Feride Tiglay Department of Mathematics Purdue University 150 N. University Street West Lafayette, IN 47907-2067 Web: http://www.math.purdue.edu/~ftiglay/ Email: ftiglay at math.purdue.edu

About my research

Often the most interesting nonlinear PDE arise naturally in mathematical physics as approximate models for a diverse range of physical phenomena. Examples include PDE from gas dynamics, hydrodynamics, elasticity, gravitation, relativity, ecology and thermodynamics. A particularly interesting example from hydrodynamics is the Navier-Stokes equation where the nonlinearity comes from inertial effects. In general, when attacking a particular nonlinear PDE there are two fundamental questions to focus on: (i) how to establish existence and uniqueness of solutions for the associated boundary value problem, and (ii) how to ''solve'' the equation in an appropriate sense.

One of the challenging and tantalizing aspects of nonlinear PDE is that very few general techniques have been developed that work for all such PDE. Indeed it is almost a rule that each particular equation requires study as a separate problem in order to successfully address questions (i) and (ii) above. On the other hand nonlinear PDE that arise from integrable systems can often be completely solved in an appropriate sense. Perhaps the best known example is the Korteweg-de Vries equation. However there is no general criteria known for verifying integrability of infinite dimensional systems modeled by nonlinear PDE, and often special tools must be developed for each particular system.

Most of my research has been focused on attacking the following two fundamental questions for nonlinear PDE of one space dimension: (i) establishing existence, uniqueness and continuous dependence on initial data for the associated Cauchy problem, and (ii) developing tools for ''integrating'' the equation.

Publications

• The periodic Cauchy problem for Novikov's equation.
  Int. Math. Res. Not., doi:10.1093/imrn/rnq267, Advance Access published December 14, 2010. journal / arXiv
  


• Euler-Poincaré equations on Lie groups and homogeneous spaces, their orbit invariants and applications. (with C. Vizman)
  Lett. Math. Phys., Vol. 97, Issue 1 (2011). journal / arXiv
  


• Integrable evolution equations on spaces of tensor densities and their peakon solutions. (with J. Lenells and G. Misiolek)
  Comm. Math. Phys., Vol. 299, no.1 (2010), 129--161. journal / arXiv
  


• Global Existence of some infinite energy solutions for a perfect incompressible fluid. (with R. Saxton)
  SIAM J. Math. Anal. Vol. 40, no.4 (2008), 1499--1515. journal / arXiv
  


• The Cauchy problem and integrability of a modified Euler-Poisson equation.
  Trans. Amer. Math. Soc. Vol. 360 (2008), 1861--1877. journal / arXiv
  


• On unique continuation for the modified Euler-Poisson equations. (with A. Himonas and G. Misiolek)
  Discrete Contin. Dyn. Syst., Special Issue on Geometric Mechanics, Vol 19, no.3 (2007), 515--529. journal
  


• The periodic Cauchy problem of the modified Hunter-Saxton equation.
  J. Evol. Equ., Vol. 5, no.4 (2005), 509--527. journal / arXiv
  

Preprints

• Fredholm determinants and breakdown for μCH equation, in preparation.


• Analytic solutions for the μ-equations, in preparation.

Teaching

• Math 36600 Ordinary Differential Equations, spring 2013, Purdue University.


• Math 34100 Foundations of Analysis, fall 2012, Purdue University.


• Math 26100M Multivariate Calculus, fall 2012, Purdue University.


• Math 1225B Methods of Calculus, spring 2012 , University of Western Ontario.


• Math 0110A Introductory Calculus, fall 2011, University of Western Ontario.


• Math 2108 Calculus II, single variable, summer 2008, University of New Orleans.


• Math 2108 Calculus II, single variable, spring 2008, University of New Orleans.


• Math 2221 Elementary Differential Equations, spring 2008, University of New Orleans.


• Math 6490 Topics in Analysis-Introduction to Hilbert Spaces, fall 2007, University of New Orleans.


• Math 2107 Calculus I, single variable, fall 2007, University of New Orleans.


• Math 2221 Elementary Differential Equations, summer 2007, University of New Orleans.


• Math 3221 Methods in Differential Equations, spring 2007, University of New Orleans.


• Math 2108 Calculus II, single variable, spring 2007, University of New Orleans.


• Math 4221 Intermediate Ordinary Differential Equations, fall 2006, University of New Orleans.


• Math 2108 Calculus II, single variable, fall 2006, University of New Orleans.


• Math 2107 Calculus I, single variable, summer 2006, University of New Orleans.


• Math 2108 Calculus II, single variable, summer 2006, University of New Orleans.


• Math 2107 Calculus I, single variable, spring 2006, University of New Orleans.


• Math 1125 Precalculus Algebra, spring 2006, University of New Orleans.


• Math 2107 Calculus I, single variable, fall 2005, University of New Orleans.


• Math 2221 Elementary Differential Equations, fall 2005, University of New Orleans.


