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.

Title:  An Introduction to Diophantine Approximation

Abstract:  Diophantine approximation is a branch of number theory that deals with the approximation of real numbers by rational numbers. Our goal will to be to de?fine what it means to be a "good" approximation and then ?find the "best" among these. Along the way, we'll develop the basics of continued fractions and even take advantage of some geometrical properties of lattices. When the smoke clears, we'll see why 22/7 is a good approximation to pi and why 355/113 is even better. We'll also see why the golden ratio is in some sense the most irrational of all the real numbers. Finally, we will reveal the secret to one of Ramanujan's famous approximations of ?pi.

Title:  How to untie a knot

Abstract:  How can you tell when a knot can be untangled?  This deceptively simple question is at the heart of Knot Theory, a field which emerged with the work of Vandermonde in the 18th century.  In this talk, we will introduce the knot classification problem, covering a few (but definitely not all) of the following ideas: knot invariants, Reidemeister moves, Conway and Jones polynomials, Gauss linking numbers, braid groups, Vassiliev invariants, the Kontsevich integral, Khovanov homology, categorification...

Title:  The brachistochrone problem and variational methods.

Abstract:  The brachistochrone is the problem of finding the path that would take a ball bearing between to points in the least amount of time. We will  construct the solution using variational methods.

Other classic problems we may talk about include describing the shape of a hanging chain and finding the shape which encloses the largest area given a set perimeter.

 

Title:  Lagrange's Four Square Theorem

Abstract:  Lagrange's Four Square theorem states that any natural number can be written as the sum of four integers. This is the best we can do since, for example, 7 cannot be written as the sum of three squares. We will prove the theorem and take a look at the natural numbers that cannot be written as the sum of 3 squares.

 

Solitons are solitary waves that are stable to perturbations.  This talk will focus on the paper "Korteweg--deVries Equation and Generalizations. VI. Methods for Exact Solution" by Gardner, Greene, Kruskal, and Miura which explores solving the Korteweg--deVries equation using inverse scattering, a method pioneered in this paper.  This allows one to write explicit solutions to the KdV equation, including solutions that demonstrate solitons.