Abstract: Let G be a finite abelian group. We say that a given subset of G is uniform if all of its Fourier coefficients are small. We'll show that uniform sets are common and explore some of their nice additive properties.
Abstract: In geometry and combinatorics we are interested in “finite objects”, which are either Riemannian manifolds with finite volume or finite complexes. One way to construct such objects is to take a nice covering space X, and divide it by a discrete subgroup \Gamma. For example, by dividing X=R by \Gamma=Z we get the circle S^1. A far more general case is when we divide a symmetric space X associated with a semisimple Lie group G by an "arithmetic subgroup" \Gamma, for example G=SLn(R), \Gamma=SLn(Z).
Let G be a graph with oriented edges. One can assign to each edge of G an integer weight. This assignment is a flow if it conserves matter at each vertex, i.e. the sum of weights on the vertex's inward edges is equal to the sum on its outward edges. Can one assign such a flow to G without using 0 as a weight? Collecting all graphs for which such a non-zero flow exists, can you bound the absolute value of the weights you use? What is the best possible bound?
In the 19th century, Hermann Schwarz studied a differential expression which became known as the "Schwarzian Derivative". Early on, it was discovered that this expression vanishes exactly for Mobius Transformations. That discovery led to many interesting results in 20th century Complex Analysis, which connect the behaviour of a given holomorphic (or meromorphic) function to the geometry of a domain on which it is defined. In this talk we will review some of these results.