Driven by recent technological advancements, behavior and brain activity can now be measured at an unprecedented resolution and scale. This “big-data” revolution is akin to a similar revolution in biology. In biology, the wealth of data allowed systems-biologists to uncover the underlying design principles that are shared among biological systems. In my studies, I apply design principles from systems-biology to cognitive phenomena. In my talk I will demonstrate this approach in regard to creative search.
Motivated by three-dimensional N=4 superconformal field theory, in 2016 Beem, Peelaers and Rastelli considered short even star-products for homogeneous symplectic singularities (more precisely, hyper-Kahler cones) and conjectured that they exist and depend on finitely many parameters. We prove the dependence on finitely many parameters in general and existence for a large class of examples, using the connection of this problem with zeroth Hochschild homology of quantizations suggested by Kontsevich.
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.
In the mid-18th century,Euler derived his famous equations of motion of an incompressible fluid, one ofthe most studied equations in hydrodynamics. More than 200 years later, in1966, Arnold observed that they are, in fact, geodesic equations on the(infinite dimensional) Lie group of volume-preserving diffeomorphisms of amanifold, endowed with a certain 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.