Research

The two main themes of my research program are multilinear operators and point configurations. Many concepts in mathematics, from boundary value problems in partial differential equations and mathematical physics to finite point configurations in geometric combinatorics, are fundamentally tied to various operator bounds. This frequently leads to the estimation of linear and multilinear operators using techniques from harmonic analysis, often combined with geometric and combinatorial principles.

A particularly rich source of inspiration is a famous question of Erdös where he asked about the least number of distinct distances determined by points in the plane. Instead of distance, which is a simple pattern, I mainly study multipoint configuration versions of this problem, both in the original discrete setting, as well as in a continuous setting where geometric measure theory plays a key role. These point configuration problems have direct connections to big data.

Geometric averaging operators, such as averaging a function over a sphere, play a key role in many of my investigations and are of independent interest. A particularly important class are multilinear analogues of linear generalized Radon transforms. In addition to applications to geometric measure theory and combinatorics, further investigation of these multilinear generalized Radon transforms led to applications to restriction theory, partial differential equations, Sobolev trace inequalities and multilinear analogues of Stein's spherical maximal operator theorem.

My work on multilinear singular integral operators goes all the way back to my thesis. I used time-frequency analysis to study both Lp estimates, variational estimates and sparse estimates for such operators. These operators are frequently motivated by, and have potential applications to, both non-linear partial differential equations and ergodic theory.

Publications

  1. E. A. Palsson, F. Romero Acosta, "A Mattila-Sjölin theorem for triangles", (2021), submitted, arXiv:2109.13429.
  2. A. Iosevich, E. A. Palsson, S. R. Sovine, "Simplex averaging operators: quasi-Banach and Lp-improving bounds in lower dimensions", (2021), submitted, arXiv:2109.09017.
  3. H. L. Fleischmann, H. B. Hu, F. Jackson, S. J. Miller, E. A. Palsson, E. Pesikoff, C. Wolf, "Distinct angle problems and variants", (2021), submitted, arXiv:2108.12015.
  4. H. L. Fleischmann, S. J. Miller, E. A. Palsson, E. Pesikoff, C. Wolf, "Optimal point sets determining few distinct angles", (2021), submitted, arXiv:2108.12034.
  5. E. Boldyriew, Elena Kim, S. J. Miller, E. A. Palsson, S. Sovine, F. T. Suarez, J. Zhao, "Tinkering with Lattices: A New Take on the Erdos Distance Problem", (2020), submitted, arXiv:2009.12450.
  6. B. Cook, K. Hughes, E. Palsson, "Supercritical discrete restriction estimates for forms in many variables", (2020), submitted, arXiv:2004.02301.
  7. A. Iosevich, E. A. Palsson, "An improved dimensional threshold for the angle problem", (2018), submitted, arXiv:1807.05465.
  8. R. F. Durst, M. Hlavacek, C. Huynh, S. J. Miller, E. A. Palsson, "Classification of crescent configurations", (2016), submitted, arXiv:1610.07836.
  9. A. Greenleaf, A. Iosevich, B. Liu, E. A. Palsson, "An elementary approach to simplexes in thin subsets of Euclidean space", (2016), submitted, arXiv:1608.04777.
  10. T. C. Anderson, A. V. Kumchev, E. A. Palsson, "Discrete maximal operators over surfaces of higher codimension", (2020), accepted for publication in La Matematica, arXiv:2006.09968.
  11. T. C. Anderson, E. A. Palsson, "Bounds for discrete multilinear spherical maximal functions", (2019), accepted for publication in Collectanea Mathematica, arXiv:1910.11409.
