For a pdf version of my thesis, click here.
My research interests lie at the applied mathematics/data analysis interface, driven by the need to provide rigorous mathematical foundations for modeling tools used in applications. My research is setting out a path for bringing together expertise from the areas of PDE analysis and data analysis, establishing new research topics at their intersection and in their own right.
Mathematical Advances in PDE Analysis
Nonlinear partial differential equations (PDEs) arising from models in physics, biology, fluid mechanics, chemistry, engineering, economics and social sciences often have hidden connections across applications. PDE analysis provides techniques to explore the structural similarities in their dynamics, giving key insights into the driving forces behind a process and providing powerful predictive tools. Nonlinearities and long-range interactions (e.g. arising from interactions between agents elsewhere), in addition to local effects pose analytical challenges that cannot be tackled with conventional PDE methods. My work focuses on developing new mathematical tools to understand the behavior of these models, informed and inspired by a range of applications.
- Long-time behavior for nonlinear and nonlocal PDEs: Often, PDE models that arise in applications are so complex that it is hopeless to find explicit solutions. Information about the behavior of the system after a very long time gives key insights into the characteristics of the governing dynamics without knowing solutions explicitly. For this reason, the long-time behavior of a given PDE is often one of the key modeling questions, and a challenging one if different forces are at play in a nonlinear and nonlocal manner. I’m interested in nonlinear nonlocal models that exhibit competing effects of attractive vs repulsive forces, with the aim of understanding long-time behavior and pattern formation.
- Development of optimal transport tools: Optimal transport theory is the mathematical framework for optimally (according to a given cost function) transforming a given distribution of mass into another. Despite the mathematical theory having advanced significantly over the past century, this has passed largely unnoticed by other areas of science. It is only recently that these techniques have made their way from the mathematics community into a broader range of applications, from machine learning, data science, quantum and classical mechanics, computer science, engineering and image processing to economics and social sciences. There is now a real demand from other scientists for mathematicians to communicate the powerful tools that optimal transport can provide, and an opportunity for applications to guide new mathematical developments in optimal transport theory. My goal is to develop new theory that serves that aim.
- Large graph limits: Graph Laplacians are discrete operators defined on graphs that are widely used to identify geometric structures in data. Their spectral properties play a central role in a number of machine learning algorithms for data analysis tasks. Recent mathematical results on large data limits of these operators open up exciting avenues for the development of novel efficient algorithms able to handle big data.
Mathematical Advances in Data Analysis
With more and more data available, machine learning approaches start to play a crucial role. Developments in machine learning are currently largely driven by outcomes rather than mathematical understanding, leading to so-called “black box” algorithms. With machine learning likely entering bigger parts of daily life in the near future, rigorous knowledge on how these algorithms work, and when they may fail, is of utmost importance; however, currently, the theory is lagging behind the practice. There are several ways how PDE techniques can play a particular role in this process. I’m interested in developing unsupervised and semi-supervised learning algorithms that are both promising in terms of outcomes, and allow for rigorous mathematical analysis on (i) consistency, and (ii) uncertainty quantification. Another promising avenue for uncertainty quantification in data-driven applications is the framework of Bayesian inference. My goal is to bring together knowledge from inverse problems, gradient flows and optimal transport tools, as well as graph Laplacians and clustering algorithms to develop novel mathematical theory for a number of data-driven applications from climate modeling to mechanics and power grid systems.
- Properties and applications of graph Laplacians: Although there is good amount of empirical evidence that spectral clustering algorithms based on graph Laplacians by far outperform simpler, more widely used methods such as k-means, a rigorous mathematical justification is still missing.
- Unsupervised and semi-supervised learning: Unsupervised learning is the task of detecting structures and patterns in data without reference information such as labelled training data (data clustering); supervised learning uses labelled data to train a model, to then classify new data points using this model (data classification); semi-supervised learning (SSL) combines the previous two, propagating labels from a small subset of labelled data to a larger data set of unlabelled points (data classification). Using graph Laplacians in SSL algorithms, one can leverage correlations and geometric information in the data set, an approach which is especially powerful if labels are difficult or expensive to obtain.
