Algebra & Discrete Mathematics Seminar
Fall 2015 Schedule

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

Previous Semesters

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.