Title: The path semigroup of a graph with applications to moduli spaces of geometric structures.

Abstract: I'll introduce some open questions about an elementary object: a commutative, finitely generated semigroup formed by the supports of (unions of) paths in a finite graph. These semigroups are related to affine algebraic schemes called character varieties; also known as moduli of flat principal bundles and moduli of geometric structures. I'll explain how answers to my questions could say something about the symplectic and tropical geometry of these moduli spaces.

Title: Topological Complexity and Graphic Hyperplane Arrangements.

Abstract: A motion planning algorithm for a topological space X is a (possibly not continuous) assignment of paths to ordered pairs of points in X. The topological complexity of X is an integer invariant which measures the extent to which any motion planning algorithm for X must be discontinuous. In practice it is difficult to compute, but it is bounded below by an integer invariant of the cohomology ring of X, called the zero-divisors-cup-length of X. When X is the complement of an arrangement of hyperplanes, this ring is the Orlik-Solomon algebra of A and the lower bound can be controlled by some combinatorial conditions. When X is the complement of a graphic arrangement, there is a connection between topological complexity and a classical result in graph theory, the Nash-Williams decomposition theorem.

Title: Unexpected Curves and Hyperplane Arrangements

Abstract: A recent paper by Cook, Harbourne, Migliore and Nagel, showed deep connections between objects known as unexpected curves in algebraic geometry and an open conjecture on Line Arrangements known as Terao's conjecture. After giving an exposition of this connection, I discuss some ongoing work in this area.  We put restrictions on when unexpected curves may occur and discuss why we might expect these curves to have certain types of symmetry.

Title: Value sets of polynomials modulo primes

Abstract: Let f be a polynomial with integer coefficients. For each prime p, we can reduce f modulo p and consider the (relative) size of the value set of f mod p. There is much to be desired concerning how small the relative sizes can be on average over p. In this talk I will discuss some results and some open problems.

Title: An introduction to Khovanskii bases

Abstract: Khovanskii bases are subsets of algebras which have nice computational properties.  They are a generalization of so-called SAGBI bases or canonical bases in an algebra, and in this sense they are meant to do for algebras what Groebner bases do for ideals. I'll give an introduction to Khovanskii bases, describe a few theorems about their structure and their existence, and describe their relationship to other interesting topics in combinatorial algebraic geometry, like tropical geometry and toric geometry.  

Title: Ideals of Geometrically Characterized Point Sets and Simplicial Complexes

Abstract: Let X be a finite set of points in an affine space. A Lagrange polynomial for X is a polynomial which vanishes at all but one of the points of X. In a way which can be made precise, not every point set admits Lagrange polynomials, but the existence of Lagrange polynomials for X guarantees that a polynomial function is completely determined by its values on X. If in addition, these Lagrange polynomials fully factor into products of linear polynomials, the set X is called geometrically characterized. In this talk, we will discuss the geometry of geometrically characterized sets, the ideals of such point sets, and will make some connections to simplicial complexes

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.