#### RG Analysis and Partial Differential Equations

# Graduate seminar on Advanced topics in PDE

## Organizers

## Schedule

##### Oct 13 - Organizational meeting

##### Oct 20 - Jianghao Zhang (Bonn)

##### Title: A Survey on Superorthogonality

##### Abstract:

This talk is about the superorthogonality phenomenon in harmonic analysis. We will mainly focus on the newest type of superorthogonality introduced by Gressman-Pierce-Roos-Yung. Some applications will also be briefly discussed.

##### Oct 27 - Kornélia Héra (Bonn)

##### Title: Hausdorff dimension of Besicovitch sets of Cantor graphs

##### Abstract:

It is well known that planar Besicovitch sets- sets containing a unit line segment in every direction- have Hausdorff dimension 2. In a joint work with Iqra Altaf and Marianna Csörnyei we consider Besicovitch sets of Cantor graphs in the plane- sets containing a rotated (and translated) copy of a fixed Cantor graph in every direction, and prove lower bounds for their Hausdorff dimension.

##### Nov 3 - Michel Alexis (Bonn)

##### Title: Some counterexamples in two-weight norm inequalities for Calderón-Zygmund operators

##### Abstract:

I will discuss some bad behavior that two-weight norm inequalities for Calderón-Zygmund operators on $L^p$ exhibit that are not present in the well-known (Muckenhoupt) one-weight theory for Singular Integrals. Namely, we will see via some counterexamples that two-weight norms lack a characterization in terms of the weights alone, are unstable under biLipschitz change of variables, and when p not 2, are only known to be characterized by some technical vector-valued conditions.

##### Nov 10 - Alexandros Eskenazis (IMJ-PRG, Sorbonne University)

##### Title: Some recent advances in discrete harmonic analysis

##### Abstract:

Boolean analysis has evolved into a multifaceted field of mathematics, blending techniques and intuition from analysis, probability and combinatorics. In this talk, we shall survey a line of recent developments in the field that has been motivated by problems in functional analysis and discrete geometry. Time permitting, selected applications in theoretical computer science will also be discussed.

##### Nov 17 - Yung-Chang (Martin) Hsu (Purdue)

##### Title: Triangular Hilbert Transform along Parabola: An alternate proof with basic Van der Corput lemma

##### Abstract:

In this talk, we go over a few recent developments on multilinear singular integrals to motivate the study of Triangular Hilbert Transform along parabola. We then browse through the original proof and comment on a few technicalities. Lastly, we will see a sketch of an alternate proof that circumvents those technicalities with a few remarks on the comparison and applications.

##### Nov 24 - Ruoyuan Liu (Edinburgh)

###### *Location: Lipschitz Saal (joint session with RG Functional Analysis, IAM)*

##### Title: Local well-posedness of a quadratic nonlinear Schroedinger equation on the two-dimensional torus

##### Abstract:

In this talk, I will present results on local well-posedness of the nonlinear Schroedinger equation (NLS) with the quadratic nonlinearity |u|^2, posed on the two-dimensional torus, from both deterministic and probabilistic points of view. For the deterministic well-posedness, Bourgain (1993) proved local well-posedness of the quadratic NLS in H^s for any s > 0. In this talk, I will go over local well-posedness in L^2, thus resolving an open problem of 30 years since Bourgain (1993). In terms of ill-posedness in negative Sobolev spaces, this result is sharp. As a corollary, a multilinear version of the conjectural L^3 -Strichartz estimate on the two-dimensional torus is obtained. For the probabilistic well-posedness, I will talk about almost sure local well-posedness of the quadratic NLS with random initial data distributed according to a fractional derivative of the Gaussian free field. I will also mention a probabilistic ill-posedness result when the random initial data becomes very rough. The first part of the talk is based on a joint work with Tadahiro Oh (The University of Edinburgh).

##### Dec 1 - Leonard Busch (University of Amsterdam)

##### Title: An Inverse Problem with Partial Neumann Data and $L^{n/2}$ Potentials

##### Abstract:

Motivated by electrical impedance tomography we consider a partial data inverse problem with unbounded potentials. Rather than rely on functional analytic arguments, we construct an explicit Green's function with which we construct complex geometric optics (CGO) solutions and show unique determinability of potentials in $L^{n/2}$ for the Schroedinger equation with partial Neumann data.

##### Dec 8- Alex Rutar (University of St. Andrews)

##### Title: Assouad dimension and the local geometry of fractal sets

##### Abstract:

The Assouad dimension is a notion of dimension which captures the worst-case localized scaling properties of sets. The Assouad dimension has originally introduced to study the problem of bi-Lipschitz embeddability of metric spaces into Euclidean space. Moreover, the Assouad dimension has strong connections with Furstenberg's notion of a microset, i.e. any limit of rescaled copies of small pieces of the original set. I will give a general introduction to the Assouad dimension and the relationship with microsets, and present these ideas using some explicit examples such as Mandelbrot percolation and self-affine sets.

