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Department of Mathematics
University of Kentucky
753 Patterson Office Tower
Lexington, Kentucky
40506-0027
Recent Research Papers:
My recent research has focussed on three areas: random Schrodinger operators, geometric analysis of real and complex hyperbolic manifolds, and resonance and eigenvalue estimates. My research is partially supported by the National Science Foundation DMS. My current doctoral students are Megan Gier, Aaron Saxton, and George Tiser. You can also check the archives for most of my recent papers.
A more complete list of my publications may be found on MathSciNet.
with Carlos Villegas-Blas, Semiclassical Szego limit of resonance clusters for the hydrogen atom Stark Hamiltonian, to appear in Asympototic Analysis.
Fundamentals of scattering theory and resonances in quantum mechanics, an expanded version of lectures delivered at Penn State in August 2010, to appear in Cubo Journal.
with J-M. Combes and F. Germinet: Conductivity and the current-current conductivity correlation measure, to appear in J. Phys. A.
with P. Muller: Uniform convergence of spectral shift functions, to appear in Publ. RIMS (Kyoto).
with D. Borthwick, T. Christiansen, and P. Perry: Resonances for manifolds hyperbolic near infinity: Optimal lower bounds on order of growth, to appear in IMRN.
with P. Perry, S.-H. Tang: CR-Invariants and the Scattering Operator for Complex Manifolds with Boundary, Analysis & PDE volume 1 #2 (2008) 197--227
with T. Christiansen: Maximal Order of Growth for the Resonance Counting Functions for Generic Potentials in Even Dimensions, Indiana University Mathematics Journal 59, 621-660 (2010).
with T. Christiansen: Resonances for Schrodinger operators with compactly supported potentials, Journees Equations aux derivees partielles, Evian, France, 2-6 juin 2008, GDR 2434 CNRS, Expose no. III, 18 pages (available on Cedram)
with Ph. Briet, G. Raikov, E. Soccorsi: Mourre estimates for a 2D quantum Hamiltonian on strip-like domains, Contemporary Mathematics, volume 500, pages 33-46, 2009.
with Peter Muller: The spectral shift function for compactly supported perturbations of Schrodinger operators on large bounded domains, Proc. AMS, volume 138, number 6, pages 2141-2150, 2010.
Lectures on random Schrodinger operators, Contemporary Mathematics, volume 476, pages 41-131, 2008.
Upcoming Workshops:
International Congress on Mathematical Physics: Funding is anticipated from the US National Science Foundation for a block grant to support the participation of young researchers at US institutions in the XVIth International Congress on Mathematical Physics and the Young Researchers Symposium both to be held in Aalborg, Denmark, during the period 3--11 August 2012. This grant will provide support for travel, lodging, and local expenses. Applicants must have a position at a US institution and be not more than 5 years beyond the PhD. Preference will be given to those without other sources of funding. Co-funding from an applicant's institution is encouraged. Women and members of under-represented groups are especially encouraged to apply. Application information is on the web page. Deadline for applications is 15 March 2012. Announcements of awards will be made by 15 April 2012.
Five-day mini-workshop at the Mathematisches Forschungsinstitut Oberwolfach: Correlations and interactions for random quantum systems, co-organized with W. Kirsch, P. Mueller, and S. Warzel, 23-29 October 2011.
Fall semester 2012: Special program at the Mittag-Leffler Institute in Djursholm, Sweden, on Hamiltonians in magnetic fields, organized by R. Benguria, A. Jensen, G. Raikov, G. Rozenblum and J. Ph. Solovej. See the institute web page for updates.
Course Material for Spring 2012:
MA 633 001 Partial Differential Equations II
Class meets MWF 10:00-10:50 in CB 345. The main goal of this course is to cover the material in chapters 5, Sobolev spaces, and chapter 6, Second-order elliptic equations, in Evans.
Course Material for Fall 2011:
MA 533 001 Partial Differential Equations
Class meets MWF 3:00-3:50 in CB 341.
Problem set 2 NEW: Due in class, Wednesday, 21 September 2011.
Midterm exam is on Friday, 28 October in class. The exam covers harmonic functions and Poisson and Laplace equations, and the heat equation up to the mean value property. No class on Monday and Wednesday, 24 and 26 October.
Problem set 5 Due in class, Wednesday, 2 November 2011. Moved to Friday, 4 November.
