# Graduate seminar on Advanced topics in PDE

### RG Analysis and Partial Differential Equations

## Organizers

## Schedule

#####
** July 12th - Joris Roos **

** Title: **

Discrete analogues of maximally modulated singular integrals of Stein-Wainger type

** Abstract: **

Stein and Wainger introduced an interesting class of maximal oscillatory integral operators related to Carleson's theorem. This talk is about the L^2 theory for discrete analogues of some of these operators. This problem features a number of new and substantial difficulties arising from a curious fusion of number theory and analysis.
Our approach is building on work of Krause (2018) and Krause-Lacey (2015). A key ingredient is a recent variation-norm estimate (Guo-Roos-Yung 2017).

#####
** July 5th - Francesco di Plinio **

** Title: **

Directional Carleson sequences: weighted estimates and applications

** Abstract: **

I will present a notion of Carleson norm for sequences indexed by families of tubes pointing along a certain set of directions.
I will discuss weighted (and unweighted) inequalities for these Carleson sequences. Applications include: the sharp form of Meyer's lemma on directional square
functions; sharply quantified Rubio de Francia type estimates for Fourier restrictions to directional sets; quantified or sharply quantified weighted and unweighted
inequalities for the maximal directional function and the maximal directional Hilbert transform. Joint work with Natalia Accomazzo (partly) and Ioannis Parissis (in full) of University of Basque Country.

#####
** June 28th - Bas Nieraeth **

** Title: **

Weighted theory and extrapolation for multilinear operators

** Abstract: **

In one of its forms in the linear case, Rubio de Francia's extrapolation theorem states that if an operator T is bounded on on $L^q(w)$ for a single $1≤ q<\infty$ for all weights w in the Muckenhoupt class $A_q$, then T is in fact bounded on $L^p(w)$ for all $1 <% p < \infty$ for all w in the Muckenhoupt class $A_p$. In recent developments, motivated by operators such as the bilinear Hilbert transform, multilinear versions of this result have appeared. In this talk I will discuss the recent multilinear extrapolation result I obtained. The proof here differs from the proofs given in the works of Cruz-Uribe, Martell [2017], Li, Martell, Ombrosi [2018], and the recent work of Li, Martell, Martikainen, Ombrosi, Vuorinen [2019] in that it does not rely on any linear off-diagonal extrapolation techniques. Rather, a multilinear analogue of the Rubio de Francia algorithm was developed, leading to an extrapolation theorem that includes the endpoints as well as a sharp dependence result with respect to the involved multilinear Muckenhoupt constants.

#####
** June 21st - Denis Brazke**

** Title: **

BMO and Carleson measures on Riemannian manifolds

** Abstract: **

Let M be a Riemannian n-manifold with a metric such that the manifold is Ahlfors-regular. We also assume either non-negative Ricci curvature, or that the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. We characterize BMO-functions on M by a Carleson measure condition of their σ-harmonic extension. As an application we show that the famous theorem of Coifman--Lions--Meyer--Semmes holds in this class of manifolds, which now follows from the arguments for commutators recently proposed by Lenzmann and Schikorra using only harmonic extensions, integration by parts, and trace space characterizations.
This is joint work with Armin Schikorra and Yannick Sire.

#####
** June 14th - Matthew de Courcy-Ireland**

** Title: **

Energy, packing, and linear programming

** Abstract: **

The sphere packing problem is to arrange equal-sized spheres without overlap so as to occupy the greatest fraction of volume possible. Energy minimization is a closely related problem where overlap is allowed, at a price, and the goal is to minimize the total cost. Linear programming gives bounds on how well one can do at both problems. The linear program has exact solutions in Euclidean space of dimension 8 and 24, found by Cohn-Kumar-Miller-Radchenko-Viazovska. The goal of this talk is to introduce these problems and describe some ongoing work joint with Henry Cohn and Ganesh Ajjanagadde concerning the high-dimensional case.

#####
** June 7th - No talk (Pre-pentecost break)**

#####
** May 31st - Friedrich Littman**

** Title: **

Concentration inequalities for bandlimited functions

** Abstract: **

This talk considers the following problem: How much of an integral norm of a function with compactly supported Fourier transform can be concentrated on a sparse set?
The resulting inequalities have explicit bounds that depend on the size of the support of the transform and on a measure of sparsity of the set. I will describe some applications from analytic number theory, signal processing, and Lagrange interpolation, and will outline existing strategies (building on work of Selberg and of Donoho and Logan) to obtain concentration inequalities.

