Abstract: In this talk we will provide a brief introduction
to the eigenfunctions of the Laplace-Beltrami operator. We will then
focus on the so-called symmetry conjecture and a natural variant
raised by Nadirashvili and Jakobson. In particular we will present
certain results (positive and negative) obtained in collaboration
with F. Torres de Lizaur, and extended by a group of students during
last summer at the Fields Undergraduate Summer Research Program
under our supervision. Time permitting we will comment on work in
progress or open questions.

Date:
Tuesday 15th March 2022, 15:00–16:00 pm CST (GMT+8), online
on Webex

Speaker: Giorgio
Stefani, Scuola
Internazionale di Studi Superiori Avanzati

Abstract: We revisit Yudovich's
well-posedness result for the 2-dimensional Euler equations
for an inviscid incompressible fluid. Existence of
global-in-time weak solutions holds in
suitable uniformly-localized versions of the Lebesgue space
and of the Yudovich space, respectively, for the vorticity,
with explicit modulus of continuity for the associated
velocity. Uniqueness of weak solutions, in contrast to
Yudovich's energy method, follows from a Lagrangian
comparison. Our entire argument relies on elementary
real-variable techniques, with no use of either Sobolev
spaces, Calderón-Zygmund theory or
Littlewood-Paley decomposition, and actually applies not
only to the Biot-Savart law, but also to more general
operators whose kernels obey some natural structural
assumptions. This is a joint work in collaboration with
Gianluca Crippa.

Date:
Tuesday 22nd March 2022, 15:00–16:00 pm CST (GMT+8), online
on Webex

Abstract: Differently from their integer versions, the
fractional Sobolev spaces $W^{\alpha,p}(\mathbb{R}^n)$ do
not seem to have a clear distributional nature. By
exploiting suitable notions of fractional gradient and
divergence already existing in the literature, we introduce
via a distributional approach the new spaces
$BV^{\alpha,p}(\mathbb{R}^n)$ of $L^p$ functions with
bounded $\alpha$-fractional variation in $\mathbb{R}^n$, for
$\alpha \in (0,1)$ and $p \in [1, \infty]$. In addition, we
define in a similar way the distributional fractional
Sobolev spaces $S^{\alpha,p}(\mathbb{R}^n)$, which extend
naturally $W^{\alpha,p}(\mathbb{R}^n)$. The properties of
these spaces have been analyzed in a series of papers in
collaboration with Giorgio Stefani, Daniel Spector, Elia
Bruè and Mattia Calzi. In this talk, we shall focus
ourselves on the absolute continuity property of the
fractional variation with respect to a suitable Hausdorff
measure and on the existence of precise representatives for
$S^{\alpha,p}$ and $BV^{\alpha,p}$ functions. Subsequently,
we shall derive fractional Leibniz rules and Gauss-Green
formulas involving $BV^{\alpha,p}$ and $S^{\alpha,p}$
functions. As an application of these results we will show
the well-posedness of the boundary-value problem for a
general $2\alpha$-order fractional elliptic operator in
divergence form.

Date:
Tuesday 29th March 2022, 15:00–16:00 pm CST (GMT+8), online
on Webex

Abstract: I In this seminar, I shall talk about
analogues of pseudo-differential operators
(pseudo-multipliers) associated with the joint functional
calculus of the Grushin operator. In particular, I shall
discuss some sufficient conditions on a symbol function,
those which imply $L^2$-boundedness of the associated
Grushin pseudo-multiplier. This talk is based on a joint
work (arXiv:2111.10098) with Sayan Bagchi.

Date: Tuesday
12th April 2022, 09:00–10:00 am CST (GMT+8), online on
Zoom

Speaker:
Andreas Seeger,
University of Wisconsin-Madison

Abstract: Consider families of spherical means where the
radii are restricted to a given subset of a compact interval. One is
interested in the Lp improving estimates for the associated maximal
operators and related objects. Results depend on several notions of
fractal dimension of the dilation set, or subsets of it. There are
some surprising statements on the shape of the possible type sets.
Joint work with J. Roos and with T. Anderson, K. Hughes and J. Roos.

Date: Tuesday 26th April 2022,
15:00–16:00 pm CST (GMT+8), online on Zoom

Abstract: We
consider integrals in the sense of Choquet with respect to the
Hausdorff content. In particular, we discuss the validity of
Poincare, Poincare-Sobolev, and Trudinger type inequalities for
functions defined on John domains in this context. This is joint
work with Petteri Harjulehto.

Date: Tuesday 3rd May 2022,
15:00–16:00 pm CST (GMT+8), online on Zoom

Title: Elimination of Cusps
in a Free Boundary Problem for Water Waves

Abstract: The study of a
2-dimensional wave profile of an inviscid, incompressible fluid
acted upon by gravity traveling at constant speed goes back to 1847
and the work of Stokes, who conjectured that there should exists a
one-parameter family of smooth wave profiles and an extreme wave of
maximal height which had a peak with a 120 degree angle. It turns
out that a wave profile in this setting is the free-boundary of a
solution to a related free-boundary problem. The first half of this
talk will serve as an incomplete and biased introduction to the
study of this free boundary problem. Particularly, it will focus
upon the ``density methods" developed by Varvaruca and Weiss. The
second half of this talk will focus upon the issue of cusps. The
aforementioned ``density methods" are unable to eliminate cusps, but
recent work (joint with Lisa Naples) has recently found a rather
simple approach to eliminating cusps.

Date: Tuesday 10rd May 2022,
15:00–16:00 pm CST (GMT+8), online on Zoom

Title: The Heat Equation With A Dynamic Hardy-Type Singular Potential

Abstract:Motivated
by the celebrated paper of Baras and Goldstein (1984), in this
talk, we study the heat equation with a dynamic Hardy- type
singular potential. In particular, we are interested in the
case where the singular point moves in time. Under appropriate
conditions on the potential and initial value, we show the
existence, non-existence and uniqueness of solutions, and
obtain a sharp lower and upper bound near the singular point.
This is a joint work with Profs. G. Hwang, J. Takahashi and E.
Yanagida.

Date: Tuesday 17rd May 2022,
15:00–16:00 pm CST (GMT+8), online on Zoom

Title: Local regularity
conditions on initial data for weak solutions to the
incompressible Navier-Stokes equations

Abstract:We
consider the regularity of weak solutions to the three
dimensional incompressible Navier-Stokes equations. We prove
that the suitable weak solution is locally regular if a local
scaled energy of the initial data is sufficiently small. As an
application, it is shown that if a global weighted L^2 norm of
the initial data is finite, then the weak solution is regular
in a region confined by space-time hypersurfaces determined by
the weight. Our result is also applied to studying energy
concentration near a possible blow-up time. This talk is based
on a joint work with
Kyungkeun Kang (Yonsei Univ.) and
Tai-Peng Tsai
(Univ. British Columbia).

Date: Tuesday 24th May 2022,
15:00–16:00 am CST (GMT+8), online on Zoom

Abstract: In
this talk I will discuss multilinear Kakeya estimates and the
nonlinear Brascamp-Lieb inequality. Both may be viewed as perturbed
versions of the classical version of the Brascamp-Lieb inequality
and emerged around 2005 in connection with the multilinear
restriction problem. Progress on these problems has relied heavily
on induction-on-scales and I will try to give an overview of the
main ideas.