  12. E. A. Palsson, S. Senger, C. Wolf, "Angle chains and pinned variants", Bulletin of the Hellenic Mathematical Society, 65 (2021), 35-53. arXiv:2104.09960
  13. S. Gunter, E. A. Palsson, B. Rhodes, S. Senger, "Bounds on point configurations determined by distances and dot products", Combinatorial and Additive Number Theory IV, 243-260, Springer Proceedings in Mathematics & Statistics, 297, Springer, Cham, 2021. arXiv:2011.15055
  14. E. A. Palsson, S. Senger, A. Sheffer, "On the number of discrete chains", Proceedings of the American Mathematical Society, 149 (2021), no. 12, 5347-5358. arXiv:1902.08259
  15. T. C. Anderson, E. A. Palsson, "Bounds for discrete multilinear spherical maximal functions in higher dimensions", Bulletin of the London Mathematical Society, 53 (2021), no. 3, 855-860. arXiv:1911.00464
  16. H. N. Brenner, J. S. Depret-Guillaume, E. A. Palsson, S. Senger, "Uniqueness of optimal point sets determining two distinct triangles", Integers, 21 (2021), #A43. arXiv:1910.00629
  17. J. Hoelscher, E. A. Palsson, "Counting restricted partitions of integers into fractions: symmetry and modes of the generating function and a connection to ω(t)", PUMP Journal of Undergraduate Research, 3 (2020), 277-307. arXiv:2011.14502
  18. S. Fish, D. King, S. J. Miller, E. A. Palsson, C. Wahlenmayer, "Crescent configurations in normed spaces", Integers, 20 (2020), #A96. arXiv:1909.08769
  19. E. A. Palsson, S. R. Sovine, "The triangle averaging operator", Journal of Functional Analysis, 279 (2020), no. 8, #108671. arXiv:1910.01282
  20. R. F. Durst, C. Huynh, A. Lott, S. J. Miller, E. A. Palsson, W. Touw, G. Vriend, "The inverse gamma distribution and Benford's law", PUMP Journal of Undergraduate Research, 3 (2020), 95-109. arXiv:1609.04106
  21. H. N. Brenner, J. S. Depret-Guillaume, E. A. Palsson, R. W. Stuckey, "Characterizing optimal point sets determining one distinct triangle", Involve, 13 (2020), no. 1, 91-98. arXiv:1910.00633
  22. J. DeWitt, K. Ford, E. Goldstein, S. J. Miller, G. Moreland, E. A. Palsson, S. Senger, "Dimensional lower bounds for Falconer type incidence theorems", Journal d'Analyse Mathématique, 139 (2019), 143-154. arXiv:1612.00539
  23. K. Cordwell, A. Epstein, A. Hemmady, S. J. Miller, E. A. Palsson, A. Sharma, S. Steinerberger, Y. N. T. Vu, "On algorithms to calculate integer complexity", Integers, 19 (2019), #A12. arXiv:1706.08424
  24. A. Epstein, A. Lott, S. J. Miller, E. A. Palsson, "Optimal point sets determining few distinct triangles", Integers, 18 (2018), #A16. arXiv:1609.00206
  25. R. Dorward, P. Ford, E. Fourakis, P. Harris, S. Miller, E. A. Palsson, H. Paugh, "Individual gap measures from Generalized Zeckendorf decompositions", Uniform Distribution Theory, 12 (2017), no. 1, 27-36. arXiv:1509.03029
  26. R. Dorward, P. Ford, E. Fourakis, P. Harris, S. Miller, E. A. Palsson, H. Paugh, "A Generalization of Zeckendorf's Theorem via Circumscribed m-gons", Involve, 10 (2017), no. 1, 125-150. arXiv:1508.07531
  27. Y. Do, R. Oberlin, E. A. Palsson, "Variation-norm and fluctuation estimates for ergodic bilinear averages", Indiana University Mathematics Journal, 66 (2017), no. 1, 55-99. arXiv:1504.07134
  28. A. Iosevich, M. Mourgoglou, E. A. Palsson, "On angles determined by fractal subsets of the Euclidean space", Mathematical Research Letters, 23 (2016), no. 6, 1737-1759. arXiv:1110.6792
  29. D. Burt, E. Goldstein, S. Manski, S. J. Miller, E. A. Palsson, H. Suh, "Crescent configurations", Integers, 16 (2016), #A38. arXiv:1509.07220
  30. B. Murphy, E. A. Palsson, G. Petridis, "The cardinality of sumsets: different summands",
    Acta Arithmetica, 167 (2015), no. 4, 375-395. arXiv:1309.2191
  31. A. Greenleaf, A. Iosevich, B. Liu, E. A. Palsson, "A group-theoretic viewpoint on Erdos-Falconer problems and the Mattila integral", Revista Matematica Iberoamericana, 31 (2015), no. 3, 799-810. arXiv:1306.3598
  32. L. Grafakos, A. Greenleaf, A. Iosevich, E. A. Palsson, "Multilinear generalized Radon transforms and point configurations", Forum Mathematicum, 27 (2015), no.4, 2323-2360. arXiv:1204.4429
  33. D. Geba, A. Greenleaf, A. Iosevich, E. A. Palsson, E. Sawyer, "Restricted convolution inequalities, multilinear operators and applications", Mathematical Research Letters, 20 (2013), no. 4, 675-694. arXiv:1209.6574
  34. Y. Do, R. Oberlin, E. A. Palsson, "Variational bounds for a dyadic model of the bilinear Hilbert transform",
    Illinois Journal of Mathematics, 57 (2013), no. 1, 105-120. arXiv:1203.5135
  35. E. A. Palsson, "Lp estimates for a singular integral operator motviated by Calderón's second commutator",
    Journal of Functional Analysis, 262 (2012), no. 4, 1645-1678. arXiv:1104.5160
  36. E. A. Palsson, "Lp estimates for a singular integral operator motivated by Calderón's second commutator",
    PhD Thesis, Cornell University, May 2011.

