M. Burak Erdoğan
Professor
Department of Mathematics,
University of Illinois,
1409 W. Green Street (MC-382)
Urbana, Illinois 61801

Office: CAB 41
Phone: (217) 265-6761
E-mail: berdogan at illinois edu

Teaching:

  • Math 542, Complex Variables I (Fall 2023), MWF 11-11:50, 119 English Building
  • Math 428, Honors Topics: Intro to Fourier Analysis (Fall 2023), MWF 12-12:50, 140 Henry

Biographical Information:

Research Interests:

Students:

External grants:

  • NSF DMS-2154031, award date: 04/05/2022
  • Simons collaboration grant 634269, award date: 9/1/2019
  • NSF DMS-1501041, award date: 5/15/2015
  • NSF DMS-1201872, award date: 5/15/2012
  • NSF DMS-0900865, award date: 6/1/2009
  • NSF DMS-0600101, award date: 5/15/2006
  • NSF DMS-0540084, award date: 9/1/2004
  • NSF DMS-0303413, award date: 7/1/2003

Publications (supported in part by the grants above):

  1. (with W. Green and K. Lamaster) " $L^p$-continuity of wave operators for higher order Schrodinger operators with threshold eigenvalues in high dimensions", preprint 2024, 23pp, submittted, [pdf]
  2. (with N. Tzirakis) "Smoothing estimates for weakly nonlinear internal waves in a rotating ocean", preprint 2024, 17pp, accepted by J. Math Physics, Analysis, Geometry , [pdf]
  3. (with M. Goldberg and W. Green) "Dispersive estimates for higher order Schrodinger operators with scaling-critical potentials", preprint 2023, 25pp, submittted, [pdf]
  4. (with C. Huynh and R. McConnell) "Talbot Effect on the Sphere and Torus for d >= 2", preprint 2023, 42pp, to appear in Math Z. [pdf]
  5. (with W. Green) "A note on endpoint Lp-continuity of wave operators for classical and higher order Schrodinger operators", J. Differential Equations 355 (2023), 144--161. [pdf]
  6. (with M. Goldberg and W. Green) "The Lp-unboundedness of wave operators for higher order Schrodinger operators", J. Funct. Anal. 285 (2023), no. 5, 110008 [pdf]
  7. (with W. Green) "The Lp continuity of wave operators for higher order Schrodinger operators", Adv. Math. 404 (2022), Paper No. 108450, 41 pp. [pdf]
  8. (with W. Green and E. Toprak) "What is the Dirac equation?", Notices Amer. Math. Soc. 68 (2021), no. 10, 1782--1785. [pdf]
  9. (with W. Green) "On the one dimensional Dirac equation with potential ", Jour. de Mathématiques Pures et Appliquées. 151 (9) (2021), 132--170. [Revised pdf]
  10. (with A. Barron and T. Harris) "Fourier decay of fractal measures on hyperboloids", Trans. Amer. Math. Soc. 374 (2021), no. 2, 1041--1075. [pdf]
  11. (with W. Green and E. Toprak) "On the fourth order Schrodinger equation in three dimensions: Dispersive estimates and zero energy resonances ", J. Differential Equations 271 (2021), 152--185. [pdf]
  12. (with M. Goldberg and W. Green) "The Massless Dirac Equation in Two Dimensions: Zero-Energy Obstructions and Dispersive Estimates", J. Spectral Theory 11 (2021), no. 3, 935--979. [pdf]
  13. (with M. Goldberg and W. Green) "Strichartz estimates for the Schrodinger equation with a measure-valued potential ", Proc. Amer. Math. Soc. Ser. B 8 (2021), 336--348. [pdf]
  14. (with B. Gurel and N. Tzirakis) "Fifth order KP-II equation on the upper half plane ", Differential and Integral Eqns 33 (2020), 555--596 [pdf].
  15. (with W. Green and E. Toprak) "Dispersive estimates for Dirac operators in dimension three with obstructions at threshold energies", Amer. J. Math. 141 (2019), no. 5, 1217--1258 [pdf]
  16. (with B. Gurel and N. Tzirakis) "Smoothing for the fractional Schrodinger equation on the torus and the real line", Indiana Univ. Math. J. 68 (2019), no. 2, 369--392. [pdf]
  17. (with G. Shakan) "Fractal solutions of dispersive partial differential equations on the torus", Selecta Math. (N.S.) 25 (2019), no. 1, Art. 11, 26 pp. [pdf]
