Caltech

Recently, the Newman-Janis shift has been revisited from the angle of scattering amplitudes in terms of the so-called "massive spinor-helicity variables," tracing back to Penrose and Perjés in the 70s. However, well-established results are limited in the same-helicity (self-dual) sector, while a puzzle of spurious poles arises in mixed-helicity sectors. This talk will outline how massive twistor theory can reproduce the same-helicity results while offering a possible solution to the spurious pole puzzle. Firstly, the Newman-Janis shift in the same-helicity sector is derived from a complexified version of the equivalence principle. Secondly, the massive twistor particle is coupled to background fields from bottom-up and top-down perspectives. The former is based on perturbations of symplectic structures in massive twistor space. The latter provides a generalization of Newman-Janis shift in generic backgrounds, which also leads to "curved massive twistor space" and its deformed massive incidence relation. Lastly, the Feynman rules of the first-quantized massive twistor particle and their physical interpretation are briefly discussed. Overall, a significant emphasis is put on the Kähler geometry ("zig-z̄ag structure") of massive twistor space, which eventually connects to a worldsheet structure of the Kerr solution.

Tom Hutchcroft and I have been working to develop a general theory of percolation on arbitrary finite transitive graphs. This extends from percolation on local approximations to infinite graphs, such as a sequence of tori, to percolation on the complete graphs - the Erdős-Rényi model. I will summarise our progress on the basic questions: When is there a phase transition for the emergence of a giant cluster? When is the giant cluster unique? How does this relate to percolation on infinite graphs? I will then sketch our proof that for finite transitive graphs with uniformly bounded vertex degrees, the supercritical giant cluster is unique, verifying a conjecture of Benjamini from 2001.

We consider the Bayesian inverse problem of inferring the initial condition of a linear dynamical system from noisy output measurements taken after the initial time. In practical applications, the large dimension of the dynamical system state poses a computational obstacle to computing the exact posterior distribution. Balanced truncation is a system-theoretic method for model reduction which obtains an efficient reduced-dimension dynamical system by projecting the system operators onto state directions which simultaneously maximize energies defined by reachability and observability Gramians. We show that in our inference setting, the prior covariance and Fisher information matrices can be naturally interpreted as reachability and observability Gramians, respectively. We use these connections to propose a balancing approach to model reduction for the inference setting. The resulting reduced model then inherits stability properties and error bounds from system theory, and yields an optimal posterior covariance approximation. 

Neural networks have shown great success at learning function approximators between spaces X and Y, in the setting where X is a finite dimensional Euclidean space and where Y is either a finite dimensional Euclidean space (regression) or a set of finite cardinality (classification); the neural networks learn the approximator from N data pairs {x_n, y_n}. In many problems arising in the physical and engineering sciences it is desirable to generalize this setting to learn operators between spaces of functions X and Y. The talk will overview recent work in this context.

Then the talk will focus on work aimed at addressing the problem of learning operators which define the constitutive model characterizing the macroscopic behaviour of multiscale materials arising in material modeling. Mathematically this corresponds to using machine learning to determine appropriate homogenized equations, using data generated at the microscopic scale. Applications to visco-elasticity and crystal-plasticity are given.

The Teichmüller space for a closed surface of genus g is the space of marked complex/hyperbolic structures on the surface. Teichmüller space also identifies with the space of Fuchsian representations of the fundamental group into PSL(2,R) (mod conjugation). Higher Teichmüller theory concerns special representations of surface (or hyperbolic) groups into higher rank Lie groups of non-compact type.

In this talk, I will give a brief introduction to level-set methods for image analysis. I will then describe an application of level-sets to the construction of filtrations for persistent homology computations. I will present several case studies with various spatial data sets using this construction, including applications to voting, analyzing urban street patterns, and spiderwebs. I will conclude by discussing the types of data which I might imagine such methods to be suitable for analyzing and suggesting a few potential future applications of level-set based computations.

The goal of the talk will be to introduce a class of functions that answer a question in topology, can be computed via analytic methods more common in the theory of dynamical systems, and in the end turn out to enjoy beautiful modular properties of the type first observed by Ramanujan. If time permits, we will discuss connections with vertex algebras and physics of BPS states which play an important role, but will be hidden "under the hood" in much of the talk.

Random matrices now play a role in many areas of theoretical, applied, and computational mathematics. Therefore, it is desirable to have tools for studying random matrices that are flexible, easy to use, and powerful. Over the last fifteen years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that balance these criteria. This talk offers an invitation to the field of matrix concentration inequalities and their applications.

Whether the 3D incompressible Euler or Navier-Stokes equations

can develop a finite time singularity from smooth initial data has been

an outstanding open problem. Here we review some existing computational

and theoretical work on possible finite blow-up of the 3D Euler equations.

We show that the geometric regularity of vortex filaments, even in an

extremely localized region, can lead to dynamic depletion of vortex

stretching, thus avoid finite time blowup of the 3D Euler equations.

Further, we perform large scale computations of the 3D Euler equations

to re-examine the two slightly perturbed anti-parallel vortex tubes which

is considered as one of the most attractive candidates for a finite time

blowup of the 3D Euler equations. We found that there is tremendous dynamic

depletion of vortex stretching and the maximum vorticity does not grow faster

than double exponential in time. Finally, we present a new class of solutions

for the 3D Euler and Navier-Stokes equations, which exhibit very interesting

dynamic growth property. By exploiting the special nonlinear structure of the

equations, we prove nonlinear stability and the global regularity of this class of solutions.