Selected Talks

  1. "How many points do you need to guarantee many patterns?",
    Colloquium,
    UMass Lowell,
    Lowell, Massachusetts, March 30, 2021.
  2. "Geometric averaging operators motivated by point configurations",
    Fourier Restriction Online 2021 Workshop,
    Online, March 13, 2021.
  3. "Discrete operators in harmonic analysis",
    AIM Workshop: Arithmetic statistics, discrete restriction, and Fourier analysis,
    Online, February 19, 2021.
  4. "Falconer type theorems: How many points do you need to guarantee many patterns?",
    Colloquium,
    The Ohio State University,
    Columbus, Ohio, April 16, 2020.
  5. "Bounds for a discrete variant of the bilinear spherical maximal operator",
    Analysis Seminar,
    University of Rochester,
    Rochester, New York, April 10, 2020.
  6. "Geometric averaging operators and point configurations",
    Analysis Seminar,
    Georgia Tech,
    Atlanta, Georgia, February 26, 2020.
  7. "Geometric averaging operators and point configurations",
    Seminar,
    Missouri State University,
    Springfield, Missouri, January 28, 2020.
  8. "On the number of discrete chains",
    Combinatorics Seminar,
    University of Rochester,
    Rochester, New York, November 7, 2019.
  9. "Falconer type theorems and integral operators",
    Caltech/UCLA joint Analysis Seminar,
    University of California, Los Angeles,
    Los Angeles, California, October 18, 2019.
  10. "Bounds for a discrete variant of the bilinear spherical maximal operator",
    AMS Special Session on Recent Developments in Harmonic Analysis,
    Fall Central Sectional Meeting,
    Madison, Wisconsin, September 14, 2019.
  11. "How many points do you need to guarantee many patterns?",
    Physics and Math Colloquium,
    Universidad de Colima,
    Colima, Mexico, May 30, 2019.
  12. "A group-theoretic viewpoint on Falconer type theorems",
    Harmonic Analysis and PDE Seminar,
    University of Virginia,
    Charlottesville, Virginia, April 9, 2019.
  13. "On the number of discrete chains",
    Combinatorics, Algebra and Geometry Seminar,
    George Mason University,
    Fairfax, Virginia, April 5, 2019.
  14. "A group-theoretic viewpoint on Falconer type theorems",
    Analysis and PDE Seminar,
    Purdue University,
    West Lafayette, Indiana, December 6, 2018.
  15. "An improved threshold for the Falconer angle problem",
    AMS Special Session on Analysis and Geometry of Fractals,
    Fall Western Sectional Meeting,
    San Francisco, California, October 27, 2018.
  16. "A group-theoretic viewpoint on Falconer type theorems",
    Analysis Seminar,
    Cornell University,
    Ithaca, New York, October 22, 2018.
  17. "A group-theoretic viewpoint on Falconer type theorems",
    AMS Special Session on Interplay between Analysis and Combinatorics,
    Fall Eastern Sectional Meeting,
    Newark, Delaware, September 29, 2018.
  18. "A group-theoretic viewpoint on Falconer type theorems",
    Pure Mathematics Seminar,
    University of Queensland,
    Brisbane, Australia, June 19, 2018.