#####
** May 24th - Ziping Rao**

** Title: **

Blowup stability for wave equations with power nonlinearity

** Abstract: **

We introduce the method of similarity coordinates to
study the stability of ODE blowup solutions of wave equations with power
nonlinearity in the lightcone. We first recall stability results in
higher Sobolev spaces. In this case, using the Lumer--Philips theorem we
obtain a solution semigroup to the Cauchy problem. Then by the
Gearhart--Prüss theorem we obtain enough decay of the semigroup to
control the nonlinearity. Then we show stability of the ODE blowup for
the energy critical equation in energy space, by establishing Strichartz
estimates in similarity coordinates. In this case the Gearhart--Prüss
theorem does not give a useful bound. Hence we need to construct an
explicit expression of the semigroup, from which we are finally able to
prove Strichartz estimates and an improved energy estimate to control
the nonlinearity in the energy space. The result in the energy critical
case in $d=5$ is by Roland Donninger and myself (the pioneering work of
the $d=3$ case is by Roland Donninger).

#####
** May 17th - Talk Cancelled**

#####
** May 10th - Felipe Gonçalves**

** Title: **

Broken Symmetries of the Schrodinger Equation and Strichartz Estimates

** Abstract: **

This will be short talk where we report some of the partial results of an
ongoing work with Don Zagier. We study the Schrodinger equation from the
point of view of Hermite and Laguerre expansions and establish a
diagonalization result for initial data with prescribed parity in 3
dimensions that present exotic and unexpected associated eigenvalues. In
particular, we derive a sharpened inequality for the one dimensional
Strichartz inequality for even initial data. For odd initial data we prove
the extremizer is the derivative of a Gaussian. We remark this is still
unfinished work, and some questions are still left to be answered, so the
audience is more than welcome to ask all sorts of questions.

#####
** May 3rd - Alexander Volberg**

** Title: **

Bi-parameter Carleson embedding

** Abstract: **

Nicola Arcozzi, Pavel Mozolyako, Karl-Mikael Perfekt, Giulia
Sarfatti recently gave the proof of a bi-parameter Carleson embedding
theorem. Their proof uses heavily the notion of capacity on bi-tree. In
this note we give one more proof of a bi-parameter Carleson embedding
theorem that avoids the use of bi-tree capacity. Unlike the proof on a
simple tree that used the Bellman function technique, the proof here
is based on some rather subtle comparison of energies of measures on
bi-tree. The bi-tree Carleson embedding theorem turns out to be very
different from the usual one on a simple tree. In particular, various
types of Carleson conditions are not equivalent in general for
bi-parameter case.

#####
** April 26th - Wiktoria Zatoń**

** Title: **

On the well-posedness for higher order parabolic equations with rough coefficients

** Abstract: **

In the first part we study the existence and uniqueness of solutions to the higher order parabolic Cauchy problems on the upper half space, given by $\partial_t u = (-1)^{m+1} \mbox{div}_m A(t,x)\nabla^m u$ and $L^p$ initial data space. The (complex) coefficients are only assumed to be elliptic and bounded measurable. Our approach follows the recent developments in the field for the case $m=1$.
In the second part we consider the $BMO$ space of initial data. We will see that the Carleson measure condition
$$\sup_{x\in \mathbb{R}^n} \sup_{r>0} \frac{1}{|B(x,r)|}\int_{B(x,r)}\int_0^{r}|t^m\nabla^m u(t^{2m},x)|^2\frac{dxdt}{t}<\infty$$
provides, up to polynomials, a well-posedness class for $BMO$. In particular, since the operator $L$ is arbitrary, this also leads to a new, broad Carleson measure characterization of $BMO$ in terms of solutions to the parabolic system.

#####
** April 19th - No talk (Karfreitag)**

#####
** April 12th - Alex Amenta**

** Title: **

Banach-valued modulation-invariant Carleson embeddings and outer measure spaces: the Walsh case

** Abstract: **

Consider three Banach spaces $X_0, X_1, X_2$, linked with a bounded trilinear form $\Pi : X_0 \times X_1 \times X_2 \to \mathbb{C}$. Given this data one can define Banach-valued analogues of the
bilinear Hilbert transform and its associated trilinear form. Using the Do-Thiele theory of outer $L^p$-spaces, $L^p$-bounds for these objects can be reduced to modulation-invariant Carleson embeddings of $L^p(\mathbb{R};X_v)$
into appropriate outer $L^p$-spaces. We prove such embeddings in the Banach-valued setting for a discrete model of the real line, the 3-Walsh group. Joint work with Gennady Uraltsev (Cornell).
## News

Corona: Maßnahmen im Mathematik-Zentrum

Hausdorff-Gedächtnispreis 2019/2020 verliehen

Bachelorpreis 2019/20 der BMG verliehen

Dr. Michael Welter erhält den Lehrpreis der Math.-Nat. Fakultät im Jahr 2020

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Prof. Dr. Valentin Blomer wurde zum Mitglied der Academia Europaea gewählt

Prof. Peter Scholze erhält Fields-Medaille 2018