Analysis and Geometry Program
This Program in the MSI is coordinated by Alan McIntosh (Room 2141, Dedman Bldg). The program includes work in the following areas:
several complex variables; Banach algebras; convex geometry in statistical learning theory; spectral theory of operators; harmonic analysis on Lie groups, manifolds and Lipschitz surfaces; microlocal analysis; non-commutative geometry; and applications to PDE.
Please click on the links below for further information about the program.
- In 2009, Mike Eastwood joined the CMA on one of the Australian Government's prestigious Federation Fellowships. In their press release, the Australian Research Council wrote "Professor Eastwood is one of the world's leading experts in conformal differential geometry. His work on the development of transform methods linking mathematical physics, differential geometry, harmonic analysis and special function theory is internationally acclaimed. Professor Eastwood's new research will focus on the interaction between geometry, differential equations and symmetry in conformal differential geometry. Advances in this area will provide essential tools in fundamental science and establish novel links between neighbouring fields of mathematics."
- Andreas Axelsson, who graduated in 2003 with a PhD under the supervision of Alan McIntosh, has been awarded the Tage Erlander Prize 2009 in science and techology, by the Swedish Royal Academy of Science "for his contributions to the development of the functional calculus in harmonic analysis, which led to the proof of Lions' conjecture." One such prize is awarded each year to a younger Swedish scientist. It is worth A$18,000 for research and an additional A$9000 for organizing a symposium or workshop. For further information, see Andreas' web page.
- Alan Carey's work with Phillips on spectral flow and their collaboration with Adam Rennie and Fydor Sukochev on index theory in noncommutative geometry has been widely recognised. Alan received the 2008 Moyal Medal for his contribution to mathematics, particularly in the area of physically motivated mathematical problems in geometry and index theory.
- Bryan Wang recently resolved a long standing puzzle in index theory, namely how to handle the Gysin map in K-theory for the non-K-oriented maps. (The K-oriented case was settled by Atiyah and Hirzebruch in 1958.) He also made a breakthrough in twisted index theory and established from a mathematical perspective what the geometric significance of the string theorists' D-branes really is.
- Alan McIntosh, together with Pascal Auscher, Andreas Axelsson and Stephen Keith, obtained quadratic estimates for operators arising in partial differential equations (PDE) and geometry, using techniques developed for the proof of the Square Root Problem of Kato. This work has applications concerning solvability of elliptic PDE with bounded measurable coefficients and square integrable boundary values.
- Andrew Hassell showed that almost every stadium domain is not quantum unique ergodic, meaning there exist sequences of eigenfunctions that concentrate unevenly in some regions of the domain. (It is known that almost all eigenfunctions have the opposite property, that is that there is a density one sequence of eigenfunctions that becames asymptotically equidistributed.) These comprise the first examples of plane domains that are quantum ergodic but not quantum unique ergodic.
- Derek Robinson co-authored the book Analysis on Lie Groups of Polynomial Growth (Birkhauser, 2003) with Nick Dungey and Tom ter Elst.
- Rick Loy, working in collaboration with Laustsen and Read, constructed a Banach space whose algebra of operators has a closed ideal lattice structure not previously encountered.
- Alexander Isaev, with Kruzhilin, solved a 10-year old problem on characterising complex n-space by its automorphism group.
There will be a course at 4th Year/PhD level in the 2nd semester, taught be Alan McIntosh and Alan Carey. It would be on the Spectral Theory and Functional Calculi of Linear Operators in Banach spaces, in particular in Hilbert spaces, and on Fredholm theory.
Prerequisite: Analysis III and some knowledge of complex analysis.
Information about graduate courses offered at the MSI can be found here. Enquiries related to PhD supervision can be directed to the postgraduate research coordinator Dr Ben Andrews. If you have a particular supervisor in mind, then please contact them directly as well.
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