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Weichang Yu: "Discriminant Analysis Distributions for Melanoma”
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”

Research areas

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

Nonlinear Harmonic Analysis Seminar

PDE Seminar

  • Probability and stochastic processes
  • Statistical theory
  • Global analysis, analysis on manifolds