MA 113 Sections 029, 030, 031, and 032 Calculus 1
This is the basic course in the calculus of functions of one real variable. The emphasis of Calculus 1 is differentiation, integration and applications. MEETS: MWF 12:00-12:50 in BE 148.
Final Exam on Wednesday, 14 December, 6:00-8:00 PM in the Student Center Theater. Review session: Monday, 12 December, 6:00-8:00 PM Room 110 CB.
Exam 3 is on Tuesday, 15 November, 7:30-9:30 PM, in the Student Center Theater. REVIEW SESSION: Monday, 14 November, 7:30 PM in Room CB 106 (Note change of room from last time).
Course Material for Spring 2011:
MA 773 001 An introduction to geometric analysis
Topic 1: Selberg trace formula for compact hyperbolic surfaces. You should be reading the notes given out in our class.
Topic 2: Isoperimetric inequalities and estimates for low lying eigenvlaues of the Laplacian on bounded domains.
Topic 3: Resonances for Schrodinger operators: Definitions, Existence and the Counting Function.
Possible paper topics:
MA 214 Section 002 Ordinary Differential Equations:
This is a basic course on ordinary differential equations (ODEs). There are three main components: 1) first-order ODEs and modeling (chapter 1-2), 2) second-order ODEs and applications to oscillators (chapter 3), and 3) the Laplace transform and applications to initial value problems.
The final grades are posted. Good luck with everything and have a good summer! you can get copies of your final and the solutions are posted below.
This course has a computer lab component using a free software package called IODE developed at the University of Illinois by P. Brinkmann, R. Jerrard, and R. Laugesen. There will be four projects during the semester. No prior knowledge of MatLab is needed. IODE also runs on GNU Octave, a free software available at: http://www.gnu.org/software/octave/. I have not used it but there are comments about using Octave on the Illinois Iode web page.
You may now download IODE from the main IODE web page. The program works on 2007, 2008, 2009 and 2010 versions of Matlab. These are on the machines in Engineering and the library. The Save feature now works but it might still be more convenient to save your work as described next: ON SAVING GRAPHS: It is easiest to save your graphs as a Word document. This is convenient because you can put two graphs on one page and add comments! Enlarge the graph using the button on the upper right so it fills the screen. Press the "Print Screen" key on the keyboard. This copies the screen into a temp directory. Open Word and simply past the document into word. You can also type in your explanations, etc. you should then save the word document to your locker on the L-drive.
Instructions on how to start-up IODE using Matlab, Spring 2011
Course Material for Fall 2010:
MA 641 Differential Geometry
This course will cover the basics of Riemannian geometry following the book Riemannian geometry by do Carmo (Birkhauser, 1988). We will study differentiable manifolds, Riemannian metrics, tangent and cotangent spaces, vector fields, geodesics, connections, and curvature. We will develop enough machinery to describe the spaces of constant curvature and complete manifolds. Riemannian geometry is built on the classical theory of curves and surfaces in space. It is recommended that the students look at a book such as Differential geometry of curves and surfaces by do Carmo to see the origins of the subject.
Course Material for Spring 2010:
MA 776 Pseudodifferential Operators and Applications:
This course will be a self-contained course on pseudodifferential operators (PsDOs). The goal is to prove a theorem of Hormander on the asymptotic behavior of the eigenvalue counting function for an elliptic, lower-semibounded, self-adjoint PsDO on a compact Riemannian manifold. This will require us to study: oscillatory integrals, stationary phase, symbol calculus, basic properties of PsDOs (products, paramatrices), and basic Fourier integral operators.
MA 214 Section 003 Ordinary Differential Equations:
This is a basic course on ordinary differential equations (ODEs). There are three main components: 1) first-order ODEs and modeling (chapter 1-2), 2) second-order ODEs and applications to oscillators (chapter 3), and 3) the Laplace transform and applications to initial value problems.
NEW: Grades are posted. You may get a copy of your final by stopping by my office. The solutions are posted below. Have a good summer!
This course has a computer lab component using a free software package called IODE developed at the University of Illinois by P. Brinkmann, R. Jerrard, and R. Laugesen. There will be four projects during the semester. No prior knowledge of MatLab is needed. IODE also runs on GNU Octave, a free software available at: http://www.gnu.org/software/octave/. I have not used it but there are comments about using Octave on the Illinois Iode web page.
