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Department of Mathematics
University of Kentucky
753 Patterson Office Tower
Lexington, Kentucky
40506-0027
UK PDE/Analysis Seminar Program Fall 2014 and Spring 2015
The most recent schedule for the PDE/Analysis Seminar may be found here: Schedule Fall 2014. Schedule Spring 2015.
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 Joseph Lindgren and Robert Wolf. Megan Gier received her PhD in May 2014 (assistant professor at Cedar Grove College) and Aaron Saxton received his PhD in August 2014 (employed at Neustar). 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 M. Krishna: Eigenvalue statistics for random Schrodinger operators with non rank one perturbations .
with F. Klopp: Optimal Wegner estimate and the density of states for N-body, interacting Schrodinger operators with random potentials, to appear in Markov Processes and Related Fields
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 N. Dombrowski and E. Soccorsi: Edge currents and eigenvalue estimates for magnetic barrier Schrodinger operators, to appear in Asymptotic Analysis
with E. Soccorsi: Edge states induced by Iwatsuka Hamiltonians with positive magnetic fields, to appear in J. Math. Analysis and Applications
with T. Christiansen: Some remarks on scattering resonances in even dimensional Euclidean scattering, to appear inTrans. AMS
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)
Lectures on random Schrodinger operators, Contemporary Mathematics, volume 476, pages 41-131, 2008.
Upcoming Workshops:
Five-day workshop at the Banff International Research Station in Banff, Canada: Geometric scattering theory, co-organized with F. Froese, R. Mazzeo, and P. Perry, 2 - 7 November 2014.
The mathematics of quantum disordered systems, 15--19 December 2014, Euler International Mathematical Institute, St. Petersburg, Russia, organized by F. Klopp, A. Fedotov, T.Vinogradova, N.Zalesskaya.
Ohio River Analysis Meeting 6, University of Kentucky, Lexington KY, Saturday and Sunday, 12 March - 13 March 2016.
Course Material for Fall 2014:
MA 575 Introduction to Analysis
This graduate class meets MWF 11:00-11:50 AM in CB 339.
The goal of this course is to provide everyone with a firm foundation in the theory of functions of a single real variable. Although a lot of this is familiar from the calculus, we’ll carefully and rigorously study properties of functions, like continuity and differentiability, and the Riemann integral. We’ll also look at questions of convergence of sequences of numbers and functions, including the important topic of uniform convergence of functions. These are fundamental ideas that all mathematicians should be familiar with. No class the week of November 3.
We will have a weekly problem session for discussing the problem sets on Friday afternoon at 3 PM. Let's meet in CB 339.NEW: make-up class on 14 and 21 November at this time in CB 339.
Homework problem sets. Problem set 9 due Friday, 5 December.
MA 114 Honors Calculus II sections 009 and 010
This is the honors Calculus II class for fall 2014 semester. Both sections 009 and 010 meet for lectures MWF 1:00-1:50 in CB 110. Note change in room!
CALENDAR: 29 Aug. We are finishing section 10.1 and will begin 10.2 on Wed. 03 Sept. Discussed limit laws and bounded monotone seq. Fri. 05 Sept. We did series and the principle of induction, will start 10.3 on Monday. Mon. 08 Sept. we did 10.3 on positive series, we'll begin 10.4 on Wed. Wed. 09 Sept. We did the alt series test. WW dates through WWH06 have been adjusted. Friday, 19 Sept. Finished power series, 10.6. First Exam on Tuesday, 23 Sept. 5-7 PM 116 BS. Wed. 24 Sept. Beginning 10.7 on Taylor's series. Fri. 26 Sept. Taylor polynomials and remainder, sec. 8.4. Mon. 29 Sept. Finished Taylor polynomials and began sec. 6.2 on computing volumes. Wed. 1 Oct. We did some volume calculations and the density. We'll begin 6.3 on Friday. We are doing volumes using the disk, washer and shell methods through Mon. Oct. 6. Work on WWh11-14. On Oct. 8 we did work, sec. 6.5. Week of 13 Oct.: We are now working in chapter 7 doing integration by parts, 7.1, and trig integrals, 7.2. Exam 2: Tuesday, 21 Oct. 5-7 PM in 116 BS. Review 20 Oct. 8 PM CP 155. Material from Taylor series through trig integrals (no trig substitutions). WWH-15 through 18 are now ready. Solutions to Test 2 are posted. Midterm grades based on tests 1 and 2 are posted. Wed. 29 Oct. We finished sec. 3.9 and 7.4 on hyperbolic trig functions. We began sec. 7.8 on numerical methods. 31 Oct. We finished our discussion of numerical approximation. WA2 is posted and due Friday, 14 Nov. We'll be doing section 8.1,8.3, and 9.1 next week. 12 Nov. We're finishing sections 9.1, 9.2, 9.3, and 9.4 on differential equations. Review for test 3 on Monday, 17 Nov. 8PM in CP 155. The Euler method and the logistic equation will NOT be on Test 3. 19 Nov. We did the logistic ODE, sec. 9.4. We'll finish 9.5 on Friday, and begin sections 11.1 and 11.2 on curves. Week of December 1: We finished parametrically defined curves, and are doing polar coordinates, sections 11.3 and 11.4. Final Review: Friday, 12 Dec. 7 PM in CB 106; Final Exam: Tuesday, 16 December, 6-8 PM in BS 116. The solutions to the final are posted below. Happy new year and Good luck during spring 2015 semester.