##### Dec 15 - Edward McDonald (Penn State)

##### Title: Lipschitz estimates for functions of operators

##### Abstract:

Given bounded self-adjoint operators $A$ and $B$ on a Hilbert space and a locally bounded Borel function $f$, the function $t\mapsto f(A+tB)$ defines an operator valued function. It is of interest to classify those functions $f$ such that $f(A+tB)$ is continuous, Lipschitz continuous, differentiable, etc. I will give an overview of the history of this theme and some recent results.

##### Dec 22 - *No Seminar*

##### Jan 12- Thomas Alazard (CNRS and ENS, Paris-Saclay) **13.00 at Lipshitzsaal**

##### Title: Paralinearization of free boundary problems in fluid dynamics

##### Abstract:

##### Jan 19 - Enno Lenzmann (University of Basel)

##### Title: The energy-critical Half-Wave Maps equations with rational data

##### Abstract:

The energy-critical Half-Wave Maps equation (HWM) is the following quasi-linear hyperbolic system $\partial_t S = S \times |D_x| S$, where $x$ belongs to the real line and $S=S(t,x)$ takes values in the two dimensional unit-sphere. This equation is known to be locally wellposed for sufficiently smooth initial data and to admit a Lax pair. However, no conservation laws controlling the higher Sobolev regularity $H^s$ for $s > 1/2$ are known. If the initial datum is a rational of $x$, the smooth solution is known to remain a rational function as long as it exists. We prove that such rational solutions are always global in time. Furthermore, under a spectral assumption on the Lax operator, we prove scattering and obtain a-priori bounds on Sobolev norms $H^s$ for any $s > 1/2$. The main ingredients are an explicit formula for the HWM-flow, involving Toeplitz operators on the Hardy space, which is inherited from a matrix-valued generalisation of the Benjamin-Ono equation. This is joint work with Patrick Gérard (Paris-Saclay) and Gaspard Ohlmann (Basel).

##### Jan 26, 14.15 - Mikel Florez Amatriain (BCAM)

##### Title: Pointwise localization and sharp weighted bounds for Rubio de Francia square functions

##### Abstract:

The Rubio de Francia square function is the square function formed by frequency projections over a collection of disjoint intervals of the real line. J. L. Rubio de Francia proved in 1985 that this operator is bounded in L^p for p\ge 2 and in L^p(w) for p > 2 with weights w in the Muckenhoupt class A_{p/2}. What happens in the endpoint L^2(w) for w\in A_1 was left open, and Rubio de Francia conjectured the validity of the boundedness. In this talk we will show a new pointwise sparse bound for the Rubio de Francia square function. This sparse bound leads to quantified weighted norm inequalities. We will also show that the weighted L^2-conjecture holds for radially decreasing even weights and in full generality for the Walsh group analogue of the Rubio de Francia square function; in general the weighted L^2 inequality is at this point still an open problem. In the first part of the talk, we will give the background of the problem while in the second part we will explain the new results mentioned above. This talk is based on a joint work with F. Di Plinio, I. Parissis and L. Roncal.

##### Jan 26, 15.15 - Jaume de Dios Pont (ETH Zürich)

##### Title: Query lower bounds for log-concave sampling

##### Abstract:

Given a density $\rho(x)$, how does one effectively generate samples from a random variable with this prescribed density? Variations of this question arise in most computational fiends, from Statistics to Computer Science to computational Physics. Significant effort has been devoted to designing more and more efficient algorithms, ranging from relatively simple algorithms, such as rejection sampling, to increasingly sophisticated such as langevin-based or diffusion based models. In this talk we will focus on the converse question: Finding universal complexity bounds that no algorithm can beat. We will do so in the case when the log-density is a strictly concave smooth function. In this case we will be able to construct tight bounds in low dimension using a modification of Perron's sprouting construction for Kakeya sets. Based on joint work with Sinho Chewi (Yale), Jerry Li (Microsoft Research), Chen Lu (MIT) and Shyam Narayanan (MIT).

##### Feb 2, 14.15 - Alan Chang (Washington University in St. Louis)

##### Title: Nikodym-type spherical maximal functions

##### Abstract:

We study $L^p$ bounds on Nikodym maximal functions associated to spheres. In contrast to the spherical maximal functions studied by Stein and Bourgain, our maximal functions are uncentered: for each point in $R^n$, we take the supremum over a family of spheres containing that point. This is joint work with Georgios Dosidis and Jongchon Kim.

##### Feb 2, 15.15- Junfeng Li (Dalian University of Technology)

##### Title: The Space Times estimate for the Schrödinger equation

##### Abstract:

In this talk, I will present our recent work on the space-time estimate for the Schrödinger equation. By using wave packet decompositions, polynomial partitioning method and a refined Strichartz estimate, we obtain a high frequency input maximal estimate for the 2D Shrouding group. By this estimate we confirmed a conjecture formulated by Planchon. For high dimensional case, we set up a high frequency input bilinear estimate, and improved the known result in high dimensional case. We also studied the 2D space-time local smoothing estimate for the SchrÃ¶dinger equation. This talk is based on the joint work with Changxing Miao and Ankang Yu.## News

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