Algebra & Discrete Mathematics Seminar
Location: Minard 308
Time: Tuesday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă
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
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.