You may now download IODE from the main IODE web page. The program works on 2007, 2008 and 2009 versions of Matlab. The Save feature now works but it might still be more convenient to save your work as described next: UPDATE on SAVING GRAPHS: If the usual save button does not work, you can save to a Word document. This is convenient because you can put two graphs on one page and add comments! Enlarge the graph using the button on the upper right so it fills the screen. Press the "Print Screen" key on the keyboard. This copies the screen into a temp directory. Open Word and simply past the document into word. You can also type in your explanations, etc.
Instructions on how to start-up IODE using Matlab, Spring 2010
Lab 1: Introduction to IODE and the first project, Spring 2010
Course Material for Fall 2009:
MA 677 Real Analysis II:
This course is a continuation of Reals I taught in the spring of 2009. We will continue with the book of Stein and Shakarchi. You should be reading Chapter 6 on abstract measure theory. You should also be focussing on your paper and projects. Please see me to discuss them! A draft of you paper is due in class on 18 November 2009.
MA 214 Section 003 Ordinary Differential Equations:
NO LAB OFFICE HOUR Wednesday, 9 Dec. 3-4. There will be a review session Tuesday, 15 December, at 3 PM in 316 CB. Office Hour on Monday, 14 December 11 AM-12 PM and 4-5 PM in my office, 753 POT. ALL GRADES ARE FINAL (except the final exam grade and the lab 3 grade) AT 4PM TODAY UNLESS YOU CONTACT ME.
We have finished all the work for our course. In chapter 6, we did sections 6.1-6.5. We will begin to review for the final exam tomorrow, Wednesday. The final exam is scheduled for: Wednesday, 16 December, 10:30-12:30, in our classroom CB 203. It is worth 180 points. I will award everyone 15 points for the lab that we were not able to do. A practice exam is posted below and you may find other practice final exams posted in pervious versions of this course below (Remember: we did not do exact ODEs.)
This course will have a lab component using a free software package called IODE developed at the University of Illinois by P. Brinkmann, R. Jerrard, and R. Laugesen. There will be four projects during the semester. No prior knowledge of MatLab is needed. IODE also runs on GNU Octave, a free software available at: http://www.gnu.org/software/octave/. I have not used it but there are comments about using Octave on the Illinois Iode web page.
UPDATE: You may now download IODE from the main IODE web page. The program works on 2007, 2008 and 2009 versions of Matlab. The only feature that does not work is the Save feature. Save your work as described next: UPDATE on SAVING GRAPHS: If the usual save button does not work, you can save to a Word document. This is convenient because you can put two graphs on one page and add comments! Enlarge the graph using the bottom on the upper right so it fills the screen. Press the "Print Screen" key on the keyboard. This copies the screen into a temp directory. Open Word and simply past the document into word. You can also type in your explanations, etc. Third project due date Friday, 4 December, in class. Please staple your pages together! CHANGE: Open lab in CB 313 on Wed. 2 Dec. from 3:15 - 4:00 PM only! You may turn in your projects on Monday.
Open Lab Hours in CB313, Fall 2009 (Please note the change on T and R mornings!)
Instructions on how to start-up IODE using Matlab, Fall 2009
Course Material for Spring 2009:
MA 676 Real Analysis I:
The grades have been posted online and the solutions to the final are below. I will return the final exam and PS #7 to your mailboxes. Have a good summer!
MA 214 Section 003 Ordinary Differential Equations:
WEEKLY PROBLEM SESSION on Wednesdays at 4PM in CB 341. We are doing chapter 6 on Laplace transforms. Read sections 6.1-6.5. No problem session on Wednesday, 29 April. We will be reviewing all week. A final exam review is scheduled for Thursday, 7 May, at 5:30PM, room CB 333.
Course grades will be posted today, Monday, 11 May. The solutions to the final are posted below. You may stop by anytime to look at your final. Have a good summer!
Course Material for Fall 2008:
MA 214 Section 03 Ordinary Differential Equations:
NO WEEKLY PROBLEM SESSION this week, 10 December.
Final grades will be posted probably by tomorrow, Tuesday, 23 December. Have a nice holiday!
MA 575 Principles of Analysis:
WEEKLY PROBLEM SESSION: Wed. 3:30-4:30 PM CB 246
Final Exam on Friday, 19 December, 8AM-10AM in CB 347. Good Luck!
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