Mathematical Analysis

at the University of Birmingham

wave front in the Heisenberg group

evolving Kakeya sets

- harmonic analysis (and its interactions with partial differential equations, combinatorial geometry, representation theory, and geometric, complex and functional analysis);
- partial differential equations (including dispersive, stochastic, elliptic and kinetic systems);
- geometric measure theory (measure-theoretic regularity and the geometry of exceptional sets arising in real and harmonic analysis);
- topological dynamics (including shadowing, symbolic dynamics and the structure of invariant subsets).

Jonathan Bennett Jonathan's interests lie in multivariable Euclidean harmonic analysis and its interactions with problems in geometric analysis and combinatorics. Recently he has been investigating the scope of heat-flow methods and induction-on-scales arguments in the analysis of geometric inequalities arising in the restriction theory for the Fourier transform. Of particular interest to Jonathan are the many ways in which oscillatory phenomena are governed by underlying geometric notions such as curvature or transversality. more info: university profile - publications (MathSciNet) |

Chris Good Chris's research is in topological dynamics and set theoretic topology. His particular interests include (in no particular order): the structure of omega-limit sets; shadowing; symbolic dynamics of tent maps; topologies making given mappings continuous; generalised metric spaces and monotonicity; Dowker spaces and normality in products; the construction of counterexamples. more info: university profile - personal webpage - publications (MathSciNet) |

Susana Gutierrez Susana's research focuses on the analysis of partial differential equations modelling physical processes using techniques and perspectives from harmonic analysis and dispersive PDEs. Susana has studied singularity formation phenomena for nonlinear Schrödinger equations and related geometric flows (Localized Induction Approximation and Landau-Lifshitz-Gilbert equations), as well as properties of the solutions of kinetic transport equations and nonlinear kinetic models of chemotaxis. more info: university profile - publications (MathSciNet) |

Olga Maleva Olga's research concerns differentiability of Lipschitz mappings between finite - and infinite - dimensional spaces and the geometry of exceptional sets. Olga is particularly interested in a range of topics related to the converse to the classical theorem of Rademacher. Namely, she has been working on establishing finer and measure-theoretic regularity properties (such as porosity, rectifiability, Hausdorff/Minkowski dimensions etc.) of universal differentiability sets and sets on which Lipschitz mappings behave in the worst possible way, as well as the behaviour of typical Lipschitz functions. more info: university profile - personal webpage - publications (MathSciNet) |

Alessio Martini Alessio's main area of research is harmonic analysis on Lie groups and their homogeneous spaces. This area is characterised by a strong interplay between techniques of functional and harmonic analysis and results from algebra and representation theory. Alessio has worked on uncertainty inequalities, on spectral theory for commuting differential operators and, more recently, on the problem of obtaining sharp spectral multiplier theorems for sub-elliptic operators. more info: university profile - personal webpage - arXiv eprints - publications (MathSciNet) |

Andrew Morris Andrew’s research concerns the development of modern techniques in harmonic analysis, functional calculus and geometric measure theory for application to partial differential equations on Riemannian manifolds and rough domains. This includes elliptic systems with rough coefficients, local T(b) techniques, first-order methods, quadratic estimates, holomorphic functional calculus, singular integral theory, layer potentials, Hardy spaces, boundary value problems and uniform rectifiability. more info: university profile - arXiv eprints - publications (MathSciNet) |

Diogo Oliveira e Silva Diogo's research focuses on harmonic analysis with special emphasis on sharp inequalities, oscillatory integrals and restriction theory for the Fourier transform. More recently Diogo became interested in uncertainty principles and nonlinear Fourier analysis. more info: university profile - personal webpage - publications (MathSciNet) |

Maria Carmen Reguera Maria works on harmonic analysis and especially on the theory of weighted inequalities for singular integral operators. She is also interested in related questions in operator theory for Bergman spaces and geometric analysis. more info: university profile - publications (MathSciNet) |

Yuzhao Wang
Yuzhao's main research interests are Nonlinear Partial Differential Equations (PDEs) and Harmonic Analysis. This includes the study of nonlinear dispersive PDEs such as nonlinear Schrödinger equations, nonlinear wave equations, and the KdV equation by using techniques from PDEs, Harmonic Analysis, and Probability theory. Yuzhao investigates well-posedness (existence, uniqueness, and stability of solutions) in both deterministic and probabilistic settings, existence of invariant measures, and Strichartz estimates in different settings. He is also interested in Fourier restriction theory and l more info: university profile - personal webpage - publications (MathSciNet) |