# CMU Algebra and Combinatorics Seminar

## Spring 2016

### Meeting Times

Wednesdays, 4–5pm, in Pearce 203.

### Abstracts

Speaker: George Grossman (Central Michigan University)
Title: Combinatorial Identities with Fibonacci and Lucas Numbers
Abstract: In this talk we explore aspects of combinatorial identities. In particular, it is shown that each Fibonacci number, up to sign change, can be represented countably many distinct ways as sum of binomial coefficients. The talk is accessible to graduate and undergraduate students.

Speaker: Adam Chapman (Michigan State University)
Title: Linkage of $p$-algebras of prime degree
Abstract: Quaternion algebras contain quadratic field extensions of the center. Given two algebras, a natural question to ask is whether they share a common field extension. This gives us an idea of how closely related those algebras are to one another. If the center is of characteristic 2 then those extensions divide into two types – the separable type and the inseparable type. It is known that if two quaternion algebras share an inseparable field extension then they also share a separable field extension and that the converse is not true. We shall discuss this fact and its generalization to $p$-algebras of arbitrary prime degree.

Speaker: Li Li (Oakland University)
Title: Combinatorics of Bases of Cluster Algebras
Abstract: Cluster Algebra is a new branch in mathematics which grows rapidly and has far-reaching implications in many fields including representation theory, geometry, combinatorics, mirror symmetry of string theory, statistical physics, etc. Lots of research of cluster algebras focuses on construction of their natural bases. Various combinatorial models are discovered in the study of bases, including snake diagrams and perfect matching, Dyck paths and compatible pairs, and in particular, a recent surprising construction of theta bases by Gross, Hacking, Keel and Kontsevich using techniques (scattering diagrams, broken lines) developed in the study of mirror symmetry of string theory. In this talk, I will discuss these combinatorial models and give explicit combinatorial enumeration of broken lines for some special classes of cluster algebras.

Speaker: Jake Levinson (University of Michigan)
Title: (Real) Schubert Calculus from Marked Points on $\mathbb{P}^1$
Abstract: I will describe a family of Schubert problems on the Grassmannian, defined using tangent flags to points of $\mathbb{P}^1$ (or more generally, a stable curve) in its Veronese embedding. For Schubert problems having a finite set of solutions, the Shapiro-Shapiro Conjecture (later proven by Mukhin-Tarasov-Varchenko) proposed that, when the marked points are all real, the solutions would be "as real as possible." More recently, Speyer gave a remarkable description of the real topology of the family in terms of Young tableaux and Schützenberger's jeu de taquin. I will give analogous results on the topology of one-dimensional Schubert problems (where the family consists of curves). In this case the combinatorics involves orbits of operations related to promotion and evacuation of Young tableaux.

Speaker: Rachel Karpman (University of Michigan)
Title: Total Positivity and Planar Networks
Abstract: A matrix is totally nonnnegative if all of its minors are nonnegative real numbers. The Lindström Lemma gives a remarkable relationship between totally nonnegative matrices and planar networks. Given a weighted planar directed network which is sufficiently "nice," we may construct a corresponding totally nonnegative matrix, and every totally nonnegative matrix arises in this way. The concept of total nonnegativity extends naturally to the Grassmannian Gr$(k,n)$, the algebraic variety whose points correspond to $k$-dimensional subspaces of a fixed $n$-dimensional complex vector space. Postnikov introduced a family of coordinate charts on the totally nonnegative Grassmannian, defined in terms of weighted planar networks; this may be viewed as a Grassmannian analogue of the Lindström Lemma. In this talk, we discuss the results of Lindström and Postnikov, and extend Postnikov's theory to the Lagrangian Grassmannian, a variety whose points correspond to isotropic subspaces of a fixed vector space with respect to a symplectic form.

Speaker: Amrita Acharyya (University of Toledo)
Title: Cofinite Graphs and their Profinite Completions
Abstract: We generalize the idea of cofinite groups, due to B. Hartley. The idea of constructing a cofinite graph starts with defining a uniform topological graph $\Gamma$ in an appropriate fashion. We endow abstract graphs with uniformities corresponding to separating filter bases of equivalence classes over $\Gamma$. It is established that for any cofinite graph there exists a unique cofinite completion. Our immediate next concern is developing group actions on cofinite graphs. Defining the action of an abstract group over a cofinite graph in the most natural way we are able to characterize a unique way of uniformizing an abstract group with a cofinite structure, obtained from the cofinite structure of the graph in the underlying action, so that the afore said action becomes uniformly continuous. This is a joint work with J. M. Corson and B. Das.

Speaker: Rajesh Kulkarni (Michigan State University)
Title: Ulrich bundles on varieties and related questions in Brauer groups
Abstract: We will discuss some classical questions in Brauer groups such as cyclicity of division algebras. We will then discuss how this is related to so the called Kummer subspaces of central simple algebras and then to Ulrich bundles on varieties. Time permitting, we will discuss recent developments in Ulrich bundles.

Speaker: David Clark (Grand Valley State University)
Title: Anti-games on affine geometries
Abstract: SET is a mathematical card game in which players compete to find collections of cards, called "sets", with special properties. The underlying mathematical structure of the game is a ternary affine geometry, which is itself beautiful and has been extensively studied. In this talk, we explore a variant of SET called Anti-SET that is inspired by the concept of "blocking sets" in finite geometries. In Anti-SET, the objective of SET is reversed: Two players take turns selecting cards from the SET deck into their hands, and the first player to hold a "set" loses the game. We will demonstrate a winning strategy that extends to $n$-dimensional affine geometries and which has links to combinatorial designs. In turn, the losing player can take advantage of substructures within affine geometries in order to put off their inevitable loss for as long as possible. We will also discuss ongoing work on the broader class of "anti-games", in which the story is not nearly so neat and tidy – and yet, these games provide excellent opportunities for undergraduate research.