## On the Fourier Spectrum of Symmetric Boolean Functions

It is well-known that any Boolean function f:{-1,+1}^n \to {-1,+1} can be written uniquely as a polynomial f(x) = \sum_{S subset [n]} f_s \prod_{i in S} x_i. The collection of coefficients (f_S's) this expression are referred to (with good reason) as the Fourier spectrum of f. The Fourier spectrum has played a central role in modern computer science by converting combinatorial and algorithmic questions about f into algebraic or analytic questions about the spectrum.

## On Functoriality; on the Correspondence; and on Their Relation, Part 1

## Edward T. Cone Concert Talk

Violinist Steven Copes, flutist Tara Helen O'Connor, and cellist Edward Arron join in a conversation about music with Institute Artist-in-Residence Derek Bermel.

## Automorphy: Automorphy Lifting Theorems I (continued)

## Analytic Geometry Over F_1

I'll talk on work in progress on algebraic and analytic geometry over the field of one element F_1. This work originates in non-Archimedean analytic geometry as a result of a search for appropriate framework for so called skeletons of analytic spaces and formal schemes, and is related to logarithmic and tropical geometry. I'll explain what analytic spaces over F_1 are, and will describe non-Archimedean and complex analytic spaces which are obtained from them.

## Automorphy: Potential Automorphy Theorems I

## Galois Representations Associated to Holomorphic Limits of Discrete Series

We attach Galois representations to automorphic representations on unitary groups whose weight (=component at infinity) is a holomorphic limit of discrete series. The main innovation is a new construction of congruences, using the Hasse Invariant, which avoids q-expansions and so is applicable in much greater generality than previous methods. Our result is a natural generalization of the classical Deligne-Serre Theorem on weight one modular forms and work of Taylor on GSp(4).

## Relativized Separations of Worst-Case and Average-Case Complexities for NP

## A Randomized Rounding Approach for Symmetric TSP

We show a (3/2-\epsilon)-approximation algorithm for the graphical traveling salesman problem where the goal is to find a shortest tour in an unweighted graph G. This is a special case of the metric traveling salesman problem when the underlying metric is defined by shortest path distances in G. The result improves on the 3/2-approximation algorithm due to Christofides for the case of graphical TSP.