  19. "Falconer type theorems for higher order point configurations",
    Southeastern Analysis Meeting,
    Atlanta, Georgia, March 24, 2018.
  20. "Falconer type theorems for higher order point configurations",
    Spring Mini Courses in Analysis and Geometry,
    Baton Rouge, Louisiana, February 9, 2018.
  21. "Falconer type theorems for simplices",
    Analysis, Logic, and Physics Seminar,
    Virginia Commonwealth University,
    Richmond, Virginia, January 26, 2018.
  22. "Dimensional lower bounds for Falconer type incidence theorems",
    Northeast Analysis Network Conference,
    Syracuse, New York, September 23, 2017.
  23. "Falconer type theorems for simplices",
    Analysis Seminar,
    Georgia Tech,
    Atlanta, Georgia, April 12, 2017.
  24. "Falconer type theorems for simplices",
    Inverse Problems and Analysis Seminar,
    University of Delaware,
    Newark, Delaware, March 3, 2017.
  25. "Falconer type theorems for simplices",
    Harmonic Analysis And Geometry Of Fractal Sets,
    Columbus, Ohio, February 4, 2017.
  26. "Estimates for a singular integral operator motivated by Calderón's second commutator",
    Colloquium,
    TIFR Bangalore,
    Bangalore, India, January 10, 2017.
  27. "Finite point configurations",
    Seminar,
    Indian Institute of Science, Bangalore,
    Bangalore, India, January 9, 2017.
  28. "Finite point configurations",
    Department of Mathematics Colloquium,
    Virginia Tech,
    Blacksburg, Virginia, November 11, 2016.
  29. "Finite point configurations",
    Analysis and Arithmetic Combinatorics Seminar,
    University of Georgia,
    Athens, Georgia, October 24, 2016.
  30. "Finite point configurations",
    Northeast Analysis Network Conference,
    Rochester, New York, September 9, 2016.
  31. "Finite point configurations",
    Summer Analysis Workshop,
    Oberlin College,
    Oberlin, Ohio, July 9, 2016.
  32. "Finite point configurations in thin sets",
    Combinatorics Seminar,
    University of Rochester,
    Rochester, New York, February 2, 2016.
  33. "Multilinear phenomena in analysis and related areas",
    Department of Mathematics Colloquium,
    University of Illinois at Urbana-Champaign,
    Urbana-Champaign, Illinois, December 10, 2015.
  34. "A look at the distinct distance problem and crescent configurations",
    Mathematics Colloquium,
    Bard College,
    Annandale-On-Hudson, New York, December 8, 2015.
  35. "Finite point configurations and multilinear Radon transforms",
    Analysis Seminar,
    Yale University,
    New Haven, Connecticut, December 1, 2015.
  36. "Finite point configurations and multilinear Radon transforms",
    Mathematics Department Seminar,
    Colgate University,
    Hamilton, New York, October 15, 2015.
  37. "Finite point configurations",
    Special Session on Geometric Aspects of Harmonic Analysis,
    Joint Meeting of the AMS, EMS and SPM,
    Porto, Portugal, June 12, 2015.
  38. "Finite point configurations",
    International Conference on Harmonic Analysis and Applications,
    The Graduate Center of City University of New York,
    New York, New York, June 1, 2015.
  39. "Variational bounds for a dyadic model of the bilinear Hilbert transform",
    Analysis Seminar,
    Brown University,
    Providence, Rhode Island, March 9, 2015.
  40. "Variational bounds for a dyadic model of the bilinear Hilbert transform",
    Analysis Seminar,
    State University of New York Albany,
    Albany, New York, November 12, 2014.
  41. "Finite point configurations and multilinear Radon transforms",
    Department of Mathematics Undergraduate Colloquium,
    Trinity College,
    Hartford, Connecticut, October 9, 2014.
  42. "Finite point configurations and multilinear Radon transforms",
    Trimester Seminar,
    Hausdorff Research Institute for Mathematics,
    Bonn, Germany, July 29, 2014.
  43. "Restricted convolution inequalities, multilinear operators and applications",
    Harmonic Analysis to celebrate Michael Cowling's 65th,
    Segovia, Spain, July 2, 2014.
