There are two very different approaches to 3-dimensional topology, the hyperbolic geometry following the work of Thurston and the quantum invariants following the work of Jones and Witten. These two...
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This talk provides a personal perspective on the way forward towards more human-like and more intelligent artificial systems. Traditionally, symbolic and probabilistic methods have dominated the domains of concept formation,...
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We discuss the problem of determining the number of generators and relations of several profinite groups of arithmetic and geometric origin. These include etale fundamental groups of smooth projective varieties,...
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I will present a new mechanism and an exact theory of electron pairing due to repulsive interaction in doped insulators. When the kinetic energy is small, the dynamics of adjacent...
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The classical question of determining which varieties are rational has led to a huge amount of interest and activity. On the other hand, one can take on a complementary perspective:...
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Calabi--Yau manifolds of a given dimension are connected by an intricate web of birational maps. This web has deep consequences for the derived categories of coherent sheaves on such manifolds,...
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Motivated by the Manin-Mumford and the Bogomolov conjecture for abelian varieties, Zhang conjectured that an analogous statement holds in a more general dynamical setting. In this talk we discuss various...
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Interesting patterns of prime numbers can arise in recursively defined sequences (such as the Fibonacci sequence). For non-linear recursions, there are intriguing connections with complex dynamics and the Mandelbrot set....
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The p-torsion in the class group of a number field K is conjectured to be small: of size at most |Disc K|^epsilon, and to have constant average size in families...
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This talk, in two parts, will discuss two (unrelated) instances of exotic quantum matter. In the first part, I will discuss quantum critical points describing possible transitions out of the...
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