Presentations
The following slides correspond to some talks about my work. They cover most of my main areas and results. References can be found at the end of this page.
MAGFweb_2022.12.01_IntIPSeminar.pdfThe Calderón problem for directionally antilocal operators
Introduction to the Calderón problem for nonlocal operators seeing cones and implications of directional antilocality [CGFR.22].
(December 2022)
MAGFweb_2022.01.20_Bienal_OPSF.pdfExceptional orthogonal polynomials and isosprectral transformations
Introduction to the double commutation method/confluent Darboux transformation and application to the construction of exceptional orthogonal (Legendre) polynomials [GFGUM.21].
(January 2022)
MAGFweb_2021UB.pdfUnique continuation and inverse problems
On applications of quantitative unique continuation and quantitative Runge approximation results to the study of the stability of inverse problems related with medical imaging. Namely, on the Calderón problem for the acoustic Helmholtz equation [GFRZ.21] and the inverse of the Hilbert transform [GFR.20].
(May 2021)
MAGFweb_2020UAMJLRdF.pdfUnique continuation, Runge approximation and applications
Overview and comparison of Runge approximation results and unique continuation properties results for local and non-local, together with some applications about prescribing qualitative properties of solutions to PDEs and the fractional Calderón problem. With references to [EGFPS.19], [GFR.20] and [GFR.19].
(October 2020)
MAGFweb_2020MinnesotaZoom.pdfUnique continuation for non-local operators
On [GFR.19] and [GFR.20]: Two new mechanisms for deducing logarithmic quantitative unique continuation estimates for certain classes of integral operators.
(July 2020)
MAGFweb_2020RSMES13.pdfUnique continuation properties for fractional Schrödinger equations
On [GFR.19]: Strong unique continuation property for solutions of higher order (variable coefficient) fractional Schrödinger operators.
(January 2020)
MAGFweb_BiotSavartBudapest.pdfThe Biot-Savart operator of a bounded domain
On [EGFPS.18b]: Integral formulation of the velocity field of an incompressible fluid with tangency boundary conditions on a bounded domain in terms of its vorticity.
(June 2019)
MAGFweb_Munich.pdfMinimal graphs with prescribed level sets
On [EGFPS.18a]: Existence of minimal graphs on bounded sets whose intersection with a hyperplane (or level set) has any prescribed geometry, up to a small deformation.
(August 2018)
MAGFweb_Defensa.pdfGlobal approximation theorems for PDEs and applications
Slides of my thesis defence with the main results contained in [EGFPS 18a], [EGFPS 18b] and especially [EGFPS.19]: Global Runge approximation theorems for general parabolic equations and existence of global solutions with a local hot spot moving along a prescribed curve for all time, up to a small error.
(July 2018)
MAGFweb_OPFSA14.pdfWronskians with classical polynomial entries
On [GFGU.15]: Formulas for the number of real zeros of the Wronskian of an arbitrary sequence of eigenfunctions of a Schrödinger’s equation and of classical orthogonal polynomials.
(July 2017)
MAGF_XOPs_seminario_junior.pdfExceptional orthogonal polynomials
Introduction to exceptional orthogonal polynomials: complete systems of orthogonal polynomials which satisfy a Sturm-Liouville problem but differ from the classical (Hermite, Laguerre and Jacobi) in that there is a finite number of degrees for which no polynomial eigenfunction exists.
(June 2015)
References
[CGFR.22] G.Covi, MAGF, A. Rüland, On the Calderón problem for nonlocal Schrödinger equations with homogeneous, directionally antilocal principal symbols, J. Differ. Equ 341 (2022) 79-149.
[EGFPS.19] A. Enciso, MAGF, D. Peralta-Salas, Approximation theorems for parabolic equations and movement of local hot spots, Duke Math. J. 168 (2019) 897-939.
[EGFPS.18a] A. Enciso, MAGF, D. Peralta-Salas, The Biot-Savart operator of a bounded domain, J. Math. Pures Appl., 119 (2018) 85-113.
[EGFPS.18b] A. Enciso, MAGF, D. Peralta-Salas, Minimal graphs with micro-oscillations, J. Differ. Equ. 265 (2018) 3339-3344.
[GFGU.15] MAGF, D. Gómez-Ullate, Oscillation theorems for the Wronskian of an arbitrary sequence of eigenfunctions of Schrödinger's equation, Lett. Math. Phys. 10 5 (2015), 551-573.
[GFGUM.21] MAGF, D. Gómez-Ullate, R. Milson, Exceptional Legendre polynomials and confluent Darboux transformations, Symmetry Integr. Geom. 17 (2021) 016.
[GFGUM.19] MAGF, D. Gómez-Ullate, R. Milson, A Bochner type characterization for exceptional orthogonal polynomials, J. Math. Anal. Appl. 472 (2019) 584-626.
[GFR.20] MAGF, A. Rüland, On Two Methods for Quantitative Unique Continuation Results for Some Nonlocal Operators, Commun. Part. Diff. Eq. 45:11 (2020) 1512-1560.
[GFR.19] MAGF, A. Rüland, Strong unique continuation for the higher order fractional Laplacian, Mathematics in Engineering 1 (2019) 715-774.
[GFRZ.21] MAGF, A. Rüland, W. Zatoń, Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation, Inverse Probl. Imaging (2021) 1-31.