Algebra & Discrete Mathematics Seminar
Fall 2015 Schedule

Location: Minard 308
Time: Tuesday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă

Previous Semesters

3 November 2015

Trevor McGuire: Gröbner Bases, Part II

Abstract: In this talk, we will cover the main points of a paper of Boffi and Logar (2007) in which they introduce a technique for computing   Gröbner bases without using the Buchberger algorithm. This technique is entirely combinatorial, making use of the bijection between pure saturated binomial ideals and vectors in the integer lattice. Time permitting, at the end of the presentation, we will cover a further generalization in which we can remove the integer lattice requirement, and thus define a  Gröbner basis of a poset under some conditions. An algebraic interpretation of this will also be provided.

27 October 2015

Trevor McGuire: Gröbner Bases, Part I

Abstract: In this talk, we will begin only with the idea of an ideal in the ring of polynomials in n variables over a field of characteristic 0. We will build up the terminology needed to define a  Gröbner basis of a given ideal. Topics covered will be term orders, a multivariable division algorithm, and the Buchberger algorithm. This is an expository presentation that is suitable for graduate and undergraduate students.

13 October 2015

Emily Gunawan (University of Minnesota): Cluster algebras from triangulations of surfaces

Abstract: The notion of cluster algebra, introduced by Fomin and Zelevinsky in 2000, links together diverse fields of study, e.g. discrete dynamical systems, Riemann surfaces, representation theory of quivers, knot theory, etc.

Cluster algebras are commutative algebras which are generated by a distinguished set of (usually infinitely many) generators, called cluster variables. Starting from a finite set $$x_1, x_2, \ldots, x_n$$, the cluster variables can be computed by an iterated elementary process. They miraculously turn out to always be Laurent polynomials in $$x_1, x_2,\ldots, x_n$$, with positive coefficients. Finding a closed-form formula for the cluster variables is one of the main problems in the theory of cluster algebra.

In this talk, I will discuss such a closed-form formula for the class of cluster algebras which can be modeled after triangulations of orientable Riemann surfaces with marked points (my running examples will be a pentagon and an annulus). The formula is given in terms of paths (called T-paths) along the edges of a fixed triangulation. The T-paths can be used to give a combinatorial proof for a natural basis (consisting of elements which are indecomposable and positive in some sense) for some types of cluster algebras.

29 September 2015

Jessica Striker: Partition and plane partition promotion and rowmotion

Abstract: In this talk, we discuss promotion and rowmotion actions on partitions and plane partitions, along with their bijective and dynamical properties. We give a previous result on the cyclic nature of rowmotion on partitions inside a box or staircase. We present a recent analog of this result, relating actions on plane partitions and increasing tableaux and exploring a new dynamical pseudo-periodicity phenomenon we call resonance. We welcome comments as to how this work may relate to monomial ideals of two and three variables.

15 September 2015

Nursel Erey: Resolutions of Monomial Ideals

Abstract: The problem to construct the minimal free resolution of a monomial ideal was raised in 1960s. While this problem remains open, much work has been done to understand invariants arising from minimal free resolutions. Thanks to polarization method, one can restrict to squarefree monomial ideals for the purpose of studying minimal free resolutions of monomial ideals.

In this talk, I will discuss the relation of squarefree monomial ideals to combinatorics. In particular, I will explain some combinatorial properties of simplicial complexes which can be used to describe the graded Betti numbers of associated ideals.

8 September 2015

Kevin Dilks: Horn's Conjecture and Related Mathematics

Abstract: A natural question in linear algebra one can ask is, if we only know the eigenvalues for two Hermitian matrices $$A$$ and $$B$$, then what possible eigenvalues can $$A+B$$ have? Trace gives us one equality that has to hold, and the min-max principle gives us some inequalities that have to be satisfied. Horn conjectured the trace condition and a certain finite set of inequalities were both necessary and sufficient, and gave a recursive formula for constructing the inequalities in arbitrarily high dimension. Horn's conjecture was later proven to be true, and it was shown that the inequalities that arise are very closely related to the structure constants that arise in the Grassmannian, symmetric functions, and representation theory of the symmetric group.

In this talk, we will go over the history of Horn's conjecture, describe its connection to the Grassmannian, and discuss how it comes up in other areas of mathematics.