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Jeremy Marzuola
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Interesting Meetings to be aware of:

Northwestern University Emphasis Year in Microlocal Analysis, 2011–12.

Workshop on Evolution Equations in honor of Terence Tao, to be held May 4-6 at Northwestern. Organizers are Jared Wunsch, Igor Rodnianski, and Gigliola Staffilani. Details to come.

Special session on
"Stability Analysis for Infinite Dimensional Hamiltonian Systems" at the Joint Mathematics Meetings to be held
Jan. 4-5, 2012 in Boston (the meeting will be 4-7 but the special session takes place on Jan. 4,5) organized by Wilhelm Schlag and Eugene Wayne.

Anyone else have any others they are aware of????

As a reminder, we are re-posting this as a dates have already passed but most are just quickly approaching!!

Important Dates for the Fall:
NSF Postdoc Fellowship Deadline - October 19, 2011.
NSF Graduate Fellowship Deadline - November 15, 2011.
GRE Subject Dates in Math - October 15 and November 12, 2011.
Berkeley Ph.D. program (as an example ... look up each school you are interested in individually) application date - December 9, 2011.
Johns Hopkins JJ Sylvester Postdoc (tends to be one of the earlier postdoc deadlines) application date - December 1, 2011

Last week we had two great seminars.

First, in the PDE seminar we had Austin Ford from Northwestern who discussed work to better understand regularity and second microlocalizations on smooth manifolds. Austin is a student of Jared Wunsch and uses a formulation of Lagrangian regularity originally due to Richard Melrose that hopefully can be generalized to more singular manifolds. Austin's expertise on singular manifolds is in the area of Euclidean cones, but his formulation may be generalizable to more general conic singularities.

The second talk in the colloquium by Ken McLaughlin from Arizona was a very nice introduction to the Gibbs Phenomenon that fourier methods while converging in an L^2 sense do not tend to converge pointwise to jump discontinuities in functions. In fact, there is an oscillatory over-shoot at the discontinuity that in certain examples can be computed explicity as an asymptotic series. While visiting the Federal University in Brazilia, Professor McLaughlin happened to meet Nigel Pitt, an analytic number theorist and was able to essentially figure out ways to sum Fourier series to give pointwise representations for solutions to linear Schrödinger equations at RATIONAL times with certain properties and IRRATIONAL times that look like certain kinds of continued fraction expansions in rational numbers with good denominators. Then, they can analyze their convergence to a discontinuous initial solution at time 0. The analytic description that comes out of the summation represents the fractal nature of the plot-able and solvable outcomes rather impressively. Interestingly, though they are harder to represent, there is apparently pointwise convergence at all irrational numbers and a description of the fractal dimension, which their work also recaptures, though this was known previously apparently in work by several authors, but in particular Kapitanski and Rodnianski were the two harmonic analysts in PDE that stood out the most to me.

Well, we've had two analysis related talks in the last two days.

First, Alex Ionescu of Princeton discussed well-posedness results for the H^1 critical defocussing nonlinear Schrödinger equation on periodic domains in 3 and 4 dimensions. Their result utilizes the profile decomposition of Keraani (a student of Patrick Gerard) and some Strichartz spaces derived in the work of Bourgain. They seem to be able to extend their results to more the full four dimensional torus (right now they are in RxT^3) due to a recently improved Strichartz estimates in four dimensions by Bourgain. Applying the profile decomposition in such a periodic setting is new and interesting, so we look forward to reading and learning more about it. Alex also gave all of us a great chance to talk deeply about symmetric spaces and dispersive equations on symmetric spaces, which was much appreciated.

The colloquium today was by Boris Khesin from Toronto who talked about
curvatures of Sobolev metrics on diffeomorphism groups. Apparently, in the right frame work, many equations from fluid dynamics (Euler, Camassa-Holm, KdV,Hunter-Saxton, ...) can be derived by looking at the geodesic equation using a certain metric in a space of well crafted metrics. This can apparently also be used to generate integrable equations, which must involve an understanding of either some kind of Lax pair structure or actual derivation of a hierarchy of conserved quantities. Again, the implications such a topological approach could have on estimates should be quite interesting, so hopefully the speaker will give us some more insight before he heads back North to let's face it, the greatest Canadian city there is.

