Abstract:
The talk presents constructions of blowup solutions to the
Keller-Segel system in \( \mathbb{R}^d \). d=2 (\(L^1\)-critical): There exist
finite time single blowup solutions that are of Type \(II\) with
finite mass. Blowup rates are quantized according to a
discrete spectrum of a linearized operator around the
rescaled stationary solution in the self-similar setting.
There is also \textit{multiple collapsing blowup solutions}
formed by a collision of multiple single solutions with
self-similarity that provides a brand new mechanism of
singularity formation.

\(d \geq 3\)(\(L^1\)-supercritical): For \(d \geq 3\),
there exist finite time blowup solutions having the form of
collapsing-ring which consists of an imploding, smoothed-out
shock wave moving towards the origin to form a Dirac mass at
the singularity. For \(d = 3,4\), we found blowup solutions with
infinite mass that are asymptotically self-similar with a
log correction to their profile.

The constructions rely on a
spectral approach for multiple-scale problems,
renormalization technique, and refined energy estimates. The
talk is based on a series of joint works with C. Collot
(Paris Cergy), T. Ghoul (NYU Abu Dhabi), N. Nouaili (Paris
Dauphine), N. Masmoudi (NYU) and H. Zaag (Paris Nord).

Date: Tuesday
19th September 2023, 15:00-16:00 TST (GMT+8)

Speaker:
Zdeněk Mihula, Czech Technical University in Prague

Abstract: In this talk,
we consider a (higher order) Sobolev inequality for the
Laplace--Beltrami operator in the ball model of the
hyperbolic space \(\mathbb{H}^n\), and we look for function spaces that
are in a sense optimal in the inequality. The inequality in
question is

\(\|u\|_{Y(\mathbb{H}^n)} \leq C \|\nabla_g^m u\|_{X(\mathbb{H}^n)} \quad \text{for every } u\in V_0^m X(\mathbb{H}^n)\);

here

\[\nabla_g^m =
\begin{cases}\Delta_g^{\frac{m}{2}} & \text{if $m$ is
even},\\\nabla_g\Delta_g^{\lfloor \frac{m}{2} \rfloor} &
\text{if $m$ is odd},\end{cases}\]

where Δg is the Laplace--Beltrami operator and ∇g is the
hyperbolic gradient; \(X(\mathbb{H}^n)\) and \(Y(\mathbb{H}^n)\) are
rearrangement-invariant spaces, and \(V_0^m X(\mathbb{H}^n)\) is a suitable
mth order Sobolev space. For a given rearrangement-invariant
space \(X(\mathbb{H}^n)\), we will describe the optimal (i.e., with the
strongest norm) rearrangement-invariant space \(Y(\mathbb{H}^n)\) on the
left-hand side.

We first discuss the general description(s)
of the optimal space. Then we turn our attention to some
concrete examples. Namely, when \(X\) is \(L^1\), \(L^\frac{n}{m}\) , or an
exponential Orlicz space ``near \(L^\infty\)''. Even in these simple
cases, the inequalities that we obtain seems to be missing
in the literature (especially, when \(m \geq 3\)).

Date: Tuesday
26th September 2023, 09:00-10:00 TST (GMT+8)

Abstract: The weak-type(1,1)
estimate for Calderón-Zygmund operators is fundamental in
harmonic analysis. We investigate weak-type inequalities for
Calderón-Zygmund singular integral operators using the
Calderón-Zygmund decomposition and ideas inspired by
Nazarov, Treil, and Volberg. We discuss applications of
these techniques in the Euclidean setting, in weighted
settings, for multilinear operators, for operators with
weakened smoothness assumptions, and in studying the
dimensional dependence of the Riesz transforms.

Date: Tuesday
3th October 2023, 15:00-16:00 TST (GMT+8) Speaker:
Andrea Cianchi, Università di Firenze

Abstract:
Local minimizers of integral functionals of the calculus of
variations are analyzed under growth conditions dictated by
different lower and upper bounds for the integrand. Growths
of non-necessarily power-type are allowed. The local
boundedness of the relevant minimizers is established under
a suitable balance between the lower and the upper bounds.
Classical minimizers, as well as quasi-minimizers are
included in our discussion. Functionals subject to so-called
p,q -growth conditions are embraced as special cases and the
corresponding sharp results available in the literature are
recovered.

