Georgia Institute of Technology College of Sciences

The conventional point of view is that the Lagrangian is a scalar object, which through the Euler-Lagrange equations provides us with one single equation. However, there is a key integrability property of certain discrete systems called multidimensional consistency, which implies that we are dealing with infinite hierarchies of compatible equations. Wanting this property to be reflected in the Lagrangian formulation, we arrive naturally at the construction of Lagrangian multiforms, i.e., Lagrangians which are the components of a form and satisfy a closure relation. Then we can propose a new variational principle for discrete integrable systems which brings in the geometry of the space of independent variables, and from this principle derive any equation in the hierarchy.

The main result to be discussed will be the boundedness from $L^\infty \times L^\infty$ into $BMO$ of bilinear pseudodifferential operators with symbols in a range of bilinear H\"ormander classes of critical order. Such boundedness property is achieved by means of new continuity results for bilinear operators with symbols in certain classes and a new pointwise inequality relating bilinear operators and maximal functions. The role played by these estimates within the general theory will be addressed.

In the course of their work on the Unique Games Conjecture, Harrow, Kolla, and Schulman proved that the spherical maximal averaging operator on the hypercube satisfies an L^2 bound independent of dimension, published in 2013. Later, Krause extended the bound to all L^p with p > 1 and, together with Kolla, we extended the dimension-free bounds to arbitrary finite cliques. I will discuss the dimension-independence proofs for clique powers/hypercubes, focusing on spectral and operator semigroup theory. Finally, I will demonstrate examples of graphs whose Cartesian powers' maximal bounds behave poorly and present the current state and future directions of the project of identifying analogous asymptotics from a graph's basic structure.

It is well known that the Horn inequalities characterize the relationship of eigenvalues of Hermitian matrices A, B, and A+B. At the same time, similar inequalities characterize the relationship of the sizes of the Jordan models of a nilpotent matrix, of its restriction to an invariant subspace, and of its compression to the orthogonal complement. In this talk, we provide a direct, intersection theoretic, argument that the Jordan models of an operator of class C_0 (such operator can be thought of as the infinite dimensional generalization of matrices, that is an operator will be annihilated by an H-infinity function), of its restriction to an invariant subspace, and of its compression to the orthogonal complement, satisfy a multiplicative form of the Horn inequalities, where ‘inequality’ is replaced by ‘divisibility’. When one of these inequalities is saturated, we show that there exists a splitting of the operator into quasidirect summands which induces similar splittings for the restriction of the operator to the given invariant subspace and its compression to the orthogonal complement. Our approach also explains why the same combinatorics solves the eigenvalue and the Jordan form problems. This talk is based on the joint work with H. Bercovici.