In the mid-18th century, Euler derived hisfamous equations of motion of an incompressible fluid, one of the most studiedequations in hydrodynamics. More than 200 years later, in 1966, Arnold observedthat they are, in fact, geodesic equations on the (infinite dimensional)Lie group of volume-preserving diffeomorphisms of a manifold, endowed with acertain right-invariant Riemannian metric.
A countable group is said to be homogeneous if whenever tuples of elements u, v satisfy the same first-order formulas there is an automorphism of the group sending one to the other. We had previously proved with Rizos Sklinos that free groups are homogeneous, while most surface groups aren't. In a joint work with Ayala Dente-Byron, we extend this to give a complete characterization of torsion-free hyperbolic groups that are homogeneous.
We study issue-by-issue voting and robust mechanism design in multidimensional frameworks where privately informed agents
have preferences induced by general norms. We uncover the deep connections between dominant strategy incentive compatibility (DIC) on the one hand,
and several geometric/functional analytic concepts on the other. Our main results are:
1) Marginal medians are DIC if and only if they are calculated
with respect to coordinates defined by a basis such that the norm is orthant-monotonic in the associated coordinate system.
Weprove eigenfunction and quasimode estimates on compact Riemannian manifolds for Schr\”odingeroperators, $H_V=-\Delta_g+V$ involving critically singular potentials $V$ which weassume tobe in $L^{n/2}$ and/or the Kato class ${\mathcal K}$. Our proof is basedon modifying the oscillatory integral/resolvent approachthat was used to study the case where $V \equiv 0$ using recently developedtechniques by many authorsto study variable coefficient analogs of the uniform Sobolev estimates ofKenig, Ruiz and the speaker. Read more about Analysis Seminar: Cancelled
Abstract: The profinite completion of a free profinite group on an infinite set of generators is a profinite group of greater rank. However, it is still unknown whether it is a free profinite group as well. I am going to present some partial results regarding this question.
Let Γ be a discrete group. A group Γ is called unitarisable if for any Hilbert space H and
any uniformly bounded representation π : Γ → B(H) of Γ on H there exists a bounded operator
S : H → H such that S^{−1}π(g)S is a unitary representation for any g ∈ Γ. It is well known that
amenable groups are unitarisable. It has been open ever since whether amenability characterises unitarisability of groups.
Dixmier: Are all unitarisable groups amenable?
One of the approaches to study unitarisability is related to the space of the Littlewood functions
Title - Measure rigidity of Anosov flows via the factorization method.
Abstract: Anosov flows are central objects in dynamics, generalizing the basic example of a geodesic flow over a Riemann surface.
In the talk we will introduce those flows and their dynamical behavior. Moreover, we show how the factorization method, pioneered by Eskin and Mirzakhani in their groundbreaking work about measure rigidity for the moduli space of translation surfaces, can be adapted to smooth ergodic theory and in particular towards the study of Anosov flows.
