Harmonic analysis on manifolds


Time and location

Classroom B115, Van Vleck Hall (math building)

Aug. 1. to Aug. 5. 2022

Madison, WI


Participants

Graduate students and recent PhD graduates


Topic

Harmonic analysis on manifolds


Modes

Participants will report pre-selected papers. Each participant will need to select one paper, read, write notes and present during the week of the summer school.


Financial support

Hotels for non-local participants will be covered. Please contact organizers if you need support for travel expenses.


Writing Camp

Some limited funding is available for participants to stay a few days longer for focused writing time for dissertations, other research projects, or job application materials.


Papers to discuss

Papers to be presented by one participant:


      1. J. Galkowski, A microlocal approach to eigenfunction concentration, J. Equi. Driv. Partielles (2018), Expos No. 3.


Wang, Jian. UNC.


      1. Blair, Matthew D. ; Sogge, Christopher D. Concerning Toponogov's theorem and logarithmic improvement of estimates of eigenfunctions. J. Differential Geom. 109 (2018), no. 2, 189--221.

Park, Chamsol. The University of New Mexico/Johns Hopkins.


      1. Sogge, Christopher D. Improved critical eigenfunction estimates on manifolds of nonpositive curvature. Math. Res. Lett. 24 (2017), no. 2, 549--570.

Jung, Hongki. IU Bloomington.


      1. Huang, Xiaoqi; Sogge, Christopher D. Quasimode and Strichartz estimates for time-dependent Schroodinger equations with singular potentials


Zhou, Zirui. Berkeley.


      1. Sogge, Christopher D. ; Toth, John A. ; Zelditch, Steve . About the blowup of quasimodes on Riemannian manifolds. J. Geom. Anal. 21 (2011), no. 1, 150--173.


Tao, Zhongkai. Berkeley.


      1. Huang, Xiaoqi ; Sogge, Christopher D. Weyl formulae for Schrodinger operators with critically singular potentials. Comm. Partial Differential Equations 46 (2021), no. 11, 2088--2133.


Quinn, Connor. Johns Hopkins.





Papers to be presented by two participants:


      1. Burq, N. ; Gerard, P. ; Tzvetkov, N. Strichartz inequalities and the nonlinear Schrodinger equation on compact manifolds. Amer. J. Math. 126 (2004), no. 3, 569--605.

Zhang, Haonan, IST Austria and Li, Franky, UW Madison.


      1. Sogge, Christopher D. ; Zelditch, Steve . Focal points and sup-norms of eigenfunctions. Rev. Mat. Iberoam. 32 (2016), no. 3, 971--994.

Sogge, Christopher D. ; Zelditch, Steve . Focal points and sup-norms of eigenfunctions II: the two-dimensional case. Rev. Mat. Iberoam. 32 (2016), no. 3, 995--999.


Johnsrude, Ben, UCLA and de Dios Pont, Jaume, UCLA.


      1. Smith, Hart F. A parametrix construction for wave equations with C^{1,1} coefficients. Ann. Inst. Fourier (Grenoble) 48 (1998), no. 3, 797--835.
        Smith, Hart F. ; Tataru, Daniel . Sharp counterexamples for Strichartz estimates for low regularity metrics. Math. Res. Lett. 9 (2002), no. 2-3, 199--204.

Green, John, The University of Edinburg/UPenn, and Duan, Baihe, UW Madison.


      1. Frank, Rupert L.; Sabin, Julien. Sharp Weyl laws with singular potentials


Brown, Madelyne, UNC and Wang, Yanfei, Johns Hopkins.


      1. Guth, Larry; Wang, Hong; Zhang, Ruixiang. A sharp square function estimate for the cone in R^3. Ann. of Math. (2) 192 (2020), no. 2, 551-581.

Gan, Shengwen, MIT and Chen, Mingfeng, Nanjing University/UW Madison.


      1. Marshall, Simon. Geodesic restrictions of arithmetic eigenfunctions. Duke Math. J. 165 (2016).


Leng, James, UCLA, and Srivastava, Rajula, UW Madison.


      1. Blair, Matthew D. ; Sogge, Christopher D. Logarithmic improvements in Lp bounds for eigenfunctions at the critical exponent in the presence of nonpositive curvature. Invent. Math. 217 (2019), no. 2, 703--748.


Denson, Jacob, UW Madison and Wang, Hong, UCLA.



Register

Please register through the following link. Deadline for registration is Apr 15, 2022.


Organizers

Acknowledgement

The summer school is supported by NSF grants of Shaoming Guo and Betsy Stovall, and by the NSF RTG grant Analysis and PDEs at Wisconsin.


A few places to see near Madison

Boating. Please google for boat rentals. I heard this website has reasonably priced ones.

Hiking. Devil's lake is what many people would recommend. I personally also like Parfrey's Glen State natural area.

Cycling. Rent a bike (BCycle, which is electronic, is quite popular) and cycle around the city and lake Monona.