• Math 20550 Calculus III for Science and Engineering, Spring 2003, University of Notre Dame.


• Math 10560 Calculus II for Science and Engineering, Fall 2003, University of Notre Dame.


• Math 10250 Elementary Calculus in Basic Science, Spring 2002, University of Notre Dame.

Conference and Seminar Talks

• Centre de Recherches Mathematiques, Workshop on Geometry and Dynamics of Fluid, May 2012, Montreal, Canada. Talk title: TBA.
• University of Toronto, Working Group in Hamiltonian Systems, November 2011, Toronto, Canada. Talk title: Periodic trajectories in pseudo-Euclidean billiards.
• University of Notre Dame, 68th Midwest PDE Seminar, November 2011, Notre Dame, Indiana. Talk title: Integrable evolution equations on spaces of tensor densities: Hamiltonian and Lagrangian approaches.
• University of Toronto, Working Group in Hamiltonian Systems, October 2011, Toronto, Canada. Talk title: Relativistic quadrics: How to construct Jacobi coordinates in pseudo-Euclidean spaces.
• Fields Institute, Workshop on Wave Breaking and Global Solutions in the Short-Pulse Dispersive Equations, May 2011, Toronto, Canada. Talk title: Integrable evolution equations on spaces of tensor densities: Hamiltonian and Lagrangian approaches.
• Yeditepe University, Department of Mathematics, Tuesday Seminars, May 2011, Istanbul, Turkey. Talk title: Integrable evolution equations on spaces of tensor densities: Hamiltonian and Lagrangian approaches.
• University of Georgia, Session on Analysis and Geometry of Nonlinear Evolution Equations, the Seventh IMACS International Conference on Nonlinear Evolution Equations and Wave phenomena: Computation and Theory, April 2011, Athens, Georgia. Talk title: Integrable evolution equations on spaces of tensor densities and their peakon solutions.
• University of Saskatchewan, Department of Mathematics & Statistics, Seminar, March 2011, Saskatoon, Saskatchewan, Canada. Talk title: The periodic Cauchy problem for Novikov's equation.
• University of Saskatchewan, Department of Mathematics & Statistics, Seminar, March 2011, Saskatoon, Saskatchewan, Canada. Talk title: Integrable evolution equations on spaces of tensor densities and their peakon solutions.
• Northeastern University, Analysis-Geometry Seminar, February 2011, Boston, Massachusetts. Talk title: Integrable evolution equations on spaces of tensor densities.
• University of Notre Dame, Special Session on Nonlinear Evolution Equations, AMS Central Section Meeting, November 2010, Notre Dame, Indiana. Talk title: The periodic Cauchy problem for Novikov's equation.
• University of Toronto, Symplectic Seminar, October 2010, Toronto, Canada. Talk title: Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications.
• Brock University, Colloquium, April 2010, St. Catharines, Ontario, Canada. Talk title: Integrable evolution equations on spaces of tensor densities and their peakon solutions.
• Macalester College, Special Session on Geometric Flows, Moving Frames and Integrable Systems, AMS Central Section Meeting, April 2010, St. Paul, Minnesota. Talk title: Integrable evolution equations on spaces of tensor densities.
• EPFL (Ecole Polytéchnique Fédérale de Lausanne), Seminar of Geometric Analysis, March 2010, Lausanne, Switzerland. Talk title: Integrable evolution equations on spaces of tensor densities and their peakon solutions.
• Technische Universität Darmstadt, Mini-Workshop on Geometric Fluid Mechanics, International Research Training Group 1529, January 2010, Darmstadt, Germany. Talk title: Euler-Poincaré equations on Lie groups and homogeneous spaces, their orbit invariants and applications to PDE.
• University of Notre Dame, Colloquium, January 2010, Notre Dame, Indiana. Talk title: Euler-Poincaré equations on Lie groups and homogeneous spaces, their orbit invariants and applications to PDE.
• EPFL (Ecole Polytéchnique Fédérale de Lausanne), Seminar of Geometric Analysis, November 2008, Lausanne, Switzerland. Talk title: Infinite energy solutions for a perfect incompressible fluid.
• Stefan Banach International Mathematical Center, Formal and Analytic Solutions of Differential Equations, August 2008, Bedlewo, Poland. Talk title: Infinite energy solutions for a perfect incompressible fluid.
• Stefan Banach International Mathematical Center, 30 Years of Bi-Hamiltonian Systems, August 2008, Bedlewo, Poland. Talk title: Degasperis-Procesi family and density modules.
• University of Georgia, Session on Well-posedness for Nonlinear Dispersive Wave Equations, the Fifth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena, April 2007, Georgia. Talk title: Infinite energy solutions for a perfect incompressible fluid.
• University of Notre Dame, International Conference in PDE, Complex Analysis, and Differential Geometry (In Honor of M. Salah Baouendi), June 2006, Notre Dame, Indiana. Talk title: The Cauchy problem for the Proudman-Johnson equation.
• University of Notre Dame, Conference on Partial Differential Equations and Applications, August 2003, Notre Dame, Indiana. Talk title: Analytic regularity of a family of evolution equations.
• Stefan Banach International Mathematical Center, Nonlinear Partial Differential Equations Workshop, August 2001, Bedlewo, Poland. Talk title: On the Cauchy problem of Hunter-Saxton equation.