Date: Tuesday
17th October 2023, 09:00-10:00 TST (GMT+8)

Abstract:
We will revisit the Theorem on Sums and use it to study
viscosity solutions of non-homogeneous equations involving
the infinite Laplacian in Euclidean Space, Riemannian
manifolds, and Carnot Groups.

Date: Tuesday
24th October 2023, 09:00-10:00 TST (GMT+8)

Abstract:
The Wigner function of a quantum state is a way of
describing the phase space distribution of a quantum
particle. The uncertainty principle from Fourier analysis
places some restriction on the allowable decay of a Wigner
function. In this talk I will give an introduction to the
Wigner function and show that rapidly decaying Wigner
functions must also be Schwartz, which can also be
interpreted as a type of uncertainty principle. This is
based on joint work with Jess Riedel.

Date: Tuesday 7th November 2023, 09:00-10:00 TST (GMT+8)

Abstract: We identify necessary and sufficient conditions on \(k\) th order linear differential operators \(\mathbb{A}\) in
terms of a fixed halfspace \(H \subset \mathbb{R}^n \) such that the Gagliardo--Nirenberg--Sobolev inequality

holds. This comes as a consequence of sharp trace theorems on \(\partial H\). Strong estimates on lower order derivatives are the best possible due to the failure of Calder\'on--Zygmund theory in \(L^1\).

Joint work with Franz Gmeineder and Jean Van Schaftingen.

Date: Tuesday
14th November 2023, 09:00-10:00 TST (GMT+8)

Abstract: We
consider maps between spheres S^n to S^\ell that minimize
the Sobolev-space energy W^{s,n/s} for some s \in (0,1) in
given homotopy class. The basic question is: in which
homotopy class does a minimizer exist? This is a nontrivial
question since the energy under consideration is conformally
invariant and bubbles can form. Sacks-Uhlenbeck theory tells
us that minimizers exist in a set of homotopy classes that
generates the whole homotopy group \pi_{n}(\S^\ell). In some
situations explicit examples are known if n/s = 2 or s=1.

In our talk we are interested in the stability of the above
question in dependence of s. We can show that as s varies
locally, the set of homotopy classes in which minimizers
exists can be chosen stable. We also discuss that the
minimum W^{s,n/s}-energy in homotopy classes is continuously
depending on s.

Joint work with K. Mazowiecka (U
Warsaw)

Date: Tuesday 21th November 2023, 15:00-16:00 TST (GMT+8)

Title:
Optimal embeddings for fractional Orlicz-Sobolev spaces

Abstract:
The optimal target space is exhibited for embeddings of
fractional-order Orlicz-Sobolev spaces. Both the subcritical
and the supercritical regimes are considered. In the former
case, the smallest possible Orlicz target space is detected.
In the latter, the relevant Orlicz-Sobolev spaces are shown
to be embedded into the space of bounded continuous
functions in \(\mathbb{R}^n\). Moreover, their optimal
modulus of continuity is exhibited. These results are the
subject of a series of joint papers with Andrea Cianchi,
Lubos Pick and Lenka Slavikova.

A.Alberico,
A.Cianchi, L.Pick and L.Slavikova, Fractional Orlicz-Sobolev
embeddings, J. de Mathematiqués Pures et Appliquées, 149
(2021).

A.Alberico, A.Cianchi, L.Pick and L.Slavikova,
Boundedness of functions in fractional Orlicz-Sobolev
spaces, Nonlinear Analysis, 230 (2023).

A.Alberico,
A.Cianchi, L.Pick and L.Slavikova, On the Modulus of
Continuity of fractional Orlicz-Sobolev functions, in
progress.

Date:Tuesday
28th November 2023, 15:00-16:00 TST (GMT+8)

Abstract:
During the talk I will discuss applications of elementary
additive combinatorics to dimensional estimates of PDE- and
Fourier-constrained measures. My main tool will be a simple
certainty principle of the following form: a set \(S \subset
\mathbb{R}^N\) contains a given finite linear pattern if
\(S\) is a spectrum of a sufficiently singular measure.

Date:Tuesday
5th December 2023, 09:00-10:00 TST(GMT+8)

Abstract:
My talk is based on two recent joint papers with Paweł Goldstein.