  18. (with M. Goldberg and W. Green) "Limiting absorption principle and Strichartz estimates for Dirac operators in two and higher dimensions", Commun. Math. Phys. 367 (2019), 241--263. [pdf]
  19. (with W. Green and E. Toprak) "Dispersive estimates for massive Dirac operators in dimension two", J. Differential Equations 264 (2018), no. 9, 5802--5837. [pdf]
  20. (with B. Gurel and N. Tzirakis) "The derivative nonlinear Schrodinger equation on the half line", Ann. Inst. H. Poincare Anal. Non Lineaire 35 (2018), 1947--1973. [pdf]
  21. (with M. Goldberg and W. Green) "On the Lp boundedness of wave operators for two-dimensional Schrodinger operators with threshold obstructions", J. Funct. Anal. 274 (2018), no. 7, 2139--2161. [pdf]
  22. (with N. Tzirakis) "Regularity properties of the Zakharov system on the half line", Comm. Partial Differential Equations 42 (2017), no. 7, 1121--1149. [pdf]
  23. (with W. Green) "The Dirac equation in two dimensions: Dispersive estimates and classification of threshold obstructions", Commun. Math. Phys. 352 (2017), no. 2, 719--757. [pdf]
  24. (with N. Tzirakis) "Dispersive partial differential equations, wellposedness and applications", London Mathematical Society Student Texts 86, Cambridge  University Press, 2016.
  25. (with N. Tzirakis) "Regularity properties of the cubic nonlinear Schrodinger equation on the half line", J. Funct. Anal. 271 (2016), no. 9, 2539--2568. [pdf]
  26. (with S. Demirbas and N. Tzirakis) "Existence and uniqueness theory for the fractional Schrodinger equation on the torus". Some topics in harmonic analysis and applications, 145--162, Advanced Lectures in Mathematics (ALM), 34, Int. Press, Somerville, MA, 2016. [pdf]
  27. (with J. Marzuola, K. Newhall, and N. Tzirakis) "The structure of global attractors for dissipative Zakharov systems with forcing on the torus", SIAM J. Appl. Dyn. Syst. 14 (2015), no. 4, 1978--1990.  [pdf]
  28. (with V. Chousionis and N. Tzirakis) "Fractal solutions of linear and nonlinear dispersive partial differential equations", Proc. Lond. Math. Soc. (3) 110 (2015), no. 3, 543--564.   [pdf]
  29. (with M. Goldberg and W. Green) "Dispersive estimates for four dimensional Schrodinger and wave equations with obstructions at zero energy",  Comm. Partial Differential Equations. 39:10 (2014), 1936--1964. [pdf]
  30. (with N. Tzirakis) "Talbot effect for the cubic nonlinear Schrodinger equation on the torus", Math. Res. Lett. 20 (2013), 1081--1090. [pdf]
  31. (with W. Green) "Dispersive estimates for matrix Schrodinger operators in dimension two", DCDS-A 33 (2013), 4473--4495. [pdf]
  32. (with N. Tzirakis) "Smoothing and global attractors for the Zakharov system on the torus", Analysis & PDE 6-3 (2013), 723--750. [pdf]
  33. (with W. Green) "A weighted dispersive estimate for Schrodinger operators in dimension two", Commun. Math. Phys. 319 (2013), 791--811.   [pdf]
  34. (with W. Green) "Dispersive estimates for Schrodinger operators in dimension two with obstructions at zero energy", Trans. Amer. Math. Soc. 365 (2013), 6403--6440  [pdf]
  35. (with N. Tzirakis) "Long time dynamics for forced and weakly damped KdV on the torus", Commun. Pure Appl. Anal. 12 (2013), 2669--2684. [pdf].
  36. (with N. Tzirakis) "Global smoothing for the periodic KdV evolution", Int. Math. Res. Notices (2013), 4589--4614.  [pdf].
  37. (with D. Hart and  A. Iosevich) "Multi-parameter projection theorems with applications to sums-products and finite point configurations in the Euclidean setting", Recent Advances in Harmonic Analysis and Applications, Springer Proceedings in Mathematics & Statistics Volume 25, 2013, 93--103.