- Inverse problems and data-driven applications: The theory of inverse problems is inherently data-driven, formulated via a model that is enhanced with observed data. We are currently developing a framework to combine inverse problems with optimal transport inspired tools to provide algorithms that not only solve an inverse problem (that is, finding the most likely parameters that generated noisily observed data), but also provide tools for uncertainty quantification. Additionally, I am interested in data-driven applications more generally, such developing a new framework for stochastic data-driven mechanics (linking to optimal transportation), and new clustering algorithms for power grids (linking to graph Laplacians).
Applications to Mathematical Modeling in Social Sciences
Many people rely on online sources for their news and information, including social media such as Facebook and Twitter. The content on social media varies in quality and trustworthiness, both of which have an enormous influence on online discussions on issues such as economic and social policy and on interactions with others of differing views; it is also known to affect actions such as voting and protests. Modeling the sharing of information and how this information impacts the opinion of individuals in a communication network can give insights about intrinsic characteristics of these dynamics.
Models for phenomena in social science can provide powerful predictive tools, however, are often challenging to formulate as there are no (or few) concrete physical laws or proven causal relationships to rely on. This generates difficult modelling questions, where I believe mathematicians can play a crucial role, providing systematic and rigorous ways of uncovering inherent properties of these models, and by testing and clarifying the impact of different modeling assumptions. My long-term goals are (i) to provide analysis for different variants and generalizations of models in the literature; (ii) to suggest mathematically informed modeling set-ups, making use of structures that are either already well-understood in other areas of mathematics, or that we can obtain understanding for; (iii) to test and compare those models using synthetic and empirical data sets; and (iv) to use these insights to provide tools for informed decision-making.
- Opinion dynamics under media influence: By augmenting random graph structures of interacting individuals with media nodes, we seeks to develop understanding how media outlets impact opinions in a communication network. Having a good measure for media impact opens up avenues both to design prevention strategies against extremism and undesirable opinion ‘echo-chambers’, as well as to maximize media impact for example in the context of advertisement campaigns.
- Macroscopic descriptions of opinion dynamics: Some of the key characteristics underlying the dynamics of opinions are macroscopic in nature, and are best observed and analyzed at a macroscopic scale, i.e. corresponding to the setting where the number of individuals in a network tends to infinity. It is not an obvious question how the network structure and its interconnection with the graph of media outlets behaves in such a limit.
- Modeling approaches in social science: I plan to investigate several variants and generalizations of opinion dynamics models in an effort to create a more realistic theory.
I believe that it is not just about what mathematical analysis can do for applications, but also what applications can do for mathematics. Applications that benefit from a PDE approach often generate new questions giving rise to novel mathematical theories, and thus providing insights not just for the application at hand, but also enabling to push boundaries in seemingly unrelated fields. I am interested in how PDE techniques can be employed to answer modeling questions across different scales, disciplines and applications.
In the past, I worked on understanding the asymptotic behavior of solutions for different types of kinetic equations and reaction-diffusion equations. Many physical systems converge to certain equilibrium states, and one is often interested to know in what sense they converge, if we can compute an explicit rate of convergence and whether all or just some solutions converge. Sometimes, the long-time behavior of an initial datum can be classified in terms of the parameters of the model similar to the critical mass dichotomy of the Patlack-Keller-Segel model for bacterial chemotaxis.
In the case of gradient flows such as the fast diffusion equation, there exist a fascinating connection between the long-time behavior of solutions and versions of the well-known Hardy-Littlewood-Sobolev inequality. I’m interested in understanding how similar connections between more complex reaction-diffusion equations and corresponding functional inequalities can help us analyze the behavior of solutions.
In the past, I have worked on parabolic and hyperbolic scaling techniques, making the connection between microscopic and macroscopic models with applications in bacterial chemotaxis and collective animal behaviour. Further, I used hypocoercivity techniques to prove existence and uniqueness of a global Gibbs state in a kinetic model describing the fibre lay-down process during non-woven textile production.