Jacek Gałęski in 2017, in the context of his research in
geometric measure theory, formulated the following conjecture:

Conjecture. Let \(1\leq m< n\) be integers and let
\(\Omega\subset\mathbb{R}^n\) be open. If \(f\in C^1(\Omega,\mathbb{R}^n)\) satisfies \(\operatorname{rank} Df\leq m\) everywhere in \(\Omega\), then \(f\) can be uniformly approximated by smooth mappings \(g\in C^\infty(\Omega,\mathbb{R}^n)\)
such that \(\operatorname{rank} Dg\leq m\) everywhere in \(\Omega\).

One can also modify the conjecture and ask about a local
approximation: smooth approximation in a neighborhood of any point.
These are very natural problems with possible applications to PDEs
and Calculus of Variations. However, the problems are difficult,
because standard approximation techniques like the one based on
convolution do not preserve the rank of the derivative. It is a
highly nonlinear constraint, difficult to deal with.

In 2018
Goldstein and Hajłasz obtained infinitely many counterexamples to
this conjecture. Here is one:

Example. There is \(f\in C^1(\mathbb{R}^5,\mathbb{R}^5)\)
with \(\operatorname{rank} Df\leq 3\) that cannot be locally and uniformly approximated by mappings \(g\in C^2(\mathbb{R}^5,\mathbb{R}^5)\) satisfying \(\operatorname{rank} Dg\leq 3\).

This example is a special case of a much more general result
and the construction heavily depends on algebraic topology including
the homotopy groups of spheres and the Freudenthal suspension
theorem.

More recently Goldstein and Hajłasz proved the
conjecture in the positive in the case when \(m=1\).
The proof is based this time on methods of analysis on metric spaces
and in particular on factorization of a mapping through metric
trees.

The method of factorization through metric trees used
in the proof of the conjecture when \(m=1\) is very different and completely unrelated to the methods of algebraic topology used in the construction of counterexamples. However, quite surprisingly, both techniques have originally been used by Wenger and Young as tools for study of Lipschitz homotopy groups of the Heisenberg group, a problem that seems completely unrelated to problems discussed in this talk.

Date:Tuesday
12th December 2023, 15:00-16:00 TST (GMT+8)

Abstract: In this talk we will consider a version of weak-type inequalities we refer to as {\em multiplier weak-type inequalities}. Given a weight \(w\) and \(1 \leq p < \infty\), the \((p,p)\) multiplier weak-type inequality for an operator \(T\) is of the form
\[ |\{ x\in {\mathbb{R}^n} : |w^{\frac{1}{p}}(x)T(w^{-\frac{1}{p}}f)(x)|> t\}| \leq \frac{C}{t^p} \int_{\mathbb{R}^n} |f(x)|^p\,dx. \]
These inequalities follow from the a strong \((p,p)\) inequality of the form
\[ \int_{\mathbb{R}^n} |Tf(x)|^pw(x)\,dx \leq C \int_{\mathbb{R}^n} |f(x)|^pw(x)\,dx \]
by mapping \(f\mapsto w^{-\frac{1}{p}}f\) and applying Chebyshev's inequality. These inequalities were first considered by Muckenhoupt and Wheeden (1977) for the maximal operator and the Hilbert transform on the real line. They showed that such inequalities hold if \(w\) is a Muckenhoupt \(A_p\) weight, but gave examples to show that the class of weights is strictly larger for these operators. Their \(A_p\) results were extended to all dimensions and all Calder\'on-Zygmund integral operators by myself, Martell, and Perez (2005). They have attracted renewed attention since they were shown to be the right way of generalizing weak-type inequalities to the setting of matrix weights (DCU, Isralowitz, Moen, Pott, Rivera-Rios, 2020).

In this talk, we will consider the problem of quantitative estimates, in terms of the \(A_p\) characteristic, for maximal operators and singular integrals. Our results extend those gotten in 2020 in the case \(p=1\) to all
\(1\leq p<\infty\) . We also show that our proofs can be adapted to prove quantitative
estimates for matrix weighted inequalities. Finally, we prove the
analogous results for the fractional integral/Riesz potential in
both the scalar and matrix weighted cases. These results are
completely new, as even qualitative results for fractional integrals
were not known.

\bigskip

This talk is joint work with Brandon Sweeting, the University of Alabama.