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Well, as a happy accident last night we also heard that the paper co-authored by Hans and Jeremy with collaborators Pierre Albin and Laurent Thomann on quasimodes for the nonlinear Schrödinger equation on compact manifolds with elliptic, periodic geodesics was accepted to Physica D: Nonlinear Phenomena. This is to leading order actually quite a linear phenomenon, but we are able to show that the nonlinear contribution is controllable and lower order. We are exploring several extensions and applications of this research, in part with some undergrads here at UNC involving questions of focussing/defocussing, existence times, etc. It is amazing how such a simple formulation can have such a complex solution and spark such a large number of questions. A link to a pre-print can be found here:

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So, this is my first attempt at a research dissemination/announcement post. It feels awfully self-serving but hey, our goal with this thing is to let people know what is going on in UNC PDE research, so here goes:

My article with Luc Hillairet entitled "Nonconcentration in partially rectangular billiards" has officially been accepted by Analysis & PDE. You can find the arxiv version here:

if interested.

It is a result that uses the so-called adiabatic ansatz to prove nonconcentration estimates that extend outside the rectangle in the Bunimovich stadium by working with different normal forms and doing a lot of ode/Green's function analysis made possible by how much Luc has thought about the spectrum of the Laplacian on domains and some neat observations of Nicolas Burq about control theory related to eigenfunctions of the Laplacian on the rectangle. Our result (which had to be refined earlier this summer because we found a dumb mistake in the final theorem of the paper though that fortunately that made us improve our techniques dramatically) improves an estimate of Burq-Hassell-Wunsch (Proceedings 2007) in the setting where you assume "non-resonance" with the eigenvalues of the rectangle. Incidentally, figuring out the size of such a "resonant" set could be an interesting problem. I don't know, I think it is a neat result and am quite proud to have Luc as a collaborator and friend of the group because of the depth of his ideas on projects like this and others.

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Some announcements regarding research related to the group:

September 26th, 2011:

The AMS Session this weekend went really well. We had a couple cancellations, but still heard ten really nice talks.

The morning sessions featured discussion of decay estimates for wave equations on surfaces with conic singularities by Matt Blair, Price's Law for the wave equation on the Kerr metric by Mihai Tohaneanu and on quasilinear wave equations on exterior domains by John Helms. The papers by Blair and Tohaneanu have appeared on arXiv's, so I encourage you to take a look. We also heard a nice introduction to ideas for wave equations on integer lattices by Michael Goldberg and a discussion on some new Morawetz-type estimates using local smoothing ideas for nonlinear magnetic Schrödinger equations by Magda Czubak, both of which are more works in progress but seem to be leading to interesting future directions. The talks by Blair, Tohaneanu and Goldberg were related to linear estimates in settings away from flat space where things such as trapping, spectral theory, scattering, etc. can inhibit one's knowledge of the linear flow of a solution and complicate estimates tremendously. The talks by Helms and Czubak talked about interesting norms in which one can control complicated nonlinear interactions using ideas from linear estimates for the underlying equations. These talks in tandem really showed the importance and difficulty in understanding and controlling underlying linear dynamics in finding the right spaces to control how nonlinear interactions alter those dynamics over time.

In the afternoon session, we talked more about strongly nonlinear interactions, namely related to nonlinear bound state dynamics and solutions for degenerate equations. Burak Erdogan gave a nice description of the smoothing effect and Sobolev norm growth bounds for KdV equations by applying some of the fascinating harmonic analysis work he has done recently with collaborators to show effectively how nonlinear interactions at high frequency cancel out in interesting ways at high frequency. This result interestingly applies without high frequency assumption, something I'd like to understand more. Also, Justin Holmer discussed a canonical and interesting new take on deriving soliton dynamics through recognizing that the soliton manifold in a Hamiltonian equation has symplectic structure. He and a student use this to derive discrete Hamiltonian phase driven dynamics between two nonlinear bound states that they can make either attractive or repulsive. Then, we also had Brian Pigott discuss his thesis work proving bounds that measure instability in a sense that perturbations of subcritical KdV solitons have a lower bound on their growth in Sobolev spaces below the orbital stability space of H^1 but above L^2. Understanding how this fits into known orbital/asymptotic stability theory seems very complex and interesting. Lastly, we had Dave Ambrose and Doug Wright speak in tandem about degenerate dispersive equations that have been designed to have "compactons," or compactly supported nonlinear traveling waves. Their work suggests that such equations are actually ill-posed (and explicitly proves it for certain examples) but leaves open many interesting questions related to analysis and computation of these poorly understood models.

