Title: Levels and Pythagoras numbers of commutative rings

Abstract:  The level s(R) of a commutative ring R is the smallest integer n such that -1 is a sum of n squares of elements in R.  Set s(R) = infinity if no such representation exists. The Pythagoras number p(R) is the smallest integer m such that every sum of squares of elements in R is already a sum of m squares in R.  Set p(R) = infinity if no such bound exists.  The study of levels and Pythagoras numbers of fields is a classical topic. Many results are known, but many open questions still remain.  The study of levels and Pythagoras numbers of arbitrary commutative rings is more recent.  I will survey known results and report on recent research with Detlev Hoffmann.

Title: Quantum error-correcting codes. II

Abstract: This talk will be a continuation of the talk from last Wednesday. Definitions from quantum computing will be formalized and algebra will finally come into play to translate quantum questions to classical coding theory.

Title: Quantum error-correcting codes

Abstract: Quantum error correction is a necessity for eventual quantum computers and unfortunately much more difficult than the classical one. In this talk, we will explore these difficulties and how to fight them with good quantum codes. The focus will be on stabilizer formalism, as a compact description of almost every known quantum code. We will use this algebraic language to translate questions raising from quantum computation to classical error correction.

Physics background is neither assumed nor required.

Title: Dimensions of secant varieties

Abstract:  A variety  is the set of solutions of a polynomial system of equations. Considering the union of all linear subspaces spanned by k points on a variety V, one obtains the k-th secant variety of V. Determining the dimension of a secant variety is an interesting and challenging problem. We illustrate this in two instances. The first one concerns the Waring rank. Any homogeneous polynomial f of degree d can be written as a sum of d-th powers of linear forms. The minimum number of summands in such a decomposition is the Waring rank of f. It admits a geometric interpretation using secant varieties. In the second instance we use linear algebra to solve the problem in some cases. The general problem (of decomposing tensors as sums of pure tensors) is open.

Title: The Waldschmidt constant

Abstract:  A (projective) variety V is a set of common zeros of the polynomials in an ideal I that is generated by homogenous polynomials. Given the generators of the ideal I,  one would like to know the minimum degree of a polynomial F such that each point of V is a root of f of a given multiplicity, say k. As this is often a difficult problem one studies first the corresponding question for large k. This leads to the Waldschmidt constant, which gives an asymptotic answer to the problem.

If I is a an ideal that is generated by squarefree monomials, then the Waldschmidt constant can be expressed as the optimal solution to a linear program or as a fractional chromatic number. This leads to the new bounds and computations of the Waldschmidt constant.

No prior knowledge of monomial ideals or graph theory is assumed. All concepts will be explained in the talk. 

Title:  Duals of Skew θ-Constacyclic Codes

Abstract:  We generalize cyclic codes to skew θ-constacyclic codes using skew polynomial rings. We provide a useful tool for exploring these codes: the circulant. In addition to presenting some properties of the circulant, we use it to re-examine a theorem giving the dual code of a skew θ- constacyclic code first presented by Boucher/Ulmer (2011). This talk includes work with Dr. Heide Gluesing-Luerssen.

Title:  Classes of modules relative to torsion theories

Abstract:  Our purpose is extend known results of some classes of modules to torsion theoretic setting in a way so that the former results are recovered when some torsion theory is chosen. For example we generalize the concept of relative pure injectivity to relative pure $\tau$-injectivity, where $\tau$ is a given hereditary torsion theory. If $\tau$ is the improper torsion theory then relative pure injectivity and relative pure $\tau$-injectivity are equivalent. This new concept retains some of the important properties of pure injectives. For instance, we show that the class of pure $\tau$-injective modules is enveloping.

Title:  The Hilbert Series of Algebras of the Veronese Type

Abstract:  Algebras of the Veronese type are semigroup algebras that have attracted considerable attention from the algebra and algebraic combinatorics communities.  In 1996, Sturmfels described Groebner bases for presentations of these algebras, and in 1997 De Negri and Hibi classified those that are Gorenstein.  In his 2005 paper "The Hilbert Series of Algebras of the Veronese Type," Mordechai Katzman added to these results by providing an explicit formula for the Hilbert series of these algebras.  In this talk, we will describe Katzman's formula and its connections to the combinatorics of a family of polytopes known as hypersimplices.

Title:  Artin's Conjecture on homogeneous forms over $\mathbb{Q}_p$

Abstract:  A field $k$ is called a $C_i$ field if any homogeneous form of degree $i$ in more than $d^i$ variables has a nontrivial zero in $k$. It is well known that finite fields are $C_1$. What about the p-adics? It was conjectured by Emil Artin that $\mathbb{Q}_p$ is $C_2$. The result turned out to be false. We will investigate some positive results.