Colloquium

The Kirchhoff and Boltzmann distributions date back to the nineteenth century, dealing respectively with the flow of electrical current in a circuit and with the distribution of particles in a gas. I will show how both can be derived as…
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Let N be a normal subgroup of a group G. When does a homomorphism from N to an abelian group A extend to a homomorphism from G to A?  When does an irreducible (projective) representation f of N extend to…
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The problem of bounding the number of rational or algebraic points of a given height in a transcendental set has a long history. In 2006 Pila and Wilkie made fundamental progress in this area by establishing a sub-polynomial asymptotic estimate…
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Let H be a self-adjoint operator defined on an infinite dimensional Hilbert space. Given some spectral information about H, such as the continuity of its spectral measure, what can be said about the asymptotic spectral properties of its finite dimensional…
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Non-Archimedean analytic spaces are analogues of complex manifolds when replacing the complex numbers by a non-Archimedean field, such as p-adic numbers or complex Laurent series. I will give several examples of situations involving degenerations in complex analysis and geometry that…
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Earlier and recent one-dimensional estimates and asymptotic relations for the cosine and sine Fourier transform of a function of bounded variation are refined in such a way that they become applicable for obtaining multidimensional asymptotic relations for the Fourier transform…
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What is the optimal way to cut a convex bounded domain K in Euclidean space R^n into two halves of equal volume, so that the interface between the two halves has least surface area? A conjecture of Kannan, Lovasz and…
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Given positive integers h and k, denote by r(h,k) the smallest integer n such that in any k-coloring of the edges of a tournament on more than n vertices, there is a monochromatic copy of every oriented tree on h…
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The family of high rank arithmetic groups is class of groups which is playing  an important role in various areas of mathematics. It includes SL(n,Z) for n>2, SL(n, Z[1/p]) for n>1, their finite index subgroups and many more. A number…
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  Consider a compact complex torus T, identified with the quotient C^n/L, where L is a lattice in C^n. Let p: C^n->T be the quotient map. Ullmo and Yafaev have recently asked the following question: Assume that X is an…
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