adjunction having codomain category  C  in the left adjoint, there is an associated notion of a monad giving rise to a category of algebras called Eilenburg-Moore algebras. In the special case that  C  is equivalent to this category of algebras, the adjunction is said to be monadic. In this talk we will discuss a base category over which the category of quasi-ordered abelian groups will be monadic.

Abstract: The theory of quasigroups has a duality (left/right) symmetry, and linear quasigroups are equivalent to representations of 2-generated groups. Recently, the concept of a hyperquasigroup was introduced to provide a more symmetrical version of quasigroup theory, carrying a triality symmetry. In this talk, linear hyperquasigroups are shown to be equivalent to general group representations.

Abstract: We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces combinatorial cell complexes (or c.c.c). We define orientability, homology and cohomology of c.c.c's, and develop enough algebraic topology in this setting to prove the Poincaré duality theorem for a c.c.c satisfying suitable regularity conditions. The definitions and proofs are completely finitary and combinatorial in nature.

Abstract: Vertex-identifying codes were originally introduced in 1998 as a way of detecting faults in multiprocessor computer systems. The main idea is that given a graph, we wish to be able to identify each vertex by only communicating with a subset of the vertices of the graph. An important problem is to find codes of minimum density. We will examine this problem on various infinite graphs and explore a connection between vertex-identifying codes and Hamming codes.

Abstract: Modes are idempotent and entropic algebras. Modals are both join semilattices and modes, where the mode structure distributes over the join. Barycentric algebras are equipped with binary operations from the open unit interval, satisfying idempotence, skew-commutativity, and skew-associativity. The talk aims to give a brief survey of these structures and some of their applications.

Abstract: In the theory of abelian groups it is often necessary, or even natural, to assume the existence of a partial ordering that "behaves well" with respect to the algebra operation. The downside to this is that some of the constructions available under normal circumstances, such as free algebras, are not available for use. In an attempt to circumvent this we introduce a new way of expressing a partially ordered abelian group in terms of a new category that captures the structure of the algebra and that of the partial ordering. Examples for creating the aforementioned free algebras and lexicographic sums will also be discussed.

Abstract: Embedding modes (idempotent and entropic algebras) as subreducts into semimodules over commutative semirings is one of main methods of representing modes. However, recent results of M. Stronkowski show that not all modes embed into such semimodules. Related to the problem of embeddability is the problem of constructing a semiring defining the variety of semimodules such that the corresponding subreducts belong to a given variety of modes and form the class of embeddable modes of this variety. We will provide a general construction of such semirings, describe some of their properties and also describe concrete semirings for selected varieties of modes.

Abstract: The theory of modular forms plays a central role in modern number theory and has many important applications. It is widely believed that a modular form with algebraic coefficients is for a congruence subgroup of the modular group if and only if its coefficients have bounded denominators. In this talk we will discuss how to approach this conjecture by using other properties of noncongruence modular forms. This is joint work with Wen-Ching Winnie Li.