For more info about any of the topics, come talk to Jeremy for a discussion about the talks themselves and/or check arxiv's or e-mail the speakers for information regarding their work.

September 18th, 2011:

Former UNC grad student Nathan Pennington wrote this one that appeared last week ... ... based off work he did in his thesis with Michael Taylor.

September 18th, 2011:

Some papers appeared on arXiv's this week related to some of the group's interests: Check out


for instance.

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So, we have been looking at lists of really fundamental papers in PDE that could be read and reported on by grad students in a PDE course by asking all colleagues here at UNC and a few elsewhere. I am not sure that everything on this list is project ready so to speak, but I think it is a good start a really nice set of very fundamental papers in the field ... though obviously not complete!!! Please comment in your favorite fundamental PDE paper and hopefully the list will grow nicely!!

MR1555394 Leray, Jean Sur le mouvement d'un liquide visqueux emplissant l'espace. (French) Acta Math. 63 (1934), no. 1, 193–248, DML Item MR1234453 (94f:35121) Camassa, Roberto; Holm, Darryl D. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71 (1993), no. 11, 1661–1664, 35Q51 (58F07 76B15 76B25)

MR1289328 (95h:35209) Pego, Robert L.; Weinstein, Michael I. Asymptotic stability of solitary waves. Comm. Math. Phys. 164(1994), no. 2, 305–349. (Reviewer: Peter L. Christiansen), 35Q53 (35B35 76B25)

MR0967634 (90d:35267) Jones, Christopher K. R. T. Instability of standing waves for nonlinear Schrödinger-type equations.Ergodic Theory Dynam. Systems 8$^*$ (1988), Charles Conley Memorial Issue, 119–138.(Reviewer: Rafael José Iório Jr.), 35Q20 (35B35 35J10 78A50)

MR1118699 (92j:35050) Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre-Louis User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. (Reviewer: P. Szeptycki),35J60 (35B05 35D05 35G20)

MR0778974 (87e:49035b) Lions, P.-L. The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223–283. (Reviewer: Gianfranco Bottaro), 49A50 (49A22)

MR0778970 (87e:49035a) Lions, P.-L. The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109–145. (Reviewer: Gianfranco Bottaro), 49A50 (49A22)

MR0649347 (84h:35091a) Cheeger, Jeff; Taylor, Michael On the diffraction of waves by conical singularities. I. Comm. Pure Appl. Math. 35 (1982), no. 3, 275–331. (Reviewer: Vesselin M. Petkov), 35L10 (58G17 78A45)

MR0658471 (84b:58109) Cheeger, Jeff; Gromov, Mikhail; Taylor, Michael Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differential Geom. 17 (1982), no. 1, 15–53. (Reviewer: Helga Baum), 58G30 (53C21)

MR0873380 (88b:35205) Sylvester, John; Uhlmann, Gunther A global uniqueness theorem for an inverse boundary value problem. Ann. of Math. (2) 125 (1987), no. 1, 153–169. (Reviewer: P. Szeptycki), 35R30 (86A20)

MR0673830 (84m:35097) Caffarelli, L.; Kohn, R.; Nirenberg, L. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982), no. 6, 771–831. (Reviewer: Tai Ping Liu), 35Q10 (76D05)

MR0097628 (20 #4096) Lax, Peter D. Asymptotic solutions of oscillatory initial value problems. Duke Math. J. 24 1957 627–646. (Reviewer: Yu Why Chen), 35.00

MR0454365 (56 #12616) Strauss, Walter A. Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977), no. 2, 149–162. (Reviewer: Alan Jeffrey), 35L05 (81.35)

MR0336122 (49 #898) Gardner, Clifford S.; Greene, John M.; Kruskal, Martin D.; Miura, Robert M. Korteweg-deVries equation and generalization. VI. Methods for exact solution. Comm. Pure Appl. Math. 27 (1974), 97–133. (Reviewer: J. Smoller), 35Q99

