Group members
Group members:
Neural networks have received a lot of attention lately, basically due to their ability to solve problems that typically only humans could solve. Despite their admirable function in certain key areas, little is actually known about why it works well. My research programme is aimed at increasing our understanding of regression models, and neural networks in particular. The idea centers around reformulating the stochastic optimization procedure used in practice into a gradient flow of a so called Dirichlet energy, this allows us to leverage our deep understanding of degenerate parabolic equations as a technical tool to understand stochastic optimization. However the optimization of neural networks for instance, gives rise to PDEs that degenerate in ways we do not fully understand. Thus the questions we wish to answer is, why does the equation degenerate?, how does it degenerate?, and what is the impact for convergence if it degenerates? Understanding more about these questions will not only further our theoretical understanding of neural networks but also help us choose better topologies, optimization algorithms, activation functions, and regularization techniques.
My research focuses on spectral theory of operators and theory of orthogonal polynomials in its various settings (real line, unit circle, multiple, matrix-valued, etc) and applications (in random matrix theory and mathematical physics). Recently we have started exploring properties of polynomials that are simultaneously orthogonal with respect to several measures, the so-called multiple orthogonal polynomials. The theory is quite new, so is rich for potential publications. The project can be tailored to fit either a bachelor, a master, or a PhD student.
I do research in Partial Differential Equations (PDEs). I am particularly interested in regularity properties for free boundary problems, non-local and degenerate PDEs, nonlinear eigenvalue problems and functional inequalities.
Free boundary problems: I have studied regularity properties of various free boundary problems. The prototype problem is the classical obstacle problem, which arises when considering an elastic membrane being pushed up from below by an obstacle. In a simplified form it can be described by the equation ∆u = f χ {u>0}. The main questions of interest are 1) the regularity of u and 2) the regularity of the free boundary, ∂{u > 0}.
Non-local non-linear problems: A natural problem in the calculus of variations in the fractional Sobolev space W s,p is to study minimizers of the W^s,p-seminorm for s between 0 and 1 and p larger than 1. The corresponding Euler-Lagrange equation is a nonlinear singular integral. This type of operator is one natural non-local version of the p-Laplace operator (formally when s = 1) and has applications in the modeling of sandpiles, image and data processing, optimal mass transportation and in optimal Hölder extensions.
Nonlinear eigenvalue problems and doubly nonlinear flows: Given u, a ground state of the Laplacian with eigenvalue λ, in a bounded and open set Ω, it is well known that if v solves the initial value problem for the heat equation with zero boundary data, then u(x) is the limit of exp(λ t) v(x, t), as t goes to infinity, unless the limit it zero. Together with Ryan Hynd (UPenn), I initiated a study of this type of results and an inverse iteration method for nonlinear eigenvalue problems and also started to developed a general theory for nonlinear eigenvalue problems in Banach spaces. Among other things we establish completely new connections between various doubly nonlinear evolution equations and nonlinear eigenvalue problems
I work in mathematical physics, i.e. the rigorous mathematical modeling of physically relevant systems. Although I develop and use mainly tools from analysis, spectral theory, functional inequalities and variational methods, I also frequently apply geometry, topology, algebra and representation theory. My research interests typically concern quantum systems with exotic symmetries such as intermediate and fractional statistics in lower dimensions (with effective particles known as anyons) as well as supersymmetric matrix models, but they also include Clifford (geometric) algebras and their applications, quantum gravity and quantum geometry.
My research is devoted to partial differential equations and to several interdisciplinary projects including data driven discovery, deep learning, blockchains and cryptocurrencies.
My interdisciplinary research currently has at least two strains. As part of the first strain of research we use machine learning, and deep learning in particular, to discover PDEs hidden in complex data sets from measurement data. I am also interested in developing and contributing to the field of continuous models/methods for Deep Neural Networks (DNN) with a particular focus on (systems of) Ordinary Differential Equations (ODEs), Stochastic Differential Equations (SDEs) and Partial Differential Equations (PDEs). My interest here is to use continuous models/methods to gain a deeper understanding of three of the key problems in deep learning: optimization through methods like Stochastic Gradient Descent (SGD), regularization through methods like dropout and the problem of generalization. The purpose of the second strain of research is to contribute to the understanding of blockchains and their cryptocurrencies and in particular to the understanding of the value formation process of cryptocurrencies and derivatives thereof. In general, having worked several years in the financial industry, I am also interested in financial mathematics and risk management.
Within the area of partial differential equations I mainly consider elliptic and parabolic, linear and non-linear partial differential equations and I am particularly interested in boundary behavior and free boundaries. To give a flavor of my research I here briefly mention a few of my recent contributions. Within each of these areas there are ample sets of open important research problems.
Together with John Lewis I have established new results concerning boundary Harnack inequalities and the Martin boundary problem, for non-negative solutions to equations of p-Laplace type with variable coefficients. The key novelty is that we consider solutions which vanish only on a low-dimensional set and this is different compared to the more traditional setting of boundary value problems set in the geometrical situation of a bounded domain. We establish our quantitative and scale-invariant estimates in the context of low-dimensional Reifenberg flat sets.
I have worked on the Kolmogorov-Fokker-Planck operator in unbounded Lipschitz type domains. The defining function of the domain satisfies a uniform Lipschitz condition, adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator, as well as an additional regularity condition formulated in terms of a Carleson measure. Among other things I have proved that the associated parabolic measure is absolutely continuous with respect to a surface measure and that the associated Radon-Nikodym derivative defines an weight with respect to this surface measure. The latter can be used to derive quantitative estimates for the parabolic measure.
Together with Pascal Auscher and Moritz Egert I have established new results concerning boundary value problems in the upper half-space for second order parabolic equations (and systems) assuming only measurability and some transversal regularity in the coefficients of the elliptic part. In addition we solve the Kato square root problem for parabolic operators with coefficients of the elliptic part depending measurably on all variables.
My research interests are currently within Fourier analysis, partial differential equations, complex analysis and complex geometry and mathematical physics.