Counting incompressible surfaces in hyperbolic 3-manifolds.

1:30pm

Marc Lackenby (Oxford)

Incompressible embedded surfaces play a central role in 3-manifold theory. It is a natural and interesting question to ask how many such surfaces are contained in a given 3-manifold M, as a function of their genus g. I will present a new theorem that provides a surprisingly small upper bound. For any given g, there a polynomial p_g with the following property. The number of closed incompressible surfaces of genus g in a hyperbolic 3-manifold M is at most p_g(vol(M)). This is joint work with Anastasiia Tsvietkova.

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The rates of growth in hyperbolic groups.

2:45pm

Koji Fujiwara (Kyoto)

For a finitely generated group of exponential growth, we study the set of exponential growth rates for all possible finite generating sets. Let G be a hyperbolic group. It turns out that the set of growth rates is well-ordered.  Also, given a number, there are only finitely many generating sets that have this number as the growth rate. I also plan to discuss the set of growth for a family of groups.

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Character varieties of random groups.

4:00pm

Emmanuel Breuillard (Oxford)

In joint work with P. Varju and O. Becker we study the representation and character varieties of random finitely presented groups with values in a complex semisimple Lie group.  We compute the dimension and number of irreducible components of the character variety of a random group. In particular we show that random one-relator groups have many rigid Zariski-dense representations. The proofs use a fair amount of number theory and are conditional on GRH. Key to them is the use of expander graphs for finite simple groups of Lie type as well as a new spectral gap result for random walks on linear groups.

We develop an algorithm for parameter-free stochastic convex optimization (SCO) whose rate of convergence is only a double-logarithmic factor larger than the optimal rate for the corresponding known-parameter setting. In contrast, the best previously known rates for parameter-free SCO are based on online parameter-free regret bounds, which contain unavoidable excess logarithmic terms compared to their known-parameter counterparts. Our algorithm is conceptually simple, has high-probability guarantees, and is also partially adaptive to unknown gradient norms, smoothness, and strong convexity. At the heart of our results is a novel parameter-free certificate for the step size of stochastic gradient descent (SGD), and a time-uniform concentration result that assumes no a-priori bounds on SGD iterates.

Additionally, we present theoretical and numerical results for a dynamic step size schedule for SGD based on a variant of this idea. On a broad range of vision and language transfer learning tasks our methods performance is close to that of SGD with tuned learning rate. Also, a per-layer variant of our algorithm approaches the performance of tuned ADAM.

This talk is based on papers with Yair Carmon and Maor Ivgi.