# Graduate seminar on Advanced topics in PDE

### RG Analysis and Partial Differential Equations

## Organizers

## Schedule

#####
** 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).

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

#####
** 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.

#####
** 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.

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

#####
** 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 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.
## News

Ausschreibung: W2-Professur Reine Mathematik (Bewerbungsschluss: 31. Juli 2019)

Hausdorff-Kolloquium im SS 2019

Toeplitz Kolloquium zur "Didaktik und Geschichte der Mathematik" im SS 2019

Berufspraktisches Kolloquium im SS 2019

Rhodes-Stipendium für Bonner Mathematikstudenten Peter Holderrieth

16.-24. Mai 2019: Felix-Klein-Lectures

Bachelorpreis 2017/18 der BMG verliehen

Prof. Jan Schröer erhält Lehrpreis der Fakultät 2018; Sonderpreis für Dr. Antje Kiesel

Prof. Peter Scholze erhält Fields-Medaille 2018

Bonner Mathematik weiterhin exzellent

Prof. Stefan Schwede zum Fellow of the AMS gewählt

Bonner Mathematik im Shanghai-Ranking auf Platz 36 und bundesweit führend

Prof. Peter Scholze neuer Direktor am MPIM

Dr. Thoralf Räsch erhält Lehrpreis der Uni Bonn

Bonner Mathematik beim CHE-Ranking wieder in Spitzengruppe

Prof. Peter Scholze erhält den Gottfried Wilhelm Leibniz-Preis 2016