Solutions to written assignment 3. There will not be a fourth written assignment.
Course Material for Spring 2014:
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. NEW: Classroom change to CB 341!!
MA 113 Calculus I sections 005, 006, 007, 008
This is the first semester of calculus. After a review of functions, we will study the derivative and integral. ALL FOUR SECTIONS MEET FOR LECTURES: MWF 12:00-12:50 in CB 114. . The Final Exam is WEDNESDAY, 7 May, at 8:30-10:30 PM in CB 106. If you have a conflict, see me or email me before 30 April 2014. The Review Session is MONDAY, 5 May, from 3:30-5:00 PM, in CB 106. If you want a copy of your final or have questions about your grades, please email me. Have a good summer!!
Course Material for Fall 2013:
MA 681 Functional Analysis
This is a course in Banach and Hilbert space theory with an emphasis on the theory of linear operators. The class meets MWF 12-12:50 in CB 343. Special class: Monday, 7 October at 4PM
MA 113 Calculus I sections 025, 026, 027, 028
This is the first semester of calculus. After a review of functions, we will study the derivative and integral. ALL FOUR SECTIONS MEET FOR LECTURES: MWF 2:00-2:50 in CB 110. The book store may still have some copies of the book. You can also order it on Amazon. NEW: Final exam, Wednesday, 18 December, 8:30--10:30 PM in CB 118. REVIEW SESSION: Tuesday, 17 December 2013 3:30-5 PM in CB 118. Good luck with the final!
Course Material for Spring 2013:
MA507 and PHY507 Mathematical Methods of Physics
Class meets MWF 12:00-12:50 in CB 345. The main goal of this course is to learn complex analysis and partial differential equations(PDE). We will begin with complex analysis and cover material through contour integration and the residue theorem. We will then study PDEs focusing on elliptic (Laplace and Poisson equations), wave and heat equations. We will cover separation of variables for PDEs and special functions in more detail. The final course grades have been posted. There was a 20 point curve on the final exam. Have a good summer!
Solutions to the Final Exam. You may pick-up your final when convenient. Have a good summer!
MA 114 Calculus II sections 005, 006, 007, 008
This is the second semester of calculus. We will study integration methods and applications. ALL FOUR SECTIONS MEET FOR LECTURES: MWF 10:00-10:50 in CP 320. NEW: Final Review: Sunday and Monday, 28 and 29 April, from 6:30-8:30 PM, in CP 153 (Sunday) and CB 102 (Monday). Final Exam, 1 May, 6-8 PM in BS 107. Alternate Final Exam, 2 May, 1-3 PM in the Mathskellar. Good luck! Room for Final Exam: BS 107 (Biological Sciences).
This is my specific Course Syllabus for my sections of MA 114, sections 005, 006, 007, and 008, Spring 2013.
The general MA 114 course syllabus for spring 2013 is posted here. You will find the recitation work sheets, the course calendar, how to use Web Work, etc.
Notes on Exponentials and Logarithms. These give a summary of all you need to know!
Course Material for Fall 2012:
MA506 and PHY506 Mathematical Methods of Physics
Class meets MWF 12:00-12:50 in CP 183. The main goal of this course is to cover ordinary differential equations, linear algebra, and Strum-Liouville theory. The course continues with MA 507 in the spring of 2013. That course will cover complex variable theory and partial differential equations. NEW: I'll return the graded finals at the first 507 class or you can stop by and pick up your final. Have a nice holiday.
Midterm exam on Friday, 19 October in class.
Final exam on Friday, 14 December 10:30 AM - 12:30 PM in CP 183. Here are some notes with a list of the topics we covered in class. We'll have a review on Wednesday, 12 December, at 3:30 PM. Meet outside our classroom.
