Title: Code-Based Cryptography and a Variant of the McEliece Cryptosystem

Abstract: After a brief overview of public-key cryptography I will turn to a specific realization of a cryptosystem that relies on the hardness of decoding a random code. These cryptosystems were introduced by McEliece in 1978, but became popular only recently when it was discovered that RSA and elliptic-curve cryptography won't be secure in the presence of quantum computers. I will discuss the workings, advantages and drawbacks of the McEliece cryptosystem and also present a variant that aims at overcoming some of its drawbacks. No prior knowledge on public-key cryptography and coding theory is assumed.

Title: Elliptic Curves over finite fields and some of its applications

Abstract: We will introduce elliptic curves and talk about (some) applications of elliptic curves, including  factorizations of integers and elliptic curve protocols. 

Title: The absence of the weak Lefschetz property

Abstract:  Mezzetti, Miro-Roig, and Ottaviani showed that in some cases the failure of the weak Lefschetz property can be used to produce a variety satisfying a (nontrivial) Laplace equation. We define a graded algebra to have a Lefschetz defect of delta in degree d if the rank of the multiplication map of a general linear form between the degree d − 1 and degree d components has rank delta less than the expected rank. Mezzetti and Mir\'o-Roig recently explored the minimal and maximal number of generators of ideals that fail to have the weak Lefschetz property, i.e., to have a positive Lefschetz defect. In contrast to this, we will discuss constructions of ideals that have large Lefschetz defects and thus can be used to produce toric varieties satisfying many Laplace equations.

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.