The regular time and place for the seminar is Tuesday at 3:30 – 4:30 pm in Math Tower 154. If you are interested, please contact Liding Yao. If you are interested to topics of mathematical analysis related to operator theory, take a look to the Analysis and Operator Theory Seminar.

date | speaker | institution | title |
---|---|---|---|

Sep 5 | Alex McDonald | OSU | The Newhouse gap lemma and patterns in products of Cantor sets (part I) |

Sep 12 | Alex McDonald | OSU | The Newhouse gap lemma and patterns in products of Cantor sets (part II) |

Sep 19 | Adam Christopherson | OSU | Weak-type regularity of the Bergman projection on generalized Hartogs triangles in C^3 |

Oct 20 | Ben Bruce (on Zoom) | UBC | Hausdorff dimension and patterns determined by curves |

Oct 31 | Gennady Uraltsev | University of Arkansas | Multilinear and uniform bounds in harmonic analysis |

Nov 7 | Terence Harris | UW-Madison | Horizontal Besicovitch sets of measure zero and some related problems |

Nov 14 | Eyvindur Ari Palsson | Virginia Tech | A restricted Falconer distance problem |

Nov 28 | Robert Fraser | Wichita State | Oscillatory integrals arising from algebraic number fields |

Apr 1 | Michael Roysdon | Case Western | Comparison Problems for Radon Transforms (Monday, at EA 160) |

Apr 15 | Yuan Zhang | Purdue Fort Wayne | Optimal $\bar\partial$ regularity on product domains and its application to the Hartogs triangle |

### Alex McDonald

Title: The Newhouse gap lemma and patterns in products of Cantor sets

Abstract: Many important problems throughout mathematics can be summarized as follows: how large must a set be in order to ensure that it exhibits some type of structure? A classic example of importance in geometric measure theory and harmonic analysis is the Falconer distance problem, which asks how large the Hausdorff dimension of a compact set must be to ensure it determines a positive measure worth of distances. More generally, one may ask for an abundance of more complicated point configurations. There has been considerable recent progress on variants of these problems where Hausdorff dimension is replaced by a quantity called thickness, which provides an alternative way to quantify the “size” of a compact set. In these talks, I will give an overview of thickness from the ground up, and discuss some applications to finding point configurations in Cantor sets.

### Adam Christopherson

Title: Weak-type regularity of the Bergman projection on generalized Hartogs triangles in C^3

Abstract: In this talk, we give a complete characterization of the weak-type regularity of the Bergman projection on rational power-generalized Hartogs triangles in C^3. In particular, we show that the Bergman projection satisfies a weak-type estimate only at the upper endpoint of L^p boundedness on both of these domains. A similar result has been observed by Huo-Wick and Koenig-Wang for the classical Hartogs triangle and punctured unit ball, respectively. This work is joint with K.D. Koenig.

### Ben Bruce

Note: Ben will give a Zoom talk on Zoom on Oct 20 11:30- 12:25, during the class Math 8210 at Scott Lab E103. Zoom link is here.

Title: Hausdorff dimension and patterns determined by curves

Abstract: In this talk, I will discuss joint work with Malabika Pramanik on the problem of locating patterns in sets of high Hausdorff dimension. More specifically, suppose Γ is a smooth curve in Euclidean space that passes through the origin. Is it true that every set with sufficiently high Hausdorff dimension must contain two distinct points x,y such that x-y \in Γ? We showed that if Γ is suitably curved then the answer is yes, while for certain flat curves the answer is no. This generalizes work of Kuca, Orponen, and Sahlsten, who answered this question affirmatively when Γ is the standard parabola in R^2.

### Gennady Uraltsev

Title: Multilinear and uniform bounds in harmonic analysis

Abstract: See the pdf here.

### Terence Harris

Title: Horizontal Besicovitch sets of measure zero and some related problems

Abstract: We show that horizontal Besicovitch sets of measure zero exist in R^3. The proof is constructive and uses point-line duality analogously to Kahane’s construction of measure zero Besicovitch sets in the plane. Some consequences and related examples are discussed for the SL_2 Kakeya maximal function.

### Eyvindur Ari Palsson

Note: Eyvi’s talk will be at Baker Systems Engineering, BE394. Also he will give a pre-talk on Nov 13 11:30-12:25 at E103, during the class Math 8210.

Title: A restricted Falconer distance problem

Abstract: The Falconer distance problem, a continuous analogue of the celebrated Erdos distance problem asks: How large does the Hausdorff dimension of a Borel set, in the plane or higher dimensions, need to be to ensure that the Lebesgue measure of its distance set is positive? This question has seen much progress in recent years, yet the conjectured threshold remains open. After a quick introduction of this problem I will give an overview of a number of variants leading to a formulation of a restricted Falconer distance problem with a particular example being a diagonally restricted distance set.

### Robert Fraser

Title: Oscillatory integrals arising from algebraic number fields

Abstract: We introduce a new class of real-variable oscillatory integrals arising as the trace of a polynomial in the algebra (R ⊗_Q K) where K is an algebraic number field. We prove bounds for such oscillatory integrals that generalize a classical result of Arhipov, Cubarikov, and Karacuba for real polynomials and a recent result of Wright for complex polynomials.

### Michael Roysdon

Note: Michael’s talk is on Monday, and the location is EA 160.

Title: Comparison Problems for Radon Transforms

Abstract: At the start of the 20th century J. Radon answered the following question: can one reconstruct a function based on its integral on lines? While this question is simple at its core, the methods involved have had a lasting effects in various scientific fields, and have even brought about two entire fields of mathematics: geometric tomography and analytic tomography. Each of these fields concern questions about determining information about of an object given lower-dimensional information; for example, information about the volume of the object knowing the volumes of sections and projections of that object onto planes.

Inspired by the famed Busemann-Petty problem from convex geometry and geometric tomography (~1954), we address more general questions of this nature in the realm of analytic tomography. We ask the very simple question: given a pair of even, non-negative, continuous and integrable functions f and g, such that the Radon transform of f is pointwise smaller than the Radon transform of g, does it necessarily follow that the L^p-norm of f is smaller than the L^p norm of g when p>1? We address this question for two types of Radon transforms: the classical Radon transform and the spherical Radon transform. The solution to this question is quite subtle and requires techniques from Harmonic Analysis and Fourier Analysis. As it turns out, this question is intimately related to the slicing problem of Bourgain (a question from Asymptotic Geometric Analysis (Geometric Probability)), reverse estimates for the Radon transform due to Oberlin and Stein from the 1980s, and finally some very recent estimate on the Radon transform due to Bennett and Tao. If time permits, we will discuss a lower-dimensional analogue of this problem.

Based on a joint work with Alexander Koldobsky and Artem Zvavitch.

### Yuan Zhang

Note: Yuan’s talk is on Monday.

Title: Optimal $\bar\partial$ regularity on product domains and its application to the Hartogs triangle

Abstract: The $\bar\partial$ problem is to study the existence and regularity of the Cauchy-Riemann equation $\bar\partial u = f$ on pseudoconvex domains. Since H\”ormander’s fundamental $L^2$ theory, there has been substantial investigation for domains exhibiting favorable geometry and/or regularity. In this talk, we shall first focus on the $\bar\partial$ problem on a specific type of Lipschitz domains — product domains, and discuss recent advancements regarding its optimal Sobolev and H\”older regularity. Then we explore its application to the optimal Sobolev regularity on the Hartogs triangle. Part of the talk is based on joint works with Yifei Pan.