Appointment: 
2017-             Evans Instructor, Rice University

Education:
2017              PhD in Mathematics, University of Missouri, Columbia. 
                       Thesis title "Topics In Spectral Theory of Differential Operators"
                       Advisors: Fritz Gesztesy and Yuri Latushkin

2012              B.Sc. in Mathematics (degree with honors), Taras Shevchenko National University of Kyiv, Ukraine                  


Research
My research interests are in mathematical physics, partial differential equations, spectral theory, and infinite dimensional dynamical systems. Specifically, my work concerns the following topics:
  • asymptotic distribution of eigenvalues of elliptic operators on bounded domains: Weyl-type asymptotic formulas, bounds for the eigenvalue counting function,
  • (in)stability of nonlinear waves: evaluation of the Morse index via the Maslov index,
  • spectral theory of differential operators with non-local boundary conditions,
  • computation of the spectral flow  via the Maslov index.
Papers and preprints:

  • A bound for the eigenvalue counting function for Krein--von Neumann and Friedrichs extensions.
    Joint with M.S. Ashbaugh, F. Gesztesy, A. Laptev, and M. Mitrea,
    Advances in Mathematics  304, 1108–1155 (2017). [PDF, 48 pages]

  • Counting spectrum via the Maslov index for one dimensional θ-periodic Schrödinger operators.
    Joint with
    C. K. R. T. Jones and Y. Latushkin,
    Proceedings of the AMS 145, 363-377 (2017). [PDF, 15 pages]
  • The Morse and Maslov indices for Schrödinger operators.
    Joint with
    Y. Latushkin and A. Sukhtayev,
    Accepted for publication in Journal d'Analyse Mathematique.
    arXiv:1411.1656 [PDF, 36 pages] 

  • A bound for the eigenvalue counting function for higher-order Krein Laplacians on open sets.
    Joint with
    F. Gesztesy, A. Laptev, and M. Mitrea,
    Mathematical  Results in Quantum Mechanics,  Proceedings of the QMath12 Conference P. Exner, W. König, and H. Neidhardt (eds), World Scientific, Singapore, 2015, pp. 3-29. [PDF, 27 pages]

  • The Maslov index and the spectra of second order elliptic operators.
    Joint with
    Y. Latushkin. Preprint  available on the ArXiv [PDF, 48 pages].

Upcoming Talks/Conferences:

  1. September 1, 2017, Mathematical Physics and Harmonic Analysis Seminar, Texas A&M, College Station, TX. 
           
    Talk:"The Maslov Index and the Spectra of Second Order Elliptic Operators"
  2. September 6, 2017, Geometry-Analysis Seminar, Rice University, Houston, TX.
         
    Talk:"The Maslov Index and the Spectra of Second Order Elliptic Operators"
  3. October 20-22, 2017 Waves, Spectral Theory and Applications – Part 2, University of North Carolina, Chapel Hill, NC.
  4. November 3-5, 2017, Texas Analysis and Mathematical Physics Symposium, University of Texas, Austin, TX 
  5. December 9-12, 2017, SIAM Conference on Analysis of Partial Differential Equations, Baltimore, MD.
Past Talks/Conferences:
  1. SIAM Conference on Applications of Dynamical Systems, Snowbird, UT, USA 
              Talk:"The Maslov Index and the Spectra of Second Order Elliptic Operators"    
    1. KUMUNU Conference on PDE, Dynamical Systems and Applications 2017, Lincoln, NE, USA  
    2. Baylor University, Analysis Seminar, Waco, TX, USA
                Talk :"The Maslov Index and the Spectra of Second Order Elliptic Operators" 
    3. The Tenth IMACS Conference on Nonlinear Waves, Athens, GA, USA      
                Talk :"The Maslov Index and the Spectra of Second Order Elliptic Operators"
    4. Conference on Methods of Modern Mathematical Physics, The Fields Institute, Toronto, Canada

    5. Frontiers in Mathematical Physics. In honor of Barry Simon's 70th birthday,University of Montreal, Canada

    6. SIAM Conference on Nonlinear Waves and Coherent Structures, Philadelphia, PA, USA
      Talk: "Counting spectrum via the  Maslov index for Schrödinger operators"

    7.  Applications of the Maslov index to multidimensional stability problems, AIM SQuaRE, San Jose, CA, USA

    8.  Mathematical Physics and Harmonic Analysis Seminar, Texas A&M University, College Station, TX, USA
      Talk: "A Bound for the Eigenvalue Counting Function for Higher-Order Krein Laplacians on Open Sets"

    9. Computational & Applied Mathematics Seminar, University Kansas, Lawrence, KS, USA
      Talk:"Counting spectrum via the  Maslov index for Schrödinger operators"

    10.  Geometric and Computational Spectral Theory, University of Montreal, Canada

    11.  KUMU PDE, Dynamical Systems and Applications, University of Kansas, Lawrence, KS, USA
      Talk:"The Morse and Maslov Indices for Schrödinger Operators"

    12.  The 1st Annual Meeting of SIAM Central States Section, Missouri University of Science and Technology, Rolla, MO, USA
      Talk:"The Morse and Maslov Indices for Schrödinger Operators"

    13.  University of North Carolina, Chapel Hill, research visit by invitation of Prof. C. K. R. T. Jones

    14.  Differential Equations Seminar, University of Missouri, Columbia, MO, USA
      Talk:"The Morse and Maslov Indices for Schrödinger Operators"

    15.  The 34th Southeastern-Atlantic Regional Conference on Differential Equations, University of Memphis, TN, USA
      Talk:"The Morse and Maslov Indices for Periodic Schrödinger Operators"

    16.  Differential Equations Seminar, University of Missouri, Columbia, MO, USA
      Talk:"A Bound for the Eigenvalue Counting Function for Higher-Order Krein Laplacians on Open Sets"

    17.  Stability of Solitary waves, Pisa, Italy

    18.  Operator Semigroups and Dispersive Equations, Heinrich-Fabri Institut, Blaubeuren, Germany
      Talk:"A Spectral Mapping Theorem for Nonlinear Schrodinger Equation"
    Teaching

    Past teaching:
    • [TA] Advanced Calculus of One Real Variable I:  Spring 2014, Fall 2016, Spring 2017.
    • [TA] Analytic Geometry and Calculus I: Fall 2013, Spring 2014, Fall 2014, Fall 2015, Spring 2017.
    • [TA] Calculus II: Spring 2015, Spring 2016, Fall 2016.