What are the noncommutative rings that are most analogous to polynomial rings? One class of such rings are the regular algebras first defined by Artin and Schelter in 1987. Since then such algebras have been extensively studied. We give a survey of these interesting examples and their associated projective geometry.

Efficient optimization has become one of the major concerns in machine learning, and there has been a lot of focus on first-order optimization algorithms because of their low cost per iteration. In 1983, Nesterov's Accelerated Gradient method (NAG) was shown to converge in $\mathcal{O}(1/k^2)$ to the minimum of the convex objective function $f$, improving on the $\mathcal{O}(1/k)$ convergence rate exhibited by the standard gradient descent methods, which is the phenomenon referred to as acceleration. It was shown that NAG limits to a second order ODE, as the time-step goes to 0, and that the objective function $f(x(t))$ converges to its optimal value at a rate of $\mathcal{O}(1/t^2)$ along the trajectories of this ODE. In this talk, we will discuss how the convergence of $f(x(t))$ can be accelerated in continuous time to an arbitrary convergence rate $\mathcal{O}(1/t^p)$ in normed spaces, by considering flow maps generated by a family of time-dependent Bregman Lagrangian and Hamiltonian systems which is closed under time rescaling. We will then discuss how this variational framework can be exploited together with the time-invariance property of the family of Bregman dynamics using adaptive geometric integrators to design efficient explicit algorithms for accelerated optimization. We will then discuss how these results and computational methods can be generalized from normed spaces to Riemannian manifolds. Finally, we will discuss some practical considerations which can be used to improve the performance of the algorithms. 

I will recall a construction of certain operator algebras arising naturally as multiparameter deformations of  operator algebras of Coxeter groups, initially motivated by the study of cohomology of groups acting on buildings. We will explain that for right-angled Coxeter groups, at a certain range of multiparameters, the resulting von Neumann algebra is a factor, thus completing earlier results of Garncarek, and of Caspers, Klisse and Larsen. This result, of interest in itself, has several consequences and interpretations for the representation theory of both right-angled Coxeter groups and of certain groups acting on buildings. I will also outline further questions/results related to the classification of the related C*-algebras.

Based on joint work with Sven Raum.

If a man is on a rocket with a certain finite speed, he will travel through the entire universe (no matter how large the universe is) in the blink of an eye. Notoriously weird and unrealistic claim from relativity, but takes just about 20 minutes to really understand this.

In many important actions of groups there are no invariant measures. For example: the action of a free group on its boundary and the action of any discrete infinite group on itself. The problem we will discuss in this talk is 'On a given action, how invariant measure can be? '. Our measuring of non-invariance will be based on entropy (f-divergence).

In the talk I will describe the solution of this problem for the Free group acting on its boundary and on itself. For doing so we will introduce the notion of ultra-limit of $G$-spaces, and give a new description of the Poisson-Furstenberg boundary of $(G,k)$ as an ultra-limit of $G$ action on itself, with 'Abel sum' measures. Another application will be that amenable groups possess KL-almost-invariant measures (KL stands for the Kullback-Leibler divergence). All relevant notions, including the notion of Poisson-Furstenberg boundary and the notion of Ultra-filters will be explained during the talk. This is a master thesis work under the supervision of Yehuda Shalom.

The Harish-Chandra-Itzykson-Zuber integral, also called spherical integral is defined as the expectation of exp(Tr(AUBU*)) for A and B two self adjoint matrices and U Haar-distributed on the unitary/orthogonal/symplectic group. It was initially introduced by Harish-Chandra to study Lie groups and it also has an interpretation in terms of Schur functions. Since then, it has had many kinds of applications, from physics to statistical learning. In this talk we will look at the asymptotics of these integrals when one of the matrices remains of small rank. We will also see how to use these asymptotics to prove large deviation principles for the largest eigenvalues for random matrix models that satisfy a sub-Gaussian bound. This talk is mainly based on a collaboration with Justin Ko. 

In geometric measure theory, the monotonicity formula for the area functional is a basic tool upon which many other basic fundamental facts depend. Some of them also follow from a weaker analytic tool, which is the Michael-Simon inequality. For anisotropic integrands (which generalize the area), monotonicity does not hold, while the latter inequality is conjectured to be true (under appropriate assumptions); actually, the latter is more essential to geometric measure theory, in that it turns out to be equivalent to the compactness of the classes of rectifiable and integral varifolds. In this talk we present some partial results, one of which is a slight improvement of a posthumous result of Almgren, namely the validity of this inequality for convex integrands close enough to the area, for surfaces in $R^3$. Our technique relies on a nonlinear inequality bounding the $L^1$-norm of the determinant of a function, from the plane to $2x2$ matrices, with the $L^1$-norms of the divergence of the rows, provided the matrix obeys some pointwise nonlinear constraints. This is joint work with Guido De Philippis (NYU).

In 2015 Greither and Popescu constructed a new class of Iwasawa modules, which are the number field analogues of $p-$adic realizations of Picard 1- motives constructed by Deligne. They proved an equivariant main conjecture by computing the Fitting ideal of these new modules over the appropriate profinite group ring. This is an integral, equivariant refinement of Wiles' classical main conjecture. As a consequence they proved a refinement of the Brumer-Stark conjecture away from 2. All of the above was proved under the assumption that the relevant prime $p$ is odd and that the appropriate classical Iwasawa $\mu$–invariants vanish. Recently, Dasgupta and Kakde proved the Brumer-Stark conjecture, away from 2, unconditionally, using a generalization of Ribet's method. We use the Dasgupta-Kakde results to prove an unconditional equivariant main conjecture, which is a generalization of that of Greither and Popescu. As applications of our main theorem we prove a generalization of a certain case of the main result of Dasgupta-Kakde and we compute the Fitting ideal of a certain naturally arising Iwasawa module. This is joint work with Cristian Popescu.

[Pre-talk at 1:20PM]