Algebra Seminar

Speaker: TBA

Title: TBA

Speaker:  Dr. Mihai D. Staic, BGSU

Title: Generalizations of the determinant map.  

Speaker:  Dr. Anibal Medina-Mardones, MPIM

Title: Effective constructions in algebraic topology and applications

Abstract: There is some tension between functoriality and constructibility in algebraic topology reaching back to its origins. For example, the cohomology of a space X could be described in terms of homotopy classes of maps from X to certain spaces, or via a cochain complex generated by a combinatorial representation of X. In this survey talk we will discuss effective constructions of finer structure enhancing the cohomology of X in combinatorial terms. This so-called E-infinity structure and associated cochain level operations on X, have become important for new applications of cohomology in the study of, for example, topological QFTs and persistent homology.

Speaker:  Dr. Van Nguyen, United States Naval Academy

Title: Quantum symmetries through the lens of linear algebra

Abstract: The McKay matrix M_V records the result of tensoring the simple modules with a finite-dimensional module V. In the case of finite groups, the eigenvectors for M_V are the columns of the character table, and the eigenvalues come from evaluating the character of V on conjugacy class representatives.

In this talk, we will explore what can be said about such eigenvectors when the McKay matrix is determined by modules over an arbitrary finite-dimensional Hopf algebra H. Here, the McKay matrix M_V encodes quantum symmetries coming from the actions of H. We illustrate these results for the small quantum group u_q(sl_2), where q is a root of unity (and generally for the Drinfeld double D_n of the Taft algebra). In these examples, the eigenvalues and eigenvectors for these matrices can be described in terms of several kinds of Chebyshev polynomials.

Speaker:  Dr. David Fernandez, University of Luxembourg

Title: Noncommutative Poisson geometry and pre-Calabi-Yau algebras

Abstract: A long-standing problem in Poisson geometry has been to define appropriate "noncommutative Poisson structures". To solve it, M. Van den Bergh introduced double Poisson algebras and double quasi-Poisson algebras that can be regarded as noncommutative analogues of usual Poisson manifolds and quasi-Poisson manifolds, respectively. Recently, N. Iyudu and M. Kontsevich found an insightful correspondence between double Poisson algebras and pre-Calabi-Yau algebras; certain cyclic A_infty-algebras that can be seen as noncommutative versions of shifted Poisson manifolds. In this talk, I will present an extension of Iyudu-Kontsevich’s correspondence to the differential graded setting. Moreover, I will explain how double quasi-Poisson algebras give rise to pre-Calabi-Yau algebras. Interestingly, they involve an infinite number of nonvanishing higher multiplications weighted by the Bernoulli numbers. This is a joint work with E. Herscovich (Grenoble).

Speaker:  Dr. Miodrag Iovanov, The University of Iowa

Rescheduled for the Fall semester

Speaker: Dr. Robin Stoll, Stockholm University

Title: Modular operads as modules over the Brauer properad

Abstract: Both modular operads and properads are variants of operads encoding certain kinds of operations. After recalling these notions and illustrating why one might care about them, I will explain how modular operads can be seen as modules over a certain simple properad, called the Brauer properad. Moreover, I will recall the Feynman transform, which is an analogue for modular operads of the (co)bar construction of an operad, and explain how in this setting it arises as a special case of the (co)bar construction of a module over a properad. Together, this yields definitions of both modular operads and the Feynman transform that are quite easy to work with. If time permits, I will also say a few words about Koszul duality for modular operads.

Speaker: Dr. Hiba Fayoumi, The University of Toledo

Title: Groupoids, Logic Algebras and Public Key Cryptography

Abstract: A Groupoid is just a set X with a binary operation *. Let Bin(X) be the collection of all groupoids (binary systems) on X, then (Bin(X), ◇) is a semigroup, where ◇ is an associative groupoid product such that (X, ∗)◇(X, •) = (X, ✷);  x✷y=(x∗y)•(y∗x) for all x, y in X. Groupoid decomposition or factorization can have a lot of application in various fields. I will give a brief overview of the substructures of Bin(X) then I will survey my recent two results about groupoid factorization in Bin(X) and their possible application to public key cryptography. The first result relates different notions of groupoids with logic algebras such as BCK/BCI/Q and d-algebras. The second one discusses a PKE scheme that computes a secret key using the first result on groupoid factorization, which is important in the theory of information. If time permits, I can briefly discuss the process we used to achieve our results.

Speaker: Dr. Yevgenia Kashina, DePaul University

Title: TBA