Mathematics

Toward a unified theory of sparse dimensionability reduction in Euclidean space (joint with Jean Bourgain)

Jelani Nelson (Harvard School of Engineering and Applied Sciences)

Abstract: This talk will discuss sparse Johnson-Lindenstrauss transforms, i.e. sparse linear maps into much lower dimension which preserve the Euclidean geometry of a set of vectors. We derive upper bounds on the sufficient target dimension and sparsity of the projection matrix to achieve good dimensionality reduction. Our bounds depend on the geometry of the set of vectors, moving us away from worst-case analysis and toward instance-optimality. Joint work with Jean Bourgain.

Fundamental limitations in formation control with localized information

Ali Belabbas (UIUC ECE)

Abstract: Formation control deals with the design of control laws for multi-agent systems in which agents are required to stabilize at prescribed distances from each other. The most interesting cases, both in theory and practice, arise when agents only have access to partial information about the state and the objective of the system — we refer to this scenario as formation control with localized information. After formally introducing formation control, we will establish several fundamental limitations stemming from the information localization: first, we will show that for many formations, local stabilization around prescribed distances requires agents to know the objectives of agents whose states they cannot observe. Second, we will show that information localization can force the appearance of undesired, yet locally stable equilibria. Finally, if time permits, we will show that an often-used gradient-type control law is not robust to mismatch in the objectives.

Integration of generalized complex structures

Michael Bailey (CIRGET/UQAM/McGill)

Abstract: Generalized complex geometry is a generalization of both symplectic and complex geometry, proposed by Nigel Hitchin in 2002, which is of particular interest in string theory and mirror symmetry. Modulo a parity condition, generalized complex manifolds locally "look like" holomorphic Poisson manifolds, though globally they may not admit a complex structure at all. Therefore, locally they should integrate to holomorphic symplectic groupoids. One can take the global integration if one passes to holomorphic "symplectic" stacks. Earlier work by Crainic defined an integration for generalized complex structures which did not capture the holomorphic nature.

Topological Twisting and applications

Sheldon Katz (Illinois Math)

Abstract: I first describe topological twistings of a quantum field theory, yielding topological quantum field theories. I explain how Seiberg-Witten theory can be twisted so as to be formulated on compact oriented Riemannian manifolds, producing the Seiberg-Witten equations which led to the solution of much of Donaldson theory. I then introduce the idea of dimensional reduction, showing how N=1 SUSY in four dimensions reduces to (2,2) SUSY in two dimensions. Finally, I formulate the non-linear sigma model of maps from a Riemann surface to a Kahler manifold X. This is a two dimensional theory with (2,2) SUSY. I perform two topological twists, the A-twist and the B-twist, leading to the A-model and the B-model. The A-model is shown to localize on holomorphic maps and have observables described by the de Rham cohomology of X. After this localization, the correlation functions can be defined and computed rigorously within mathematics in terms of Gromov-Witten invariants.

To Be Announced

Chenxi Wu (Cornell)