Page 62 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 62
INVITED SPEAKERS
istence theorems obtained in this way come at a high price: solutions are highly irregular, non-
differentiable, and very much non-unique as there is usually infinitely many of them. Therefore
this technique has often been thought of as a way to obtain mathematical counterexamples in
the spirit of Weierstrass’ non-differentiable function, rather than advancing physical theory;
"pathological", "wild", "paradoxical", "counterintuitive" are some of the adjectives usually as-
sociated with solutions obtained via convex integration. In this lecture I would like to draw on
some recent examples to show that there are many more sides to the story, and that, with proper
usage and interpretation, the convex integration toolbox can indeed provide useful insights for
problems in hydrodynamics.
HMS categorical symmetries and hypergeometric systems
Špela Špenko, spela.spenko@vub.be
Université Libré de Bruxelles, Belgium
Hilbert’s 21st problem asks about the existence of Fuchsian linear differential equations with a
prescribed "monodromy representation" of the fundamental group. The first (slightly erroneous)
solution was proposed by a Slovenian mathematician Plemelj. A suitably adapted version of
this problem was solved, depending on the context, by Deligne, Mebkhout, Kashiwara-Kawai,
Beilinson-Bernstein, ... The solution is now known as the Riemann-Hilbert correspondence.
Homological mirror symmetry predicts the existence of an action of the fundamental group
of the "stringy Kähler moduli space" (SKMS) on the derived category of an algebraic variety.
This prediction was established by Halpern-Leistner and Sam for certain toric varieties. The
decategorification of the action found by HLS yields a representation of the fundamental group
of the SKMS and in joint work with Michel Van den Bergh we show that it is given by the
monodromy of an explicit hypergeometric system of differential equations.
Quadrature error estimates for layer potentials evaluated near curved
surfaces in three dimensions
Anna Karin Tornberg, akto@kth.se
KTH Royal Institute of Technology, Sweden
Coauthors: Ludvig af Klinteberg, Chiara Sorgentone
When numerically solving PDEs reformulated as integral equations, so called layer potentials
must be evaluated. The quadrature error associated with a regular quadrature rule for evaluation
of a layer potential increases rapidly when the evaluation point approaches the surface and the
integral becomes nearly singular. Error estimates are needed to determine when the accuracy is
insufficient and a more costly special quadrature method should be utilized.
In this talk, we start by considering integrals over curves in the plane, using complex analy-
sis involving contour integrals, residue calculus and branch cuts, to derive such error estimates.
We first obtain error estimates for layer potentials in R2, for both complex and real formulations
of layer potentials, both for the Gauss-Legendre and the trapezoidal rule. By complexifying the
parameter plane, the theory can be used to derive estimates also for curves in in R3. These
results are then used in the derivation of the estimates for integrals over surfaces. The esti-
mates that we obtain have no unknown coefficients and can be efficiently evaluated given the
60
istence theorems obtained in this way come at a high price: solutions are highly irregular, non-
differentiable, and very much non-unique as there is usually infinitely many of them. Therefore
this technique has often been thought of as a way to obtain mathematical counterexamples in
the spirit of Weierstrass’ non-differentiable function, rather than advancing physical theory;
"pathological", "wild", "paradoxical", "counterintuitive" are some of the adjectives usually as-
sociated with solutions obtained via convex integration. In this lecture I would like to draw on
some recent examples to show that there are many more sides to the story, and that, with proper
usage and interpretation, the convex integration toolbox can indeed provide useful insights for
problems in hydrodynamics.
HMS categorical symmetries and hypergeometric systems
Špela Špenko, spela.spenko@vub.be
Université Libré de Bruxelles, Belgium
Hilbert’s 21st problem asks about the existence of Fuchsian linear differential equations with a
prescribed "monodromy representation" of the fundamental group. The first (slightly erroneous)
solution was proposed by a Slovenian mathematician Plemelj. A suitably adapted version of
this problem was solved, depending on the context, by Deligne, Mebkhout, Kashiwara-Kawai,
Beilinson-Bernstein, ... The solution is now known as the Riemann-Hilbert correspondence.
Homological mirror symmetry predicts the existence of an action of the fundamental group
of the "stringy Kähler moduli space" (SKMS) on the derived category of an algebraic variety.
This prediction was established by Halpern-Leistner and Sam for certain toric varieties. The
decategorification of the action found by HLS yields a representation of the fundamental group
of the SKMS and in joint work with Michel Van den Bergh we show that it is given by the
monodromy of an explicit hypergeometric system of differential equations.
Quadrature error estimates for layer potentials evaluated near curved
surfaces in three dimensions
Anna Karin Tornberg, akto@kth.se
KTH Royal Institute of Technology, Sweden
Coauthors: Ludvig af Klinteberg, Chiara Sorgentone
When numerically solving PDEs reformulated as integral equations, so called layer potentials
must be evaluated. The quadrature error associated with a regular quadrature rule for evaluation
of a layer potential increases rapidly when the evaluation point approaches the surface and the
integral becomes nearly singular. Error estimates are needed to determine when the accuracy is
insufficient and a more costly special quadrature method should be utilized.
In this talk, we start by considering integrals over curves in the plane, using complex analy-
sis involving contour integrals, residue calculus and branch cuts, to derive such error estimates.
We first obtain error estimates for layer potentials in R2, for both complex and real formulations
of layer potentials, both for the Gauss-Legendre and the trapezoidal rule. By complexifying the
parameter plane, the theory can be used to derive estimates also for curves in in R3. These
results are then used in the derivation of the estimates for integrals over surfaces. The esti-
mates that we obtain have no unknown coefficients and can be efficiently evaluated given the
60
