Matthew D. Blair |
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Professor Address: E-mail: Phone: |
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Research |
Research Statement (Fall 2017) My research concerns harmonic analysis and its interactions with partial differential equations of wave and Schrödinger 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. Much of 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. More recently, I have also investigated L^p estimates on eigenfunctions of the Laplacian on a compact Riemannian manifold. Of particular interest is to investigate how the geometry of the manifold influences the growth of L^p norms in the high frequency limit. This line of work illuminates the size and concentration properties of these vibrational modes, and is thus of great interest, due in part to its close relationship with problems arising in quantum physics. Many of the relevant methods here involve modern approaches to oscillatory integrals, such as multilinear estimates, stemming from recent progress on the restriction and Bochner-Riesz conjectures. Here are links to preprints of my work, which has been partially supported by the National Science Foundation, grants DMS-0801211, DMS-1001529, DMS-1301717, and DMS-1565436: Strichartz estimates for wave equations with coefficients of Sobolev regularity, Communications in Partial Differential Equations, 31 (5), 2006, pp. 649-688. Spectral cluster estimates for metrics of Sobolev regularity, Transactions of the AMS, 361 (3), 2009, pp. 1209-1240. Strichartz estimates for Schrödinger operators in compact manifolds with boundary (with H. Smith and C. Sogge), Proceedings of the AMS., 136 (1), 2008, pp. 247-256. On multilinear spectral cluster estimates for manifolds with boundary (with H. Smith and C. Sogge), Mathematical Research Letters, 15 (3), 2008, pp. 419-426 Strichartz estimates for the wave equation on manifolds with boundary (with H. Smith and C. Sogge), Annales de l'Institut Henri Poincare, Analyse Non Lineaire, 26, 2009, pp. 1817-1829. Strichartz estimates for the Schrödinger equation on polygonal domains (with G. A. Ford, S. Herr, and J. L. Marzuola), Journal of Geometric Analysis, 22 (2), 2012, 339-351. Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary (with H. Smith and C. Sogge), Mathematische Annalen, 354 (4), 2012, 1397-1430. Strichartz estimates for the wave equation on flat cones (with G. A. Ford and J. L. Marzuola), International Mathematics Research Notices, 2013 (3), 2013, 562-591. L^q bounds on restrictions of spectral clusters to submanifolds for low regularity metrics, Analysis & PDE, 6 (6), 2013, 1263-1288. On refined local smoothing estimates for the Schrödinger equation in exterior domains, Communications in Partial Differential Equations, 39 (5), 2014, 781-805. On Kakeya-Nikodym averages, L^p-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions (with C. Sogge), Journal of the European Mathematical Society, 17 (10), 2015, 2513-2543. Refined and microlocal Kakeya-Nikodym bounds for eigenfunctions in two dimensions (with C. Sogge), Analysis & PDE, 8 (3), 2015, 747-764. Strichartz and Localized Energy Estimates for the Wave Equation in Strictly Concave Domains, American Journal of Mathematics, 139 (3), 2017, 817-861. Refined and Microlocal Kakeya-Nikodym Bounds of Eigenfunctions in Higher Dimensions (with C. Sogge), Communications in Mathematical Physics, 356 (2), 2017, 501-533. L^p bounds on spectral clusters associated to polygonal domains (with G. A. Ford and J. L. Marzuola), Revista Matemática Iberoamericana, 32 (3), 2018, 1071-1091. Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions (with C. Sogge), Journal of Differential Geometry, 109 (2), 2018, 189-221. On logarithmic improvements of critical geodesic restriction bounds in the presence of nonpositive curvature, Israel Journal of Mathematics, 224 (1), 2018, 407-436. Logarithmic improvements in L^p bounds for eigenfunctions at the critical exponent in the presence of nonpositive curvature (with C. Sogge), Inventiones mathematicae, 217 (2), 2019, 703-748. Quasimode, eigenfunction and spectral projection bounds for Schrödinger operators on manifolds with critically singular potentials (with C. Sogge and Y. Sire), to appear, Journal of Geometric Analysis. Slides:Strichartz estimates for the Schrödinger equation in exterior domains (Beijing Conference in Harmonic Analysis and Partial Differential Equations, IAPCM, May 17-23, 2010) Strichartz estimates in polygonal domains and cones, March 16, 2011 (Seminar talk at Michigan State University) L^p norms and global harmonic analysis, September 14, 2017 (Colloquium talk at UNM Fall 17) |
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Conferences and seminars organized |
UNM Analysis Seminar Fall 2015 Special Session on Harmonic Analysis and Dispersive Equations, AMS Western Spring Sectional Meeting 2014, Albuquerque, NM Special Session on Harmonic Analysis and Partial Differential Equations, AMS Western Spring Sectional Meeting 2010, Albuquerque, NM |
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Teaching |
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