Page 75 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 75
EMS PRIZE WINNERS
Discrete monodromy groups and Hodge theory
Simion Filip, sfilip@math.uchicago.edu
University of Chicago, United States
The monodromy of differential equations has been studied for a long time, providing motiva-
tion for many of the concepts in complex analysis, algebraic geometry, and dynamical systems.
After providing the necessary background, I will present some applications of ideas from dy-
namics to the study of monodromy groups of families of algebraic varieties, and explain how
they relate to differential equations. I will also describe some connections between interesting
classes of discrete subgroups of Lie groups and Hodge theory.
Zero sets of Laplace eigenfunctions
Alexandr Logunov, alogunov@princeton.edu
Princeton University, United States
In the beginning of 19th century Napoleon set a prize for the best mathematical explanation
of Chladni’s resonance experiments. Nodal geometry studies the zeroes of solutions to elliptic
differential equations such as the visible curves that appear in Chladni’s nodal portraits. We will
discuss the geometrical and analytic properties of zero sets of harmonic functions and eigen-
functions of the Laplace operator. For harmonic functions on the plane there is an interesting
relation between local length of the zero set and the growth of harmonic functions. The larger
the zero set is, the faster the growth of harmonic function should be and vice versa. Zero sets of
Laplace eigenfunctions on surfaces are unions of smooth curves with equiangular intersections.
Topology of the zero set could be quite complicated, but Yau conjectured that the total length
of the zero set is comparable to the square root of the eigenvalue for all eigenfunctions. We will
start with open questions about spherical harmonics and explain some methods to study nodal
sets, which are zero sets of solutions of elliptic PDE.
On primes, almost primes and the Möbius function in short intervals
Kaisa Matomäki, ksmato@utu.fi
University of Turku, Finland
In this talk I will introduce some fundamental concepts in analytic number theory such as
primes, the Riemann zeta function and the Möbius function. I will discuss classical results
concerning them and connecting them to each other.
Then I will move on to discussing the distribution of primes, almost primes and the Möbius
function in short intervals. In particular I will describe some recent progress on the topic.
73
Discrete monodromy groups and Hodge theory
Simion Filip, sfilip@math.uchicago.edu
University of Chicago, United States
The monodromy of differential equations has been studied for a long time, providing motiva-
tion for many of the concepts in complex analysis, algebraic geometry, and dynamical systems.
After providing the necessary background, I will present some applications of ideas from dy-
namics to the study of monodromy groups of families of algebraic varieties, and explain how
they relate to differential equations. I will also describe some connections between interesting
classes of discrete subgroups of Lie groups and Hodge theory.
Zero sets of Laplace eigenfunctions
Alexandr Logunov, alogunov@princeton.edu
Princeton University, United States
In the beginning of 19th century Napoleon set a prize for the best mathematical explanation
of Chladni’s resonance experiments. Nodal geometry studies the zeroes of solutions to elliptic
differential equations such as the visible curves that appear in Chladni’s nodal portraits. We will
discuss the geometrical and analytic properties of zero sets of harmonic functions and eigen-
functions of the Laplace operator. For harmonic functions on the plane there is an interesting
relation between local length of the zero set and the growth of harmonic functions. The larger
the zero set is, the faster the growth of harmonic function should be and vice versa. Zero sets of
Laplace eigenfunctions on surfaces are unions of smooth curves with equiangular intersections.
Topology of the zero set could be quite complicated, but Yau conjectured that the total length
of the zero set is comparable to the square root of the eigenvalue for all eigenfunctions. We will
start with open questions about spherical harmonics and explain some methods to study nodal
sets, which are zero sets of solutions of elliptic PDE.
On primes, almost primes and the Möbius function in short intervals
Kaisa Matomäki, ksmato@utu.fi
University of Turku, Finland
In this talk I will introduce some fundamental concepts in analytic number theory such as
primes, the Riemann zeta function and the Möbius function. I will discuss classical results
concerning them and connecting them to each other.
Then I will move on to discussing the distribution of primes, almost primes and the Möbius
function in short intervals. In particular I will describe some recent progress on the topic.
73
