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:

  • The Maslov index and the spectra of second order elliptic operators.
    Joint with Y. Latushkin
    Advances in Mathematics 329, 422--486 (2018). [PDF, 64 pages]
  • 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]
  • 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. GesztesyA. 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]
  • 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]

Upcoming Talks/Conferences:

  1. March 5-9, 2018, Arizona School of Analysis and Mathematical Physics, University of Arizona, Tucson.
         Talk: "A Bound for the Eigenvalue Counting Function for Higher-Order Krein Laplacians on Open Sets"
  2. March 23, 2018, PDE Seminar, University of Houston, 
         Talk: "The Maslov Index and the Spectra of Second Order Elliptic Operators"
  3. April 14-15, 2018, Portland State University, Portland, OR (Spring Western Sectional Meeting)  
         Talk: "A Bound for the Eigenvalue Counting Function for Higher-Order Krein Laplacians on Open Sets"
  4. June 11-14, 2018, SIAM Conference on Nonlinear Waves and Coherent Structures, Anaheim, CA
         Talk: tba 
  5.  July 23–27, 2018, Conference on Mathematics of Wave Phenomena, Karlsruhe Institute of Technology
         Talk: tba
Past Talks/Conferences:
  1. February 18-19, 2018, 36th Western States meeting of Mathematical Physics, UC Irvine.
         Talk: "The Maslov Index and the Spectra of Second Order Elliptic Operators"
  2. SIAM Conference on Analysis of Partial Differential Equations, Baltimore MD
              
    Talk:"The Maslov Index and the Spectra of Second Order Elliptic Operators"
  3. Texas Analysis and Mathematical Physics Symposium 2017, Austin, TX
              
    Talk:"The Maslov Index and the Spectra of Second Order Elliptic Operators"
  4. Waves, Spectral Theory and Applications – Part 2, University of North Carolina, Chapel Hill, NC.
  5. Geometry-Analysis Seminar, Rice University, Houston, TX.
             
    Talk:"The Maslov Index and the Spectra of Second Order Elliptic Operators"
  6. Mathematical Physics and Harmonic Analysis Seminar, Texas A&M, College Station, TX. 
              Talk:"The Maslov Index and the Spectra of Second Order Elliptic Operators"
  7. 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
    Current teaching: 
    • Math 211 (differential equations and linear algebra)
    Past teaching:
    At Rice University, Fall 2017-
    • Math 211 (differential equations and linear algebra)  Fall 2017 
    • Math 321 (introduction to analysis) Fall 2017
    At the University of Missouri, Columbia, Fall 2013- Spring 2017: 
    • [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.