Conferences, Workshops, Programs Attended

• Geometric and Algebraic Structures in Mathematics: a conference to celebrate Dennis Sullivan's 70th birthday, May--June 2011, organized by A. Bonifant, J. Bowman, M. Lyubich, and S. Sutherland, Stony Brook University, New York.
• Workshop on Wave Breaking and Global Solutions in the Short-Pulse Dispersive Equations, May 2011, organized by D. Pelinovsky, Fields Institute, Toronto, Canada.
• The Seventh IMACS International Conference on Nonlinear Evolution Equations and Wave phenomena: Computation and Theory, April 2011, Session on Analysis and Geometry of Nonlinear Evolution Equations organized by D. Geba and A. Himonas, University of Georgia, Athens, Georgia.
• Thematic Program on Dynamics and Transport in Disordered Systems, January--June 2011, organized by D. Dolgopyat, K. Khanin, R. de la Llave, A. Neishtadt, J. Quastel and B. Toth, Fields Institute, Toronto, Canada.
• AMS Central Section Meeting, Special Session on Nonlinear Evolution Equations, November 2010, organized by A. Himonas and G. Misiolek, University of Notre Dame, Notre Dame, Indiana.
• Thematic Program on Asymptotic Geometric Analysis, July--December 2010, organized by V. Milman, V. Pestov, N.Tomczak-Jaegermann, Fields Institute, Toronto, Canada.
• AMS Central Section Meeting, Special Session on Geometric Flows, Moving Frames and Integrable Systems, April 2010, organized by G. Mari-Beffa and P. Olver, St. Paul, Minnesota.
• Mini-Workshop on Geometric Fluid Mechanics, January 2010, International Research Training Group 1529, Technische Universität Darmstadt, Darmstadt, Germany.
• Formal and Analytic Solutions of Differential Equations, August 2008, organized by W. Balser, G. Lysik and S. Michalik, Stefan Banach International Mathematical Center, Bedlewo, Poland.
• 30 Years of Bi-Hamiltonian Systems, August 2008, organized by M. Blaszak and A. Panasyuk, Stefan Banach International Mathematical Center, Bedlewo, Poland.
• The 3rd Symposium on Analysis & PDEs, May 2007, organized by D. Danielli, N. Garofalo, A. Petrosyan and A. Yip, Purdue University, West Lafayette, Indiana.
• The Fifth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena, Well-Posedness for Nonlinear Dispersive Wave Equations, April 2007, session chaired by J. Gorsky and D. P. Nicholls, University of Georgia, Georgia.
• International Conference in PDE, Complex Analysis, and Differential Geometry (In Honor of M. Salah Baouendi), June 2006, organized by J. Cao, A. Himonas, G. Misiolek, M. Shaw and N. Stanton, University of Notre Dame, Notre Dame, Indiana.
• Thematic Program on Partial Differential Equations, May 2004, organized by W. Craig, N. Ercolani and C. Sulem at Fields Institute, Toronto.
• 53rd Midwest PDE Seminar, March 2004, organized by S. Hofmann, D. Mitrea and M. Mitrea at University of Missouri-Columbia, Columbia, Missouri.
• Conference on Partial Differential Equations and Applications, August 2003, organized by M. Alber, Q. Han, A. Himonas, B. Hu, G. Misiolek and D. Nicholls, University of Notre Dame, Notre Dame, Indiana.
• Thematic Program on Groups and Geometry, Mini Program on Infinite Dimensional Lie Groups, Oct.-Nov. 2001, organized by N. Kamran and B. Khesin at CRM, Université de Montréal, Montréal.

Editorial and refereeing activities

• Referee for
  Journal of Mathematical Physics,
  Symmetry, Integrability and Geometry: Methods and Applications (SIGMA),
  International Mathematics Research Notices,
  Electronic Journal of Differential Equations,
  Zeitschrift fuer Angewandte Mathematik und Physik (ZAMP),
Journal of Physics A: Mathematical and Theoretical.

Other professional activities

• Co-organizer of AMS Special Session on Nonlinear Evolution Equations, Analysis, and Geometry, Joint Mathematics Meetings, New Orleans, 2011.
  
• Reviewer for the Mathematical Reviews (MR) Database.
  
• Co-organizer of the Mathematics Department Colloquium, University of New Orleans, 2007-2008.
  

Useful links

• Fields Institute
• UWO mathematics department
• University of Toronto - Mathematics
• Google
• AMS
• EMS
• Mathscinet
• arXiv
• NUMDAM archives
• SIAM
• The Chronicle of Higher Education
• Mathematics departments
• Mathematics genealogy project