Spring 2016

Fridays, 3:00-4:00

Location: MB124

The Algebra, Geometry, Cryptology (AGC) Seminar is co-organized by Zach Teitler and Marion Scheepers. For more information contact zteitler@boisestate.edu.

##### January 15:

**Zach Teitler** (Boise State University)

**Strassen’s additivity conjecture and bounds for Waring rank**

The Waring rank of a complex homogeneous form is the least number of terms in an expression of the form as a sum of powers of linear forms. Waring rank, tensor rank, and various generalized ranks are interesting for a range of applications including secant varieties and geometric complexity theory, but they are difficult to compute. In particular we do not know the maximum Waring rank among forms in a given number of variables and a given degree; it is not even known whether forms of greater than generic rank exist. I will present upper and lower bounds for Waring rank and generalized ranks that narrow the possible ranges of maximum values of generalized ranks (joint work with Grigoriy Blekherman) and in some new cases show the existence of forms of above-generic Waring rank (joint work with Jaros{\l}aw Buczy\’nski). I will present a sufficient condition for forms to satisfy a strong form of Strassen’s additivity conjecture, which asserts that the Waring rank of a sum of forms in independent variables is the sum of their ranks.

##### January 22:

Cancelled due to illness

##### January 29:

**Zach Teitler** (Boise State University)

**Random graphs**

Introduction to random graphs and overview; Crossing Lemma; application to combinatorial geometry.

##### February 5:

##### February 12:

**Hirotachi Abo** (University of Idaho)

**Eigenvectors of tensors**

Eigenvectors of tensors were first introduced by L. Qi and L.-H. Lim independently in 2005. The set of eigenvectors of a tensor forms a variety called the eigenvariety of the tensor. The purpose of this talk is to study the algebra and geometry of eigenvarieties of tensors.

The eigenvariety of a general tensor consists of a finite number of eigenvectors (up to scaling) and the formula for counting the number of distinct eigenvectors of such a tensor is known. It is therefore a very natural question to ask: “What is the condition(s) for a finite set of points to be the eigenvariety of a tensor?’’ In this talk, I discuss a characterization of eigenvarieties of 3-mod ternary tensors.

##### February 19:

##### February 26:

**Jason Smith** (Boise State University)

**Using graphs to count**

By counting triangles in complete graphs and edges in complete bipartite graphs we give combinatorial proofs for identities for sums of the first n positive integers, square integers, and cubed integers. An open problem will be given. Time permitting purely combinatorial proofs for these identities will also be given.

##### March 4:

**Marion Scheepers** (Boise State University)

**Sorting Permutations**

Sorting is often the first step in preparing data for processing by search and other algorithms. As such sorting algorithms have been studied extensively. In this talk we feature two specific sorting operations that have emerged as important to achieving efficiency in sorting.

##### March 11:

**TBA**

##### March 18:

**TBA**

##### March 25:

No meeting due to Spring Break

##### April 1:

**Joe Champion** (Boise State University)

**Integral Arithmagons**

Arithmagons are informally presented in some school mathematics curricula as polygonal figures with integer labeled vertices and edges in which, under a binary operation, adjacent vertices equal the included edge. Though providing an interesting opportunity for a range of problem solving tasks, arithmagons have only recently been formally defined and characterized. By considering the group of automorphisms for the associated graph, we will count the number of integral arithmagons whose exterior sum or product equals a fixed number.

##### April 8:

##### April 15:

**Ellen Veomett** (St. Mary’s College of California)

**The Edge Isoperimetric Inequality for a Graph on Z^n**

Isoperimetric inequalities raise very natural questions about the shape of a set and the resulting size of the boundary. The Euclidean isoperimetric inequality in the plane asks: given a closed loop, what shape should I make so that the loop encloses the largest possible area? We will discuss how the Brunn-Minkowski inequality can be used to prove the isoperimetric inequality in R^n. We will then discuss how one can define an isoperimetric inequality for a graph and give various examples of results on graphs. Interestingly, we can also use the Brunn-Minkowski inequality to solve a continuous formulation of the edge isoperimetric inequality for a graph on Z^n, which should in turn help us to solve the edge isoperimetric inequality on this graph.

##### April 22:

**Bruce Reznick** (UIUC)

Sums of powers of polynomials

##### April 29:

**TBA**

Other semesters:

- Fall 2015
- Spring 2015
- Fall 2014
- Spring 2014
- Fall 2013
- Spring 2013
- Fall 2012
- Spring 2012
- Fall 2011
- Spring 2011
- Fall 2010