The Waring rank of a homogeneous form is the number of terms needed to write the form as a sum of powers of linear forms. I will give an introduction and overview of the subject, centered on complexity of the determinant and permanent. I will describe progress in the last ten years on lower and upper bounds for Waring rank; on Strassen’s conjecture, which asserts that Waring rank is additive; and the new study of high-rank loci, initiated in the last 4 years.

A matroid is an abstract structure that generalizes the idea of linear independence in a vector space. I will define matroids and give motivation for studying them, then look at some examples of matroids and their properties.

In 1982 Michael Freedman classified simply connected topological 4-manifolds up to homeomorphism. The geography problem asks which of the topological 4-manifolds admits a smooth structure, the botany problem asks to describe the different structures if there is at least one. The geography problem has been solved up to a complete solution of the so called 11/8 conjecture, which is still open.

We will take a look at the Poincare Conjecture and the classification of spheres. Uwe Kaiser may talk a little, in particular about the most recent status. But most of the time we will show John Milnor's talks from 1965 and 2011 on the topic.

We will watch together videos on the Geometry of Matroids, and Algebraic Structures on Polytopes.

Let $\F_p$ denote the finite field of order $p$ and $\F$ its algebraic closure. Classifying $\F$-representations of the group $\Z/p\Z \times \Z/p\Z$  leads to a simply stated geometric problem involving two dimensional $\F_p$-subspaces of $\F$.  The classification of these representations is naturally described by a countable family of polynomials $f_{m,p}(t) \in\F_p[t]$.

These polynomials have a number of remarkable properties and I will describe their surprising connections with a wide variety of topics.

The polynomials may be described uniformly as expressions in $t$ and $p$. This description allows us to generalize to any integer value of $p$.  Specializing to $p=1$ we recover {\em Morgan-Voyce polynomials} which form two classical orthogonal families of polynomials in $\Z[X]$. These polynomials  were first defined and studied in the context of electrical resistance.   Our point of view yields a new compact description of the Morgan-Voyce polynomials in terms of an infinite sequence of binary vectors.

Interpreting these binary vectors as the vertices of a real polytope yields the {\em zigzag order polytopes}. These polytopes were considered by Richard Stanley who showed they have a number of interesting properties and strong connections with certain families of permutations.

For prime values of $p$, the polynomials $f_{m,p}(t)$ are related to the Fibonacci series, the order of appearance of prime numbers in the Fibonacci sequence, Artin's Primitive Root Conjecture, and the factorization of trinomials over $\F_p[X]$.

This talk is elementary and will be suitable for graduate students and senior undergaduates.

We will watch Michael Hopkins's 2011 Abel Prize Lecture on "Bernoulli numbers, homotopy groups, and Milnor".

The discriminant of a polynomial is a polynomial of the coefficients of the polynomial that is zero if and only if the polynomial has a multiple root. The concept of the discriminant of a polynomial can very naturally be extended to a more general system of polynomials. We call the zero locus of the discriminant of such a polynomial system the discriminant locus. In this talk, we explore the geometric properties of the discriminant locus.

Matroids are an abstraction of the properties of linear independence observed in vector spaces. Gian-Carlo Rota liked the name "combinatorial geometries" instead of "matroids" (which he thought was atrocious) because "combinatorial geometries" appeared to retain so many of the geometric properties of the linear spaces that motivated their study. In this talk I will connect the geometry of Grassmannians and flag varieties to study the Chow rings of matroids (linearly realizable or not). Adiprosito, Huh and Katz proved that Chow rings of matroids satisfied a suite of results ("the Hodge package") and used these results to prove that the characteristic polynomial of any matroid has a log-concave sequence of coefficients. I will explain joint work with Hunter Spink (Stanford) and Dennis Tseng (MIT) on how the above connections give rise to stronger results yielding the log-concavity of matroid h-vectors.

In physics, there are examples of differential equations, such as Maxwell's equations, which have degrees of freedom that result purely from the mathematics, and are generally regarded as "nonphysical" degrees of freedom. The techniques used for suppressing these degrees of freedom rely on choosing solutions which are unique up to some kind of symmetries that preserve the equation. In the terminology used by physicists, the solution that we pick to solve the equation is called a gauge, and the symmetries that preserve the equation are called the gauge transformations. The study of gauge theory has motivated a vast body of tools that have been used in differential geometry to prove some fascinating results, such as the construction of 4-manifolds which are homeomorphic, but not diffeomorphic to the Euclidean space R^4. In this talk, we will discuss some of the elementary tools of mathematical gauge theory, which essentially becomes the connections and curvature over principal bundles and vector bundles.

For a fixed Lie group G and base manifold X the moduli space M(X,G) is the set of all flat principal G-bundles on X up to gauge equivalence. We sketch the proof of the following folklore result: The holonomy along a loop map defines a bijection from M(X,G) to the representation space Hom(fundamental group of X,G)/G. This is used to define a topology and a symplectic structure on M(X,G). In this way, M(X,G)$ becomes the configuration space of a topological quantum field theory.