Pure Mathematics Research
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.
Emeritus Professor
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.
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.
My research interests focus on differential geometry and geometrical analysis. Recently I studied the geometrical flows in Kaehler/Sasaki geometry.
Operator Algebras
Noncommutative Geometry
Functional Analysis
Quantum Information
My current research is a blend of operator theory, C*-algebras, K-theory, numerical algorithms, mostly aimed at better computer modeling of of topological phases such as topological metals, higher order topological insulators. Some of my research is in collaboration with the Nanostructure Physics department at the Center for Integrated Nanotechnologies, Sandia National Laboratories.
More information on my research is listed on the research page of my webpage.
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 developing the theory of Hardy and BMO spaces on spaces of homogeneous type.
In the past I have done some research in wavelet theory, specifically building divergence free wavelets and exploring some of their applications.
Here is a link to my publications .
For more information you can visit my webpage .
Prof. Skripka has been doing research on noncommutative analysis with applications to mathematical physics and noncommutative geometry, and recently started research on quantum and high-dimensional statistics. Some of her contributions to multilinear operator integration and its applications are explained in the featured article "Untangling Noncommutativity with Operator Integrals" in the Notices of the AMS and summarized in the research monograph "Multilinear Operator Integrals: Theory and Applications". Skripka's research has been recognized with an NSF CAREER award and the Ruth I. Michler Memorial Prize.
My research is in the areas of partial differential equations, analysis and geometry. I am interested in problems concerning local and non-local differential equations, calculus of variations and unique continuation, geometric analysis (special holonomy geometries, conformal geometry, contact, CR and quaternionic contact structures). I also have used harmonic and complex analysis in questions related to fluid dynamics, local zeta functions and unique continuation. See https://www.unm.edu/~vassilev/ for further details.