  44. "Multilinear phenomena in analysis and related areas",
    Department of Mathematics Colloquium,
    The University of Alabama,
    Tuscaloosa, Alabama, February 18, 2014.
  45. "Finite point configurations and multilinear Radon transforms",
    Mathematics and Statistics Department Colloquium,
    Williams College,
    Williamstown, Massachusetts, January 28, 2014.
  46. "Multilinear phenomena in analysis and related areas",
    Analysis Seminar,
    University of Rochester,
    Rochester, New York, November 8, 2013.
  47. "Falconer type theorems for simplices",
    Special Session on Harmonic Analysis and Applications,
    Joint Meeting of the AMS and the RMS,
    Alba Iulia, Romania, June 29, 2013.
  48. "Restricted convolution inequalities, multilinear operators and applications",
    Pure Mathematics Seminar,
    University of Queensland,
    Brisbane, Australia, June 4, 2013.
  49. "k-point configurations and multilinear generalized Radon transforms",
    Maths Colloquium,
    University of Queensland,
    Brisbane, Australia, June 3, 2013.
  50. "k-point configurations and multilinear generalized Radon transforms",
    Sydney-UNSW Joint Colloquium,
    University of Sydney,
    Sydney, Australia, May 31, 2013.
  51. "k-point configurations and multilinear generalized Radon transforms",
    Colloquium,
    University of Rochester,
    Rochester, New York, May 2, 2013.
  52. "Multilinear generalized Radon transforms",
    Analysis Seminar,
    Brown University,
    Providence, Rhode Island, November 19, 2012.
  53. "Multilinear generalized Radon transforms",
    Analysis Seminar,
    Indiana University Bloomington,
    Bloomington, Indiana, October 25, 2012.
  54. "Variational bounds for a dyadic model of the bilinear Hilbert transform",
    AMS Special Session on Harmonic Analysis and Convexity,
    Fall Central Sectional Meeting,
    Akron, Ohio, October 21, 2012.
  55. "Variational bounds for a dyadic model of the bilinear Hilbert transform",
    AMS Special Session on Wavelet and Frame Theoretic Methods in Harmonic Analysis and Partial Differential Equations in Memory of Daryl Geller,
    Fall Eastern Sectional Meeting,
    Rochester, New York, September 29, 2012.
  56. "On multilinear generalized Radon transforms",
    Analysis Seminar,
    Cornell University,
    Ithaca, New York, March 26, 2012.
  57. "On multilinear generalized Radon transforms",
    Analysis and PDE Seminar,
    University of California Los Angeles,
    Los Angeles, California, January 13, 2012.
  58. "On multilinear generalized Radon transforms",
    AMS Special Session on Radon Transforms and Geometric Analysis (in honor of Sigurdur Helgason's 85th birthday),
    Joint Mathematics Meetings,
    Boston, Massachusetts, January 7, 2012.
  59. "On multilinear generalized Radon transforms and angles",
    Analysis Seminar,
    University of Rochester,
    Rochester, New York, December 9, 2011.
  60. "Lp estimates for a singular integral operator motviated by Calderón's second commutator",
    Incompressible Fluids, Turbulence and Mixing. In honor of Peter Constantin's 60th birthday.,
    Carnegie Mellon University,
    Pittsburgh, Pennsylvania, October 15, 2011.
  61. "Lp estimates for a singular integral operator motviated by Calderón's second commutator",
    AMS Session on Topics in Analysis,
    Joint Mathematics Meetings,
    New Orleans, Louisiana, January 9, 2011.
  62. "Lp estimates for a singular integral operator motviated by Calderón's second commutator",
    Analysis Seminar,
    Georgia Tech,
    Atlanta, Georgia, December 8, 2010.
  63. "Lp estimates for a singular integral operator motviated by Calderón's second commutator",
    Analysis Seminar,
    Cornell University,
    Ithaca, New York, October 25, 2010.
  64. "Lp estimates for a singular integral operator motviated by Calderón's second commutator",
    Analysis Seminar,
    University of Rochester,
    Rochester, New York, October 22, 2010.
  65. "Lp estimates for a singular integral operator motviated by Calderón's second commutator",
    Calderón-Zygmund Analysis Seminar,
    University of Chicago,
    Chicago, Illinois, May 3, 2010.