# Algebra and Combinatorics Seminar, Spring 2018

### Wednesdays, 3:45–4:35. Cuneo 218

Webpages from previous semesters

Organizers:  Emily Peters, Peter Tingley

 February 14 Tony Giaquinto (Loyola) Deformations to Twists and Back Again 21 Aaron Lauve (Loyola) Too Many Hopf Algebras! 28 No talk March 7 Spring break!!!!! 14 Alexander Sistko Maximal Subalgebras: Classifications, Applications, and Questions 21 No seminar due to Rataj Colloquium 28 Marius Radulescu (Loyola) Classification of Algebraic Vector Bundles Over an Elliptic Curve April 4 Canceled 11 Ryan Vitale (Indiana) Planar Algebra Presentations for a Family of Tensor Categories 18 Yajnaseni Dutta (Northwestern) Fujita's conjecture: An effective way to map projective manifolds into projective spaces 25 Sebastian Olano (Northwestern) Stringy Hodge numbers

Directions

IES (the institute for environmental sustainability) is located at the corner of W. Sheridan and N. Kenmore Avenues, Chicago, IL (map)

Parking is available on-campus for $7 in the Parking Garage (building P1 on the Lake shore campus map). To get to the Parking Garage, enter campus at the corner of West Sheridan Road and North Kenmore Avenue. Abstracts February 14: Tony Giaquinto (Loyola) Deformations to Twists and Back Again This talk will trace a path from a case by case approach to constructing algebraic deformations to a uniform method called twisting. Numerous classical and state of the art examples will be given to illustrate the theory. February 21: Aaron Lauve (Loyola) Too Many Hopf Algebras! Since Giancarlo Rota's groundbreaking work on the subject in 1979, one endeavor of Algebraic Combinatorists has been to use Hopf algebras to study combinatorial structure. The viewpoint is this: if your favorite combinatorial species gadgets carry with them natural ways to combine two---and break into two---of the same species, then you should treat this as a notion of product and coproduct. Since that time, we have studied the Hopf algebras of planar binary trees, Dyck paths, parking functions, Feynman graphs, permutations, rooted forests, posets, and more, (many with both (co)commutative and non(co)commutative versions). What does this perspective buy you? Formulas! I'll give some examples. In this talk, reporting on joint work with Mitja Mastnak, we take a step back and ask for some way of organizing this menagerie of exotic beasts. We introduce the notion of Hopf algebra coverings and highlight the transfer of structure they provide. March 14: Alexander Sistko (Iowa) Maximal Subalgebras: Classifications, Applications, and Questions A common trick in the study of (not-necessarily associative) algebras is to understand their structure through the structure of their maximal subalgebras. Although certain classes of associative algebras have been studied from this vantage point, no complete classification for their maximal subalgebras currently exists. In this talk, I will present classification theorems for maximal subalgebras of finite-dimensional associative algebras over a field. For important classes of algebras, this gives us nice presentations of subalgebras. Trivial and separable extensions feature prominently in our classification, and allow us to connect our (essentially ring-theoretic) classification problem to representation theory. We discuss how this classification yields insights into other questions related to finite-dimensional algebras, for instance: counting minimal generating sets of algebras; constructing representations of their automorphism groups; and understanding isomorphism classes of subalgebras. March 28: Marius Radulescu (Loyola) Classification of Algebraic Vector Bundles Over an Elliptic Curve In a famous article, "Vector Bundles over an Elliptic Curve," M. F. Atiyah classified indecomposable vector bundles of given rank and degree by identifying the set of their equivalence classes with the Jacobian variety of the elliptic curve, which is the curve itself. Atiyah's main result will be used for classifying indecomposable rank 2, 3 and 4 vector bundles over an elliptic curve. April 11: Ryan Vitale (Indiana) Planar Algebra Presentations for a Family of Tensor Categories We give a family of planar algebra presentations for subcategories of representations of$\mathbb{Z}$and$\mathbb{F}_p\$. These are similar in flavor to examples of small index subfactors, but differs slightly, for example the categories are not semisimple. There are some connections to the fundamental theorems of invariant theory; using planar algebra techniques we can compute generators and relations for the ring of invariants corresponding to the representation assigned to a strand in the planar algebra.

April 18: Yajnaseni Dutta Fujita's conjecture: An effective way to map projective manifolds into projective spaces.

One of the most natural questions in algebraic geometry is whether there are effective ways to embed a smooth projective variety into a projective space using its volume forms. Fujita in 1985 conjectured that such conditions should only depend on the dimension of the variety. Despite many important breakthroughs by Demailly, Reider, Ein, Lazarsfeld, Kawamata, Helmke et al., the conjecture remains unproved as of today. We will walk through this history and discuss some recent developments towards the relative analog of Fujita's conjecture.

April 25: Sebastian Olando Stringy Hodge numbers

Stringy Hodge numbers are a generalization of the usual Hodge numbers of a smooth projective variety. Batyrev defined them for projective varieties with mild singularities, and conjectured they are nonnegative. The nonnegativity of these numbers is a numerical constraint of the exceptional divisors in a resolution of singularities. We will discuss how to interpret it, and give a positive answer to Batyrev's conjecture in some cases.