CMU Algebra and Combinatorics Seminar

Spring 2015

Organizers

Meeting Times

Wednesdays, 4–5pm, in Pearce 126.

Webpages from previous semesters:

Schedule

Date Speaker Affiliation Title
1/21 Yi Su University of Michigan Electrical Networks and Electrical Lie Theory
1/28 Sivaram Narayan Central Michigan University When Leading Imply All, Mixed Matrices, and Koteljanskii Inequalities
2/4 Pete Vermeire Central Michigan University Betti Tables of Reducible Algebraic Curves
2/18 Peter Tingley Loyola University of Chicago Crystals, PBW bases, and related combinatorics
3/25 Ryan Kaliszewski Drexel University Combinatorial Fillings and a Proof of the Hook-Length Formula
4/1 C.-Y. Jean Chan Central Michigan University Dimension, vector bundles, Euler characteristic and etc. in algebraic geoemtry
**4/3** William Fulton University of Michigan Degeneracy Loci
4/15 Sidney Graham Central Michigan University Recent Results on Bounded Gaps Between Primes–A Survey
4/22 Joshua Hallam Michigan State University Applications of Quotient Posets
4/29 Carolina Benedetti Michigan State University TBD

Abstracts

Speaker: Yi Su (University of Michigan)
Title: Electrical Networks and Electrical Lie Theory
Abstract: Curtis-Ingerman-Morrow studied the space of circular planar electrical networks. Lam and Pylyavskyy introduced an electrical Lie algebra such that the positive part of the corresponding electrical Lie group $E_{A_{2n}}$ acts on circular planar electrical networks via some natural combinatorial operations. Inspired by these operations Lam gave a cell decomposition of the space of electrical networks, which is the analogue of type A Bruhat decomposition. In this talk, I will introduce mirror symmetric circular planar electrical networks, and present some analogous results about this new class of networks and its relation to type B electrical Lie theory. If time permitting, I will talk about the structure of electrical Lie algebras of type C and D.

Speaker: Sivaram Narayan (Central Michigan University)
Title: When Leading Imply All, Mixed Matrices, and Koteljanskii Inequalities
Abstract: An $n$-by-$n$ real matrix $A$ enjoys the leading implies all (LIA) property, if, whenever $D$ is a diagonal matrix such that $A+D$ has positive leading principal minors, then all principal minors of $A$ are positive. Symmetric and $Z$-matrices are known to have this property. We discuss a new class of matrices called mixed matrices that both unifies and generalizes these two classes and their special diagonal equivalences by also having the LIA property. Nested implies all (NIA) property is also enjoyed by this new class. We also give an inductive characterization of the LIA property.

It is natural to ask what other properties of M-matrices and positive definite matrices are enjoyed by mixed matrices as well. We show that mixed P-matrices satisfy a broad family of determinantal inequalities, the Koteljanskii inequalities, previously known for those two classes. In the process, other properties of mixed matrices are developed, and consequences of the Koteljanskii inequalities are given.

This is a joint work with Charles R. Johnson.

Speaker: Pete Vermeire (Central Michigan University)
Title: Betti Tables of Reducible Algebraic Curves
Abstract: We will discuss the problem of finding free resolutions of the ideal of a projective variety. In particular, we will discuss the case where the variety in question is the union of lines which can be deformed to a smooth curve. This allows us to attempt a purely combinatorial description of the generators and syzygies of the ideal in terms of the dual graph of the configuration of lines. This was part of an REU project from two summers ago.

Speaker: Peter Tingley (Loyola University of Chicago)
Title: Crystals, PBW bases, and related combinatorics
Abstract: Kashiwara's crystals are combinatorial objects used to study complex simple Lie groups and their highest weight representations. A "crystal" consists of an underlying set, which roughly indexes a basis for a representation, along with combinatorial operations related to the Chevalley generators. Here we are interested in types A and D, where the crystals are realized using Young tableaux and Kashiwara-Nakashima tableaux respectively. However the combinatorics is not always enough, and one would like to understand an actual basis corresponding to the crystal. This can be done using Lusztig's theory of PBW bases, and so there is a unique structure-preserving bijection between the PBW basis and the combinatorially constructed crystal. We describe that bijection explicitly. In type A the answer was to some extent known to experts, but in type D we believe it is new. I will explain as much of this as I can without assuming any prior knowledge of crystals or PBW bases, mainly through examples. This is joint work with John Claxton, Ben Salisbury and Adam Shultze.

Speaker: Ryan Kaliszewski (Drexel University)
Title: Combinatorial Fillings and a Proof of the Hook-Length Formula
Abstract: Inspired by the long-standing open problem to combinatorially characterize the Macdonald symmetric functions, Austin Roberts recently introduced a new combinatorial structure and proved that the Lascoux-Schutzenberger charge/ tableaux formulation for the $q=0$ Macdonald polynomials can instead be written using these combinatorial objects with the major index statistic. We have discovered that a variation on these new objects newly characterize Schur and Grothendieck polynomials (representatives for cohomology and $K$-theory classes, respectively), and lends itself to an alternate view on classical combinatorial formulas such as the Littlewood-Richardson rule and the hook-length formula for standard Young tableaux.

Speaker: C.-Y. Jean Chan (Central Michigan University)
Title: Dimension, vector bundles, Euler characteristic and etc. in algebraic geoemtry
Abstract: In this talk, we will define some terms that are commonly used in algebraic geometry such as those listed in the title. The approach is aimed at providing intuitions from both algebraic and geometric viewpoints whenever possible. Some background in the ring theory would be helpful but I will keep the assumption to the minimum. This seminar will be more or less a continuation of the Graduate Student Seminar on Tuesday, March 31, and hopefully serve as a preparation for Professor Fulton's seminar lecture on Friday, April 3, noon – 1 pm. It should be noted, however, those who are unable to attend all lectures, should not be discouraged to attend their continuations. Fulton is in the building and we are in for a treat !!!

Speaker: William Fulton (University of Michigan)
Title: Degeneracy Loci
NOTE THE SPECIAL DAY AND TIME!!!
Abstract: This talk will be a continuation of the colloquium, concentrating on the problem of finding formulas for degeneracy loci, and the combinatorics that has grown up to solve this problem.

Speaker: Sidney Graham (Central Michigan University)
Title: Recent Results on Bounded Gaps Between Primes–A Survey
Abstract: I will discuss recent work of Zhang, Maynard, and the Polymath Group on bounded gaps between primes. Many of you saw a popular account of these results in the movie "Counting from Infinity: Yitang Zhang and the Twin Prime Conjecture." I will give a slightly more technical account of these results, with emphasis on the underlying combinatorial constructions.

Speaker: Joshua Hallam (Michigan State University)
Title: Applications of Quotient Posets
Abstract: Suppose that we have a poset and an equivalence relation on the elements satisfying certain conditions. We form the quotient poset by ordering the equivalence classes in a way related to the original poset. In this talk we will discuss applications of quotient posets to the Mobius function, the characteristic polynomial, and the Whitney numbers of a poset.

Speaker: Carolina Benedetti (Michigan State University)
Title: TBD
Abstract: Coming soon …

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