MR0296602 (45 #5661) Fefferman, Charles The multiplier problem for the ball. Ann. of Math. (2) 94 (1971), 330–336. (Reviewer: R. Larsen), 42A18 (42A92 47B99)

MR0867665 (88d:35169) Floer, Andreas; Weinstein, Alan Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69 (1986), no. 3, 397–408. (Reviewer: Pierre-Louis Lions), 35Q20 (58C15 58F05)

MR0708966 (85m:81040a) Simon, Barry Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions. Ann. Inst. H. Poincaré Sect. A (N.S.) 38 (1983), no. 3, 295–308. (Reviewer: Jean Michel Combes), 81C12 (35P20)

MR0783974 (86i:35130) Weinstein, Michael I. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16 (1985), no. 3, 472–491. (Reviewer: Woodford W. Zachary), 35Q20 (78A45 82A45)

MR0820338 (87f:35023) Weinstein, Michael I. Lyapunov stability of ground states of nonlinear dispersive evolution equations.Comm. Pure Appl. Math. 39 (1986), no. 1, 51–67. (Reviewer: Roman Stankiewicz), 35B35 (35Q20)

MR1105875 (93d:35034) Journé, J.-L.; Soffer, A.; Sogge, C. D. Decay estimates for Schrödinger operators. Comm. Pure Appl. Math. 44 (1991), no. 5, 573–604. (Reviewer: P. Szeptycki), 35J10 (35B45 35P05)

MR1972492 (2004k:58046) Blue, P.; Soffer, A. Semilinear wave equations on the Schwarzschild manifold. I. Local decay estimates. Adv. Differential Equations 8 (2003), no. 5, 595–614, 58J45 (35Q75 83C57) NOTE: ORIGINAL VERSION CONTAINS SOME MISTAKES!!

MR1789924 (2001j:35180) Smith, Hart F.; Sogge, Christopher D. Global Strichartz estimates for nontrapping perturbations of the Laplacian. Comm. Partial Differential Equations 25 (2000), no. 11-12, 2171–2183. (Reviewer: Ya-Guang Wang), 35L15 (35B45 35L20)

MR2217314 (2007f:35201) Metcalfe, Jason; Sogge, Christopher D. Long-time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods. SIAM J. Math. Anal. 38 (2006), no. 1, 188–209 (electronic). (Reviewer: Nikos I. Karachalios), 35L70 (35B45)

MR1945285 (2003m:35167) Keel, Markus; Smith, Hart F.; Sogge, Christopher D. Almost global existence for some semilinear wave equations. Dedicated to the memory of Thomas H. Wolff. J. Anal. Math. 87 (2002), 265–279. (Reviewer: Albert J. Milani),35L70

MR0634248 (84a:35083) Gidas, B.; Ni, Wei Ming; Nirenberg, L. Symmetry of positive solutions of nonlinear elliptic equations in Rn.Mathematical analysis and applications, Part A, pp. 369–402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London,1981. (Reviewer: D. E. Edmunds), 35J60 (53C05 58G20)

MR0544879 (80h:35043) Gidas, B.; Ni, Wei Ming; Nirenberg, L. Symmetry and related properties via the maximum principle.Comm. Math. Phys. 68 (1979), no. 3, 209–243. (Reviewer: È. M. Saak), 35J25 (35B50)

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Important posts regarding deadlines:

1. Important Dates for the Fall:
NSF Postdoc Fellowship Deadline - October 19, 2011.
NSF Graduate Fellowship Deadline - November 15, 2011.
GRE Subject Dates in Math - October 15 and November 12, 2011.
Berkeley Ph.D. program (as an example ... look up each school you are interested in individually) application date - December 9, 2011.
Johns Hopkins JJ Sylvester Postdoc (tends to be one of the earlier postdoc deadlines) application date - December 1, 2011.

2. Also, just in case anyone would like to check out some of their talks, here is a link to the Duke applied math/analysis seminar ...

They have a few interesting speakers on the schedule already.

3. As Fall Break approaches a reminder that there is no seminar this week, but that we have Alex Ionescu from Princeton coming next week at the regularly scheduled time. Also, we just learned about this opportunity for interested grad students/postdocs:

It is in the middle of the semester, which is tough for most but if there is any interest you should apply!!

