Colloquiums and Public Lectures 2018

We provide a probabilistic representations of the solution of some semilinear hyperbolic and high-order PDEs based on branching diffusions. These representations pave the way for a Monte-Carlo approximation of the solution, thus bypassing the curse of dimensionality. We illustrate the numerical implications in the context of some popular PDEs such as the Burger's equation, the nonlinear Klein-Gordon equation, a simplified scalar version of the Yang-Mills equation, a fourth-order nonlinear beam equation and the Gross-Pitaevskii PDE as an example of nonlinear Schrödinger equations.

The spectral theorem for symmetric matrices, and its offspring, the singular value decomposition, is used by facebook and no doubt many others in facial recognition.

We don’t have any clear notion of how we store and retrieve faces from memory; one may speculate that perhaps the brain does singular value decomposition also. This was pure speculation until Prof. Tsao (Caltech) showed that something similar is used by macaque monkeys. In Prof Tsao’s words:

"In linear algebra, you learn that if you project a 50-dimensional vector space onto a one-dimensional subspace, this mapping has a 49-dimensional null space," Tsao says. "We were stunned that, deep in the brain's visual system, the neurons are actually doing simple linear algebra. Each cell is literally taking a 50-dimensional vector space—face space—and projecting it onto a one-dimensional subspace. It was a revelation to see that each cell indeed has a 49-dimensional null space; this completely overturns the long-standing idea that single face cells are coding specific facial identities. Instead, what we've found is that these cells are beautifully simple linear projection machines.”

I will explain the connection between the variational proof of the spectral theorem, singular values, variance and covariance matrix, and applications of these methods to facial recognition and other problems.