  38. (with Dan Oberlin) " Restricting the Fourier transform of measures to curves in R^2", Can. Math. Bull. 56 (2013), 326--336. [pdf]
  39. (with N. Tzirakis and V. Zharnitsky) "High frequency perturbation of cnoidal waves", SIAM J. Math. Anal. 44 (2012), no. 6, 4147--4164. [pdf]. 
  40. (with J. Chapman, D. Hart, A. Iosevich and D. Koh) "Pinned distance sets, k-simplices, Wolff's exponent in finite fields and sum-product estimates", Math Z 271 (2012), 63--93, [pdf]
  41. (with N. Tzirakis and V. Zharnitsky) "Near-linear dynamics of nonlinear dispersive waves", Physica D 240 (2011), 1325--1333. [pdf]
  42. (with D. Hundertmark and Y.-R. Lee) " Exponential decay of dispersion managed solitons for vanishing average dispersion", Math. Res. Lett. 18 (2011), no. 01, 11--24,   [pdf]
  43. (with N. Tzirakis and V. Zharnitsky) "Near-linear dynamics in KdV with periodic boundary conditions", Nonlinearity 23 (2010) 1675--1694.  [pdf]
  44. (with W. Green) "Dispersive estimates for the Schrodinger equation for $C^{\frac{n-3}{2}}$ potentials in odd dimensions", Int Math Res Notices 2010:13, 2532--2565. [pdf]
  45. (with R. Oberlin) " Estimates for the X-ray transform restricted to 2-manifolds", Rev. Mat. Ibero. 26 (2010), 91--114.   [pdf]
  46. (with M. Goldberg and W. Schlag) "Strichartz and smoothing estimates for Schrodinger operators with almost critical magnetic potentials in three and higher dimensions", Forum Math. 21 (2009), 687--722. [pdf]
  47. (with V. Zharnitsky) "Quasi-linear dynamics in nonlinear Schr\" odinger equation with periodic boundary conditions", Commun. Math. Phys. 281 (2008), 655--673.  [pdf]
  48. (with M. Goldberg and W. Schlag) "Strichartz and smoothing estimates for Schrodinger operators with large magnetic potentials in R^3", J. Eur. Math. Soc. (JEMS) 10 (2008), no. 2, 507--531. [pdf]
  49. (with M. Christ) "Mixed norm estimates for certain generalized Radon transforms", Trans. Amer. Math. Soc. 360 (2008), no. 10, 5477--5488. [pdf]
  50. (with W. Schlag) "Dispersive estimates for Schrodinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II",  J. d'Analyse Math. 99 (2006), 199--248. [pdf]
  51. "A bilinear Fourier extension theorem and applications to the distance set problem", Int. Math. Res. Not. 2005:23, 1411--1425. [pdf]
  52. "On Falconer's distance set conjecture", Rev. Mat. Ibero. 22 (2006), 649--662. [pdf]
  53. (with W. Schlag) "Dispersive estimates for Schrodinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: I", Dynamics of PDE 1 (2004), 359--379. [pdf]
  54. "A note on the Fourier transform of fractal measures", Math. Res. Lett. 11 (2004), 299--313. [pdf]
  55. (with R. Killip and W. Schlag) "Energy growth in Schrodinger's equation with Markovian forcing", Commun. Math. Phys. 240 (2003), 1--29. [pdf]
  56. "Mapping properties of the elliptic maximal function", Rev. Mat. Ibero. 19 (2003), 221--234. [pdf]
  57. (with M. Christ)"Mixed norm estimates for a restricted X-ray transform", J. d'Analyse Math. 87 (2002), 187--198. [pdf]
  58. "Mixed norm estimates for a restricted X-ray transform in R^4 and R^5", Int. Math. Res. Not. 2001, No.11, 575--600. [pdf]
  59. "Analytic and asymptotic properties of non-symmetric Linnik's densities", J. Fourier Anal. Appl. 5 (1999), 523--544.
  60. (with I.V. Ostrovskii) "On mixture representation of the Linnik density", J. Austral. Math. Soc. Ser. A 64 (1998), 317--326.
  61. (with I.V. Ostrovskii) "Analytic and asymptotic properties of generalized Linnik Probability densities", J. Math. Anal. Appl. 217 (1998), 555--579.
  62. (with I.V. Ostrovskii) "Non-symmetric Linnik Distributions", C.R. Acad. Sci. Paris Serie I 325 (1997), 511--516.

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