Research_

Analysis and probability

Taming infinite and random processes

Developing the mathematical underpinnings that improve our understanding of critical or random phenomena in real-world problems.

Probability is the mathematical study of random phenomena. It has applications in diverse fields such as biology, finance, telecommunications and traffic networks, and is the theoretical basis for statistics and the fields of Machine Learning and Artificial Intelligence.

Analysis is a broad area of modern mathematics that has grown out of and maintains connections with multiple diverse topics in science, engineering, and technology.

Our group pursues several lines of investigation, including basic research in pure mathematics to improve general understanding of Analysis and Probability and their interactions, as well as development of mathematical results that lay the groundwork for real-world applications in engineering, social and natural sciences, medicine and healthcare.

*Header image: Weichang Yu: "Discriminant Analysis Distributions for Melanoma”*

Key researchers: Florica Cirstea, Daniel Daners, Ben Goldys, Georg Gottwald, Daniel Hauer, Nalini Joshi, Robert Marangell, Martin Wechselberger, Haotian Wu, Zhou Zhang.

Partial differential equations (PDEs) play a key role in modelling real-world phenomena occurring in physics, chemistry, biology, and economics. In a given model, PDEs represent the mathematical description of different laws in a system interacting with each other.

Within the fundamental goal of solving PDEs of different types (elliptic, parabolic, or hyperbolic), mathematical questions that arise include determining the existence, uniqueness, and qualitative properties of solutions.

To solve these problems, our research group contributes with new results by investigating:

- the existence and non-existence of solutions;
- the regularity properties of solutions (boundedness, Harnack inequalities, and gradient estimates);
- the local and global asymptotic profile of solutions (Singularity Theory);
- the stability of solutions with respect to singular domain perturbations;
- the positivity properties of solutions;
- the (long-time) stability of solutions;
- eigenvalue problems and isoperimetric inequalities;
- the Dirichlet-to-Neumann operator.

To achieve our results, we often exploit geometric (e.g. concavity/convexity) properties of solutions, apply maximum principles, abstract functional analytic concepts such as linear and nonlinear semigroup theory, and use concepts from stochastic analysis and optimal transport theory. This often leads to the development of new analytical methods.

**Seminars**

Asia-Pacific Analysis and PDE Seminar

Key researchers: Jennifer Chan, Clara Grazian, Uri Keich, Linh Nghiem, John Ormerod, Shelton Peiris, Michael Stewart, Qiuzhuang Sun, Garth Tarr, Qiying Wang, Rachel Wang.

Statistical methods are the bedrock upon which quantitative science is built. These in turn are built upon probabilistic statistical models, which determine which methods are appropriate and/or valid in different applied contexts.

Beyond the scientific sphere, understanding what is and is not possible for different statistical procedures is also crucial for society more broadly: a statistically literate population is the best weapon against questionable statistical claims (“lies, damned lies and statistics”).

Research into statistical theory defines the capabilities and limits of statistical methods and reasoning. Development of the statistical theory underlying today’s highly complex data models is a major challenge.

At the University of Sydney, we have made significant progress expanding the theory for models appropriate for classification, clustering (and other machine learning methods), dependence and extremes, high-dimensional inference, model selection, survival analysis and time series analysis.

These methods are in turn applicable to a vast range of applications: biological, econometric, educational, environmental, financial, industrial, marketing, medical, psychological, and many others.

**Seminars**

- Probability and stochastic processes
- Global analysis, analysis on manifolds