 Matthew Blair [web site]
My research deals with harmonic analysis and its interactions with partial differential equations of wave and Schroedinger type. Much of my work involves finding effective ways to represent solutions to these equations and understanding the oscillatory integrals that appear as a result. In particular, I am interested in the development of regularity and norm estimates for solutions that can be obtained this way. This includes Strichartz, local smoothing, and squarefunction inequalities, which are all families of space-time integrability (L^p) estimates. They are important for various nonlinear equations and can also have applications to eigenfunction problems.
Most recently, my work has focused on establishing these inequalities for solutions over domains with a boundary. Here the boundary conditions can influence the development of waves and affect the flow of energy. Understanding this phenomena often involves a connection with the field of microlocal analysis. Here one studies waves by carefully localizing them both in space and in direction of propagation. One effective method in this direction is to represent waves as superpositions of "wave packets", approximate solutions which are highly concentrated in both space and in frequency. Wave packet methods continue to be influenced by ideas from both microlocal and harmonic analysis.
 Charles P. Boyer [web site]
My research interests range over a fairly broad part of modern differential geometry. It interfaces with the theory of Lie groups and both algebraic geometry and algebraic topology. Recently there have been two main threads. The first involves the study of the topology of moduli spaces of holomorphic maps, the so-called sigma-models, and of the moduli spaces of instantons.
The second uses contact geometry together with Lie group theoretical methods to give explicit constructions of Einstein manifolds of positive scalar curvature, and then to study the topology of such manifolds. Very recent work uses methods of algebraic geometry, more specifically Mori theory, to construct Einstein metrics on 5-manifolds, as well as homotopy spheres in all odd dimensions. This work appears in my monograph Sasakian Geometry with K. Galicki published by Oxford University Press
My earlier work was also two-tiered. The first involved group representation theory applied to problems of Mathematical Physics, in particular, to the problem of separation of variables of the partial differential equations that appear in Mathematical Physics. The second used Lie group methods to study complex geometry with special applications to general relativity.
 Alexandru Buium [web site]
I am interested in number theory and algebraic geometry. My recent work is devoted to establishing an arithmetic analogue of the theory of (ordinary/partial) differential equations. In the (ordinary) arithmetic theory the ``independent time variable'' t is replaced by a fixed prime integer p. Smooth real functions, x(t), are replaced by integer numbers, a, or, more generally, by integers in various (completions of) number fields.
The derivative operator on functions x -> dx/dt is replaced by a "Fermat quotient operator'' d which, on integer numbers, acts as da:=(a-a^p)/p. For details see my research monograph ``Arithmetic Differential Equations", Math. Surveys and Monographs 118, AMS, 2005.
 Terry Loring [web site]
I primarily conduct research in functional analysis and mathematical physics. In particular, I study approximate and exact solutions to equations and relations, both in finite matrices and in general C*-algebras. There are connections between the K-theory of C*-algebras and condensed matter physics. Much of my current work is directed to symmetries in C*-algebras with applications to topological insulators.
I have developed some new algorithms in structured linear algebra to enable numerical index studies of 2D and 3D topological insulators.
 Michael Nakamaye [web site]
My research is in complex algebraic geometry with a focus on questions of positivity. More specifically, given a line bundle on a projective variety I am interested in studying properties of its base locus and, in particular, finding relationships between the positivity of the bundle and the complexity of the base locus.
Much of the motivation for these questions comes from number theoretic questions where strong constraints on base loci of bundles is imposed by certain arithmetic hypotheses. Thus much of my work is linked to foundational issues in diophantine geometry and transcendence theory.
 Maria Cristina Pereyra [web site]
My research area is harmonic analysis, I am interested in the study of Calderon-Zygmund operators, their properties and generalizations, in particular boundedness properties on weighted Lebesgue spaces, the theory of extrapolation and dyadic harmonic analysis.
Currently I am working on weighted theory for certain dyadic model operators, some of the technicalities dissapear in this martingale approach but the problems are sufficiently challenging to be of interest.
In the past I have done some research in wavelet theory.
 Dimiter Vassilev [web site]
My research is in the areas of partial differential equations, analysis and geometry. I am working on problems concerning special holonomy geometries, conformal geometry, contact structures and analysis of nonlinear partial differential equations, such as the Yamabe equation.
Another area of my work involved geometric complex analysis and fluid dynamics. I also have used harmonic and complex analysis in questions related to local zeta functions and unique continuation.
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