Abstract: Consider a smooth, compact
Riemannian manifold with no boundary, endowed with a smooth
metric. A famous theorem of Courant states that the k-th
eigenfunction for the Laplace-Beltrami operator can have at
most k nodal domains. Nodal domains are the open and
connected sets where the eigenfunction does not vanish. H.
Donnelly and Fefferman obtained some 30 years ago a local
version of this theorem. Improvements were made by
Chanillo-Muckenhoupt and others. In this talk we obtain the
optimal local version of the local Courant theorem. We also
relate this result to conjectures of S.-T. Yau on nodal
sets, that is the zero set of eigenfunctions. The results of
our talk have been obtained jointly with A. Logunov, E.
Mallinikova and D. Mangoubi.
Tuesday 12th October 2021, 10:00–11:00 JST (UTC+9), online
on Zoom
Title:Finite point configurations and
the Vapnik-Chervonenkis dimension Abstract:
The Vapnik-Chervonenkis (VC) dimension was
invented in 1970 to study learning models. This notion has
since become one of the cornerstones of modern data science.
This beautiful idea has also found applications in other
areas of mathematics. In this talk we are going to describe
how the study of the VC-dimension in the context of families
of indicator functions of spheres centered at points in sets
of a given Hausdorff dimension (or in sets of a given size
inside vector spaces over finite fields) gives rise to
interesting, and in some sense extremal, point
configurations.
Tuesday 19th October 2021,
16:00–17:00 JST (UTC+9), online on Zoom
Abstract: We study John—Nirenberg-type
spaces where oscillations of functions are controlled via
covering lemmas. Our methods give new surprising results and
clarify classical inequalities. Joint work with Mario Milman
(Florida and Buenos Aires).
Tuesday 2nd November 2021, 16:00–17:00 JST (UTC+9), online
on Zoom
Abstract: Compensated Integrability is a recent tool of Functional
Analysis, which extends both the Gagliardo Inequality and
the Isoperimetric Inequality. It concerns the determinant of
positive symmetric tensors whose row-wise Divergence is
controlled in the space of bounded measures. It is somehow
dual to Brenier's Theorem of Optimal Transport. Its
applications cover several domains in Mathematical Physics
and in Differential Geometry.
Tuesday 16th November 2021, 10:00–11:00 JST (UTC+9), online
on Zoom
Abstract: I will
introduce fractional integral operator and its related
maximal operator. After developing some of the relevant
background, we will discuss its boundedness on Lebesgue
spaces and various related inequalities of Hedberg and
Welland. We will also cover endpoint bounds and applications
to Sobolev-Poincare inequalities.
Tuesday 30th November 2021, 10:00–11:00 JST (UTC+9), online
on Zoom
Abstract:
In this talk, we will
describe some new ways of characterising Sobolev norms,
using sizes of superlevel sets of suitable difference
quotients. They provide remedy in certain cases where some
critical Gagliardo-Nirenberg interpolation inequalities
fail, and lead us to investigate real interpolations of
certain fractional Besov spaces. Some connections will be
drawn to earlier work by Bourgain, Brezis and Mironescu.
Joint work with Haim Brezis, Jean Van Schaftingen, Qingsong
Gu, Andreas Seeger and Brian Street.
★SPECIAL LECTURE Part 2/3
Wednesday
1st December 2021, 10:00–11:00 JST (UTC+9), online on Zoom
Abstract: In
this talk we will cover the one weight inequalities for the
fractional integral operator and related fractional maximal
operator. We will discuss the background of A_p weights and
A_{p,q} weights and go over the dyadic decomposition of the
fractional integral operator. We will also cover auxiliary
results like sharp constants.
★DISTINGUISHED LECTURE
Tuesday 7th
December 2021, 16:00–17:00 JST (UTC+9), online on Zoom
Abstract:
In this talk
we will cover the two weight inequalities for the fractional
integral operator and related fractional maximal operator.
We will discuss the background of two-weight inequalities
and Sawyer’s testing conditions and two weight
characterization. We will also discuss bump conditions and
some open questions.
★SPECIAL LECTURE Part 1/2
Monday 13th December 2021, 15:00–17:30 JST
(UTC+9), online on Zoom
