Title: The Maroni Invariant of Trigonal Chains of Loops

Title: Markov Bases of Hierarchical Models

Abstract: We will start by discussing the significance of Markov Bases for investigating Hierarchical Models occurring in Algebraic Statistics. Markov Bases are often very large and hard to compute. This talk is going to introduce an alternative way of thinking about them using tools from Algebraic Geometry and present some of the latest computational results.

Title: The strange consequences of Siegel zeros



Abstract: If you believe the Generalised Riemann Hypothesis, then there are no zeros of L-functions with real part bigger than 1/2, but unfortunately we don't know how to show this. A `Siegel zero' is a putative strong counterexample to GRH, and if such exceptional zeros do exist, then there are many strange consequences for the distribution of prime numbers. However, prime numbers would also become very regular, and this allows us to prove things which go beyond even the consequences of GRH, if these exceptional zeros exist! I will survey some of these results, including recent joint work showing we can prove results towards the horizontal Sato-Tate conjecture for Kloosterman sums in this alternative world where Siegel zeros exist.

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