Title: An Introduction to the Bieberbach Conjecture
Abstract: Please see 7th floor bulletin board for abstract
Title: An Introduction to the Bieberbach Conjecture
Abstract: Please see 7th floor bulletin board for abstract
Title: Constructing Simplicial Spheres Using Algebra
Abstract: Simplicial complexes are a central subject of study in combinatorics. They are also of interest geometrically since that abstract simplicial complex can be realized in real space as a geometric object. One interesting question is when is a simplicial complex homeomorphic to a sphere. We call these simplicial spheres. Using the connection of the Stanley-Reisner ring we can study simplicial complexes by looking at a corresponding ring. In particular the Gorenstein property occurs in these rings exactly when they are simplicial spheres. Hence we can construct simplicial spheres by constructing certain Gorenstein rings. This talk is meant to show the usefulness of the Stanley-Reisner ring connection between combinatorics and algebra and should be accessible to a general audience.
(Pizza at 4:00, talk at 4:15)
Title: Presenting Sperner's Lemma
Abstract: In this talk I will prove Sperner's Lemma on simplices. This is a well-known result that highlights a connection between the areas of combinatorics and analysis (passing through topology). The talk is meant to be accessible to all graduate students in mathematics regardless of their chosen field.
Title: Polar Self-Dual Polytopes and Mahler Volume
Abstract: The dual of a polytope is a well known topic in polyhedral geometry. There are several notions of what it means for a polytope to be self-dual. Perhaps the most common idea of self-dual would be that a polytope is combinatorially equivalent to its dual (face poset isomorphism). In Stephen's recent work it has been helpful to consider the strictest form of self-duality called polar self-dual, where we require that an embedded polytope be exactly equal to its dual as a subset of real space. We will consider some of the results and conjectures relating to polar self-dual polytopes.
Another famous problem from the theory of convex duality is finding the extremizers of the Mahler volume. This is defined as the product of the area of the convex figure and the area of its dual. Using Steiner symmetrization, we will show the maximizer of this volume is the circle.
Title: Winding Numbers and Simplicial Convex d-Polytopes
Abstract: For a simplicial convex d-polytope, the winding number of a region counts how many times we wrap around that speci?fic region. It has been shown that the winding number is a nonnegative integer. While proving that the winding number is an integer is straightforward in all dimensions, showing that the winding number is nonnegative proves much more ?difficult for d>=3. In this talk, we will reference concrete examples as we prove that the winding number is a nonnegative integer for d=2.
Title: Integer-Point Transforms of Rational Polygons and Rademacher--Carlitz Polynomials (joint work with Matthias Beck)
Abstract: We introduce and study the Rademacher--Carlitz polynomial. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view the Rademacher--Caritz polynomial as a polynomial analogue (in the sense of Carlitz) of the Dedekind--Rademacher sum which appears in various number-theoretic, combinatorial, geometric, and computational contexts.
Our results come in three flavors: we prove a reciprocity theorem for Rademacher--Carlitz polynomials, we show how they are the only nontrivial ingredients of integer-point transforms ---which is a way of encoding integer points as a polynomial--- of any rational polyhedron P, and (if time allows) we derive a novel reciprocity theorem for Dedekind--Rademacher sums, which follows naturally from our setup.
Title: Pick's Theorem for Convex Polygons
Abstract: In this talk, we will prove Pick's theorem which relates the area of an integral convex polygon to the number of lattice points contained in it. Using this theorem, we will discuss the lattice point enumerator and the Ehrhart series of integral convex polygons.
Title: Pick's Theorem for Convex Polygons
Abstract: In this talk, we will prove Pick's theorem which relates the area of an integral convex polygon to the number of lattice points contained in it. Using this theorem, we will discuss the lattice point enumerator and the Ehrhart series of integral convex polygons.
Title: L_2 and pointwise a posteriori error estimates for FEM for elliptic PDE on surfaces
Abstract: Surface Finite Element Methods (SFEM) are popular numeric methods for solving PDE. A posteriori error estimators are computable measures of the error and are used to implement adaptive mesh refinement. In this talk we will introduce the basics of finite element methods and a posteriori estimates. We will end with an example of an adaptive finite element method based on point-wise a posteriori error estimate solving the Laplace-Beltrami equation over a surface.
