# Spring 2024 Analysis Seminar

Organizer: Shaoming Guo

Email: shaomingguo (at) math (dot) wisc (dot) edu

Time: Wed 3:30--4:30

Room: B223

We can use B223 from 4:30 to 5:00 for discussions after talks.

All talks will be in-person unless otherwise specified.

In some cases the seminar may be scheduled at different time to accommodate speakers.

If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu

Date | Speaker | Institution | Title | Host | ||

1 | We, Jan. 24, 2024 | |||||

2 | We, Jan. 31 | Sunggeum Hong | Chosun University | The Hörmander multiplier theorem for n-linear operators and its applications | Andreas | |

3 | We, Feb. 7 | Donald Stull | University of Chicago | Dimensions of pinned distance sets in the plane | Betsy, Shaoming, and Jake F. | |

4 | We, Feb. 14 | |||||

Fr, Feb. 16 | Jack Lutz | Iowa State University | Algorithmic Fractal Dimensions | Shaoming | department colloquium, 4-5pm | |

5 | We, Feb. | Andrei Martinez-Finkelshtein | Baylor | Zeros of polynomials and free probability | Sergey | |

6 | We, Feb. 28 | Alex Rutar | University of St. Andrews | Dynamical covering arguments via large deviations and non-convex optimization | Andreas | |

7 | We, Mar. 6 | Song-Ying Li | UC-Irvine | Sup-norm estimates for d-bar and Corona Problems | Xianghong | |

8 | We, Mar. 13 | |||||

9 | We, Mar. 20 | Xiaoqi Huang | LSU | Curvature and growth rates of log-quasimodes on compact manifolds | Shaoming | |

10 | We, Mar. 27 | Spring recess | spring recess | spring recess | ||

11 | We, Apr. 3 | Shengwen Gan | UW Madison | On the maximal Bochner--Riesz operator | analysis group | |

12 | We, Apr. 10 | Victor Bailey | University of Oklahoma | Frames via Unilateral Iterations of Bounded Operators | Betsy | |

13 | We, Apr. 17 | Jianhui (Franky) Li | Northwestern University | Weighted Fourier restriction estimate | Betsy | |

14 | We, Apr. 24 | Sergey Denisov | UW Madison | Fixed point theorems in Analysis: getting strong asymptotics of MOP | analysis group | |

15 | We, May 1 | Yunfeng Zhang | Peking University | Bounds of restriction of characters to submanifolds | Simon |

## Abstracts

### Donald Stull

Title: Dimensions of pinned distance sets in the plane

Abstract: In this talk, we discuss recent work on the Hausdorff and packing dimension of pinned distance sets in the plane. Given a point x in the plane , and a subset E , the pinned distance set of E with respect to x is the set of all distances between x and the points of E . An important open problem is understanding the Hausdorff, and packing, dimensions of pinned distance sets. We will discuss ongoing progress on this problem, and present improved lower bounds for both the Hausdorff and packing dimensions of pinned distance sets. We also discuss the computability-theoretic methods used to achieve these bounds.

### Jack Lutz

Title: Algorithmic Fractal Dimensions

Algorithmic fractal dimensions are computability theoretic versions of Hausdorff dimension and other fractal dimensions. This talk will introduce algorithmic fractal dimensions with particular focus on the Point-to-Set Principle. This principle has enabled several recent proofs of new theorems in geometric measure theory. These theorems, some solving long-standing open problems, are classical (meaning that their statements do not involve computability or logic), even though computability has played a central in their proofs.

### Andrei Martinez-Finkelshtein

Title: Zeros of polynomials and free probability

Abstract: I will discuss briefly the connections of some problems from the geometric theory of polynomials to notions from free probability, such as free convolution. More specifically, I will illustrate it with two examples: - real zeros of some hypergeometric polynomials, their monotonicity, interlacing, and asymptotics; - flow of zeros of polynomials under iterated differentiation.

### Alex Rutar

Title: Dynamical covering arguments via large deviations and non-convex optimization

Abstract: Most classical notions of fractal dimensions (such as the Hausdorff, box, and Assouad dimensions) are defined in terms of optimal covers, or families of balls minimizing some form of cost function of their radii. For general sets, the optimal covers can be forced to essentially have arbitrary complexity. But for sets satisfying some form of dynamical invariance (which is the case for the majority of well-studied ‘fractal’ sets), one hopes that the underlying dynamics can be used to inform the optimal choice of cover in a meaningful way. In this talk, I will present some techniques drawing on insights from large deviations theory and continuous optimization theory which have proven to be useful technical tools in dimension theory. To highlight these techniques, I will discuss a recent result (joint with Amlan Banaji, Jonathan Fraser, and István Kolossváry) on the dimension theory of sets invariant under certain families of affine transformations in the plane.

### Song-Ying Li

Title: Sup-norm estimates for d-bar and Corona problems

In this talk, we will present some development of Corona problem of several complex variables and discuss its relation to the solution of the sup-norm estimate for d-bar, the Berndtsson conjecture and its application to Corona problem. We will also discuss the application of Hormander weighted L^2 estimates for d-bar to Corona problem.

### Xiaoqi Huang

Title: Curvature and growth rates of log-quasimodes on compact manifolds

Abstract: We will discuss the relation between curvature and L^q norm estimates of spectral projection operators on compact manifolds. We will present a new way that one can hear the shape of compact space forms, if the shape refers to curvature, and the radios used are the L^q-norm of quasimodes. We will also discuss decay rates of L^2-norms over shrinking geodesic tubes for quasimodes under various curvature assumptions. This is based on joint work with Christopher Sogge.

### Jianhui (Franky) Li

Title: Weighted Fourier restriction estimate

Abstract: We established an $L^p$ weighted Fourier restriction estimate in higher dimensions. In this talk, I will share the intriguing secret behind the crazy exponents found in our induction formula. Surprisingly, these exponents can be computed directly using an intuitive but rough argument. This is a joint work with Xiumin Du, Hong Wang and Ruixiang Zhang.

### Victor Bailey

Title: Frames via Unilateral Iterations of Bounded Operators

Abstract: Dynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the action of a bounded linear operator. Recent works in this area consider questions such as when can a given frame for a separable Hilbert Space, $\{f_k\}_{k \in I} \subset H$, be represented by iterations of an operator on a single vector and what are necessary and sufficient conditions for a system, $\{T^n \varphi\}_{n=0}^{\infty} \subset H$, to be a frame? In this talk, we will discuss the connection between frames given by iterations of a bounded operator and the theory of model spaces in the Hardy-Hilbert Space as well as necessary and sufficient conditions for a system generated by the orbit of a pair of commuting bounded operators to be a frame. This is joint work with Carlos Cabrelli.

### Yunfeng Zhang

Title: Bounds of restriction of characters to submanifolds

Abstract: As the eigenvalue goes to infinity, how much can the eigenfunctions of the Laplace-Beltrami operator on a compact manifold concentrate? One way to explore this question is to understand the growth of Lp norms of restriction of the eigenfunctions to submanifolds. In this talk, the compact Riemannian manifold will be a compact Lie group which has a lot of conjugate points, the eigenfunctions will be matrix coefficients and in particular characters of irreducible representations of the group, and the submanifolds will be submanifolds of maximal tori as well as their orbits under the conjugation action of the group. I will demonstrate the sharp Lp restriction bounds in this setting, using the Weyl character formula and the rough idea that eigenfunctions concentrate on conjugate points.