4. Jeremy will be teaching a reading course (Math 699) on decay estimates for Schrödinger equations in the Spring. Anyone interested in enrolling needs permission, but all are welcome. The goal will be to establish the analytic tools necessary to write down dispersive, space-time and smoothing estimates for solutions to the Schrödinger equation and possibly to look at some applications later in the semester.

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A collection of old posts about UNC's seminar:

September 2nd, 2011:

Our first seminar speaker of the year was Jason Metcalfe from UNC who spoke on quasilinear Schrödinger equations relating to work done with J. Marzuola and D. Tataru. The talk clarified a lot of issues regarding the nature of the difficulty of solving quasilinear equations, why certain summability conditions are required in the function spaces and important differences between wave equations and Schrödinger equations.

September 7th, 2011:

Our second seminar today was given by Nate Totz, a new postdoc at Duke who studied at Michigan with Sijie Wu. His talk was on his thesis result, which establishes a cubic NLS equation as the envelope dynamics for the surface an infinite depth fluid model in 2-dimensions (1d interface). It uses the "normal form" analysis initiated by Wu to cancel quadratic interactions in the fluid equations without surface tension that are obstructions to long-time well-posedness. The result leaves open lots of interesting questions about higher dimensional analogues the speaker has some ideas how to pursue. We hope this will open up a lot of dialogue and progress as he hopefully continues to come to seminars and drag along other colleagues from our neighbor down 15/501.

September 9th, 2011:

The UNC colloquium last night by A. Szenes had to do with rational recursions of complex numbers on 2d lattices. The talk was nice and accessible to non-experts, though certainly out of our field but related to symmetries in mathematical physics. However, some of our colleagues think it might have related to solutions of the wave equation on a lattice, an interesting observation. Any comments or clarifications are quite welcome.

September 28th, 2011:

Great talk this week by Carla Cederbaum from Duke looking at defining a notion of mass and center of mass that make sense in relativistic space-times and limit in some large speed of light limit to the canonical Newtonian quantities. The result works for now on stationary space times but there are a tremendous number of open questions regarding extensions to non-stationary space times, other Newtonian quantities, systems with charge, etc. The key idea comes down to writing generalizing the Newtonian gravitational potential function to a relativistic uniquely defined quantity that can be "limited" along different coordinate frames using a technique originated by Ehlers. Lots of open questions about relativistic effects that limit to Newtonian effects from a coordinate independent point of view. I found this link to her very nice slides ...

October 6th, 2011:

The talk by Ha Pham last night was very interesting. She nicely demonstrated how using the structure of anti-de Sitter space allows a compactification that introduces effectively a wedge-like boundary. Then, she was able for her thesis to essentially invert the Klein-Gordon operator near such a boundary and construct a parametrix to prove propagation of singularities in that case. She is interested in proving resolvent estimates since then one can say even more about solutions to the wave equation on such backgrounds. The geometric picture is pretty hard for me to wrap my head around at this point, but there were some similarities to work done by Vasy and others on manifolds with corners and some with the work of Datchev and Vasy on gluing estimates together on hyperbolic manifolds. It is quite a result and we are all trying to process bits of it as we go along and look forward to learning more about it in the future.

October 12th, 2011:

So, today we had Gustav Holzegel give a GR related talk, which Jeremy unfortunately missed because he was at Northwestern attending a meeting and giving a talk on crystal evolution. The good news is that he learned a fair bit about related fields in applied math and computation, plus worked today and will get to work for several days with his collaborator Jon Weare to hopefully finish their description of how fully nonlinear fourth order PDEs arise in continuous limits of crystal surface evolution assuming broken bond models. He also got to talk to Peter Smereka who is quite knowledgeable regarding molecular dynamics and learned a fair bit about the subject. In addition, Ridgway Scott gave a nice talk on modeling the hydrophyllic nature of proteins with direct implications for cancer drug design that relies on simple pseudodifferential operators in the model. He claims there are interesting nonlinear versions of these models that more accurately capture the physics, which would be rather interesting to learn more about.

However, he also loves learning about GR and hopes to get caught on the talk Holzegel gave today and related topics that arose during his visit upon his return next week. Did anyone attend the talk and have any highlights to share???

Don't forget that next week the talk is Monday afternoon due to Fall Break. We'll have Betsy Stovall from UCLA talking about blow-up for Klein-Gordon equations. I strongly encourage everyone to attend!
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