Page 112 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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COMPLEX ANALYSIS AND GEOMETRY (MS-17)
Angular Derivatives and Boundary Values of H(b) Spaces of Unit Ball of
Cn
Sibel S¸ ahin, sibel.sahin@msgsu.edu.tr
Mimar Sinan Fine Arts University, Turkey
In this talk we will consider a special subclass of the Hardy-Hilbert space H2(Bn), namely
deBranges-Rovnyak spaces H(b), in the setting of the unit ball of Cn. One of the main problems
in the study of H(b) functions is their representation and in the first part of this talk we will see
how we can represent these classes through the Clark measure on Sn associated with b. In
the second part we will give a characterization of admissible boundary limits in relation with
finite angular derivatives and we will see the interplay between Clark measures and angular
derivatives showing that Clark measure associated with b has an atom at a boundary point if and
only if b has finite angular derivative at the same point. More detailed analysis of the concepts
mentioned in this talk can be found in the following study:
S¸ ahin, S. Angular Derivatives and Boundary Values of H(b) Spaces of Unit Ball of Cn, Complex
Varaibles and Elliptic Equations, doi:10.1080/17476933.2020.1715373, (2020).
Seiberg-Witten equations and pseudoholomorphic curves
Armen Sergeev, sergeev@mi-ras.ru
Steklov Mathematical Institute, Russian Federation
Seiberg–Witten equations (SW-equations for short) were proposed in order to produce a new
kind of invariant for smooth 4-dimensional manifolds. These equations, opposite to the confor-
mally invariant Yang–Mills equations, are not invariant under scale transformations. So to draw
a useful information from these equations one should plug the scale parameter λ into them and
take the limit λ → ∞.
If we consider such limit in the case of 4-dimensional symplectic manifolds solutions of
SW-equations will concentrate in a neighborhood of some pseudoholomorphic curve (more
precisely, pseudoholomorphic divisor) while SW-equations reduce to some vortex equations in
normal planes of the curve. The vortex equations are in fact static Ginzburg–Landau equations
known in the superconductivity theory. So solutions of the limiting adiabatic SW-equations are
given by families of vortices in the complex plane parameterized by the point z running along
the limiting pseudoholomorphic curve. This parameter plays the role of complex time while the
adiabatic SW-equations coincide with a nonlinear ∂¯-equation with respect to this parameter.
The CR Ahlfors derivative and a new invariant for spherically equivalent
CR maps
Duong Ngoc Son, son.duong@univie.ac.at
Universität Wien, Austria
Coauthor: Bernhard Lamel
In this talk, I will discuss a notion of the Ahlfors derivative for CR maps. This notion pos-
sesses several important properties similar to those of the conformal counterpart and provides a
new invariant for spherical equivalent CR maps from strictly pseudoconvex CR manifolds into
110
Angular Derivatives and Boundary Values of H(b) Spaces of Unit Ball of
Cn
Sibel S¸ ahin, sibel.sahin@msgsu.edu.tr
Mimar Sinan Fine Arts University, Turkey
In this talk we will consider a special subclass of the Hardy-Hilbert space H2(Bn), namely
deBranges-Rovnyak spaces H(b), in the setting of the unit ball of Cn. One of the main problems
in the study of H(b) functions is their representation and in the first part of this talk we will see
how we can represent these classes through the Clark measure on Sn associated with b. In
the second part we will give a characterization of admissible boundary limits in relation with
finite angular derivatives and we will see the interplay between Clark measures and angular
derivatives showing that Clark measure associated with b has an atom at a boundary point if and
only if b has finite angular derivative at the same point. More detailed analysis of the concepts
mentioned in this talk can be found in the following study:
S¸ ahin, S. Angular Derivatives and Boundary Values of H(b) Spaces of Unit Ball of Cn, Complex
Varaibles and Elliptic Equations, doi:10.1080/17476933.2020.1715373, (2020).
Seiberg-Witten equations and pseudoholomorphic curves
Armen Sergeev, sergeev@mi-ras.ru
Steklov Mathematical Institute, Russian Federation
Seiberg–Witten equations (SW-equations for short) were proposed in order to produce a new
kind of invariant for smooth 4-dimensional manifolds. These equations, opposite to the confor-
mally invariant Yang–Mills equations, are not invariant under scale transformations. So to draw
a useful information from these equations one should plug the scale parameter λ into them and
take the limit λ → ∞.
If we consider such limit in the case of 4-dimensional symplectic manifolds solutions of
SW-equations will concentrate in a neighborhood of some pseudoholomorphic curve (more
precisely, pseudoholomorphic divisor) while SW-equations reduce to some vortex equations in
normal planes of the curve. The vortex equations are in fact static Ginzburg–Landau equations
known in the superconductivity theory. So solutions of the limiting adiabatic SW-equations are
given by families of vortices in the complex plane parameterized by the point z running along
the limiting pseudoholomorphic curve. This parameter plays the role of complex time while the
adiabatic SW-equations coincide with a nonlinear ∂¯-equation with respect to this parameter.
The CR Ahlfors derivative and a new invariant for spherically equivalent
CR maps
Duong Ngoc Son, son.duong@univie.ac.at
Universität Wien, Austria
Coauthor: Bernhard Lamel
In this talk, I will discuss a notion of the Ahlfors derivative for CR maps. This notion pos-
sesses several important properties similar to those of the conformal counterpart and provides a
new invariant for spherical equivalent CR maps from strictly pseudoconvex CR manifolds into
110
