Algebra & Discrete Mathematics Seminar
Location: Minard 308
Time: Tuesday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă
3 November 2015
Trevor McGuire: Gröbner Bases, Part II
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
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
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
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
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
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
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.