Spring 2017

Fridays, 3:00-3:50

Location: MB139

The Algebra, Cryptology, Geometry Seminar meets on selected Fridays, dates indicated below. Everybody interested is welcome to join.The seminar is co-organized by Zach Teitler and Marion Scheepers. If you wish to present in the seminar or need information, please contact mscheepe@boisestate.edu.

**January 13:**

No meeting

**January 20:**

**Zach Teitler**, Boise State University

**Geometry of high rank loci**

Matrix rank, tensor rank, and other related notions of rank are highly important throughout mathematics, statistics, and many areas of engineering and sciences. Elements of high rank can have surprising applications, for example to questions in computational complexity, yet high rank elements are usually exotic and in many cases the locus of high rank elements is poorly understood — in fact, it is not generally known whether the highest possible rank locus is empty, arbitrarily large, or somewhere in between. I will describe one of the first general studies of high rank loci, showing for example that they are nested and giving meaningful dimension bounds. This is joint work in preparation with Jaroslaw Buczynski, Kangjin Han, and Massimiliano Mella. This talk will not assume familiarity with generalized ranks or rank loci; these ideas will be explained and illustrated with examples.

**January 27:**

**February 3:**

**Zach Teitler**, Boise State University

**Arrangement apolarity**

I will present work in progress with Jaroslaw Buczynski, Stefan Tohaneanu (U. Idaho), and Alexander Woo (U. Idaho) on apolarity of hyperplane arrangements. A hyperplane arrangement is a finite set of codimension 1 linear subspaces; from an algebraic perspective, it just corresponds to a polynomial that can be factored into linear factors. We study interrelationships between the geometrical properties of the arrangement and algebraic properties of the factored polynomial. This talk will consider apolarity, which roughly corresponds to: describing and counting the derivatives of the polynomial, and determining which differential operators annihilate the polynomial — that is, what differential equations have the given polynomial as a solution. I will describe some results and ongoing work.

**February 10:**

**February 17:**

**Jonny Comes**

**An analog of jellyfish partition categories for finite special linear groups**

The partition category P(n) and the jellyfish partition category JP(n) model tensor representations for the symmetric group S_n and alternating group A_n respectively. Generalizing the construction of P(n), Knop introduced a monoidal category K(n,q) that models tensor representations for the finite general linear group GL(n,q). In this talk I will explain how to “add jellyfish” to Knop’s category, which produces a category JK(n,q) that models tensor representations for the finite special linear group SL(n,q).

**February 24:**

**March 3:**

**Zach Teitler**, Boise State University

**Lefschetz properties, hyperplane arrangements, inclusion matrices**

I will review a very pleasant classical combinatorial result of Gottlieb: Fix integers m, i, j. Let M be the matrix with rows indexed by the i-subsets of [m] and columns indexed by the j-subsets, with (I,J) entry 1 if I is a subset of J, otherwise 0. Then Gottlieb’s result is a computation of the rank of M, namely, that M has full rank. I will use this to show that apolar algebras of certain hyperplane arrangements have the Weak Lefschetz Property, which will be defined and explained in the talk.

**March 10:**

**March 17:**

No seminar meeting

**March 31:**

No seminar meeting

**April 7:**

**Zach Teitler**, Boise State University

**The Gessel-Viennot theorem**

“The aliens will destroy Earth unless we answer their three questions!” shouted the general. “First, given lattice points E, N, W, S on the east, north, west, and south sides of a rectangle, we have to find how many ways there are to draw lattice paths from W to N, and from S to E.”

“Oh, no problem,” said the secret agent. “I can solve that using binomial coefficients. You know, Pascal’s triangle.”

“You didn’t let me finish! The paths have to be disjoint. Once you pick one of the paths, it affects how many options there are for the other path.”

“Hmm. What were the other two questions?”

“Second, we have to find the area of a triangle in R^3. But we’re not given the vertices of the triangle — just the areas of its three projections onto the xy, xz, and yz planes. Third, we have to show that the matrix whose (i,j) entry is the binomial coefficient \binom{a_i}{b_j}, for fixed nonnegative integers a_1<…<a_n and b_1<…<b_n, has a nonnegative determinant which vanishes if and only if some a_i<b_i."

"Well, is that all," smirked the agent. "We'll have these alien questions answered for you in no time. The Earth is as good as saved."

The Gessel-Viennot theorem, 1985 (also due independently to Lindstrom, 1973 and Karlin-McGregor, 1959) relates determinants to nonintersecting path systems in graphs. Its numerous applications include counting nonintersecting lattice paths; a proof of the Cauchy-Binet formula, which in turn implies a generalization of Pythagoras's theorem; and studying determinants of matrics of binomial coefficients. Other applications include showing different definitions of Schur polynomials are equivalent.

This talk will move quickly through a statement and a simple proof of the Gessel-Viennot theorem, and as many applications as we can fit into the time. Students will be able to follow the talk which will require only some familiarity with determinants and graphs.

Attend the seminar, learn some math, save the Earth from aliens!

**April 14:**

**April 21:**

TBA

**April 28:**

TBA