Notes on Exponentials, Logs, and Complex Numbers-everything you need to know!
Notes by Professor Perry on linear algebra. We'll be discussing these during the last two weeks of class.
Problem set 3 Due in class, Friday, 5 October 2012. The two problems on page 377 will be part of PS 4.
Problem set 4 Due in class, Monday, 15 October 2012.
Problem set 5. You don't have to do part d of 8.3.1. concerning convergence. Due in class, Friday, 2 November 2012.
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.
Problem set 3 Due in class, Wednesday, 29 February 2012. No kidding, no leaping!
Problem set 5 Due in class, Wednesday, 11 April 2012. Postponed! Please turn it in Friday, 20 April.
Midterm exam on Friday, 23 March, over the material in Chapter 5.
Course Material for Fall 2011:
MA 533 001 Partial Differential Equations
Course Syllabus for MA 533 001, Fall 2011. NEW! We'll have the final on Wednesday, 14 December. We'll meet in 745 POT at 1PM.
Class meets MWF 3:00-3:50 in CB 341.
Problem set 1 NEW: Due in class, Wednesday, 9 September 2011. Some typos are corrected and problem 3 is saved for problem set 2.
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.
This is my specific Course Syllabus for my sections of MA113, sections 029, 030, 031, and 032, Fall 2011.
Notes on Exponentials and Logarithms. These give a summary of all you need to know!
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:
the Yamabe problem: Given a compact Riemannian manifold (X,g) in three or more dimensions, can one find a metric, conformal to g, with constant scalar curvature? see the review article by Lee and Parker
isopsectral sets for compact Riemann surfaces, especially the work of Osgood, Phillips, and Sarnack, see the review of Perry
Ray and Singer introduced a notion of analytic torsion for a compact, oriented Reimannian manifold and conjectured that it is equal to the Reidemeister-Franz torsion (a topological invariant). This conjecture was later proved independently by W. Muller and Cheeger.
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.
Course Syllabus for MA214 002, Spring 2011. Lab office hours: Monday afternoon 4PM in CB 313. Wednesday afternoon in POT 753.
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
Open Lab Hours in CB313, Spring 2011. You may use the computers in the lab during any open lab hour. I will hold my office hour in CB 313 on Mondays 4-5 PM starting 31 January 2001. We can discuss the labs and homework.
Project 1: Due in class, Wednesday, 9 February 2011. You can use MatLab in the RG Anderson building computer labs, in the Science and Engineering Library lab (King South), and in CB 313 during open lab hours (schedule to be posted.) You cannot print in CB 313 so save your graphs in your locker using word, for example, and print them in another lab.
Comments on Project 1 compiled from reading the project reports. Please read through these comments and think about them as you do Project 2. GRADING: The lab was worth 15 points: the first ODE 3 points, and the remaining 3 ODEs were worth 4 points.
Comments on Project 2 compiled from reading the project reports. Please read through these comments and think about them as you do Projects 3 and 4. GRADING: The lab was worth 15 points with each of the 5 problems worth 3 points.
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.
Problem set 6 Due Wednesday, 15 December 2010.New version: A factor of 2 was missing in problem 1. Thanks Murat for finding this!
Notes written by Professor Perry on multivariable differential calculus, including the inverse and implicit function theorems.
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
Open Lab Hours in CB313, Spring 2010. I will hold my office hour in CB 313 on Wednesdays 4-5 PM. We can discuss the labs and homework.
Lab 1: Introduction to IODE and the first project, Spring 2010
Project 1: Due in class, Friday, 5 February 2010. You can use MatLab in the RG Anderson building computer labs, in the Science and Engineering Library lab (King South), and in CB 313 during open lab hours (schedule to be posted.) You cannot print in CB 313 so save your graphs in your locker using word, for example, and print them in another lab.
Comments on Project 1 compiled from reading the project reports. Please read through these comments and think about them as you do Project 2. GRADING: The lab was worth 15 points: the first ODE 3 points, and the remaining 3 ODEs were worth 4 points.
Comments on Project 2 compiled from reading the project reports. Please read through these comments and think about them as you do Projects 3 and 4. GRADING: The lab was worth 15 points.
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
Project 1: NEW! due in class, Wednesday, 23 September 2009 (read above). Try to get to CB313. Save you graphs in your locker using word, for example, and print them in another lab.
Comments on Project 1 compiled from reading the project reports. Please read through these comments and think about them as you do Project 2. GRADING: I will add 5 points to your score on Project 1 since it should have been worth 15 points.
Comments on Project 2 compiled from reading the project reports. Please read through these comments and think about them as you do Project 3.
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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