Week of February 25, 2024

Mon Feb 26, 2024
4:00pm to 5:00pm - RH306 - Applied and Computational Mathematics
Bhanu Kumar - (NASA Jet Propulsion Laboratory, CalTech)
4th Body-Induced Secondary Resonance Overlap Inside Normally Hyperbolic Invariant Manifolds in Planet-Moon Systems: a Jovian Case Study

Mean-motion resonances (MMRs) are regions of a celestial system’s phase space where the orbital periods of two smaller bodies (e.g. a spacecraft and a moon) revolving around a common large central mass (e.g. a planet) are rationally commensurate. Each resonant region contains both stable and unstable orbits, the latter of which form a cylindrical normally hyperbolic invariant manifold and have attached stable and unstable manifolds. Heteroclinics between unstable orbits contained in different MMRs, also known as mean-motion resonance overlapping, result in natural trajectories between orbits of different sizes. This in turn is useful for low or zero-propellant space mission design. 

While most related prior work on finding such spacecraft pathways uses a planar circular restricted 3-body problem model (PCRTBP) that accounts for the gravity of the planet and a single moon, tours of multi-moon systems require using resonant orbits whose motion may be strongly affected by not just one but by two moons. In this case study, we investigate Jupiter-Ganymede unstable 4:3 mean motion resonant orbits in a concentric circular restricted 4-body Jupiter-Europa-Ganymede model. We show that despite their high order, secondary resonances between the 4:3 orbit periods and that of Europa have a large effect on the dynamics inside the 4:3 resonance's normally hyperbolic invariant manifold. Computing separatrices for the secondary resonances definitively confirms their overlap over a wide range of the orbit family, which causes a complete structural change of the higher-energy unstable 4:3 orbits that are most useful for faster orbit transfers.

4:00pm to 5:00pm - RH 340N - Geometry and Topology
Tommy Murphy - (CSU Fullerton)
Generalizations of homogeneous Einstein manifolds.

I will explain something of the theory of homogeneous Einstein metrics and why certain generalizations of this equation occur naturally in the study of homogeneous spaces.

4:00pm to 5:00pm - RH 306 - Dynamical Systems
Bhanu Kumar - (JPL, Caltech)
4th Body-Induced Secondary Resonance Overlap Inside Normally Hyperbolic Invariant Manifolds in Planet-Moon Systems: a Jovian Case Study
4:00pm to 5:20pm - RH 340 - Logic Set Theory
Julian Eshkol - (UC Irvine)


This is the first of series of seminars that surveys the results of ineffability and its use in forcing extensions.

The first talk will be about the results in Magidor's Thesis where the fundamental notions were introduced. 


Tue Feb 27, 2024
3:00pm to 4:00pm - RH 306 - Analysis
Ruixiang Zhang - (UC Berkeley)
A new conjecture to unify Fourier restriction and Bochner-Riesz

Abstract: The Fourier restriction conjecture and the Bochner-Riesz conjecture ask for Lebesgue space mapping properties of certain oscillatory integral operators. They both are central in harmonic analysis, are open in dimensions $\geq 3$, and notably have the same conjectured exponents. In the 1970s, H\"{o}rmander asked if a more general class of operators (known as H\"{o}rmander type operators) all satisfy the same $L^p$-boundedness as in the above two conjectures. A positive answer to H\"{o}rmander's question would resolve the above two conjectures and have more applications such as in the manifold setting. Unfortunately H\"{o}rmander's question is known to fail in all dimensions $\geq 3$ by the work of Bourgain and many others. It continues to fail in all dimensions $\geq 3$ even if one adds a ``positive curvature'' assumption which one does have in restriction and Bochner-Riesz settings. Bourgain showed that in dimension $3$ one always has the failure unless a derivative condition is satisfied everywhere. Joint with Shaoming Guo and Hong Wang, we generalize this condition to arbitrary dimension and call it ``Bourgain's condition''. We unify Fourier restriction and Bochner-Riesz by conjecturing that any H\"{o}rmander type operator satisfying Bourgain's condition should have the same $L^p$-boundedness as in those two conjectures. As evidences, we prove that the failure of Bourgain's condition immediately implies the failure of such an $L^p$-boundedness in every dimension. We also prove that current techniques on the two conjectures apply equally well in our conjecture and make some progress on our conjecture that consequently improves the two conjectures in higher dimensions. I will talk about some history and some interesting components in our proof.

4:00pm - ISEB 1200 - Differential Geometry
Lihan Wang - (CSU Long Beach)
How rare are simple Steklov eigenvalues?

Steklov eigenvalues are eigenvalues of the Dirichlet-to-Neumann operator which are introduced by Steklov in 1902 motivated by physics. And there is a deep connection between the extremal Steklov eigenvalue problems and the free boundary minimal surface theory in the unit Euclidean ball as revealed by Fraser and Schoen in 2016. In the talk, we will discuss the question of how rare simple Steklov eigenvalues are on manifolds and its applications in nodal sets and critical points of eigenfunctions.

Wed Feb 28, 2024
2:00pm - 510R Rowland Hall - Combinatorics and Probability
Mark Rudelson - (University of Michigan)
Approximately Hadamard matrices and random frames

We will discuss a problem concerning random frames which arises in signal processing. A frame is an overcomplete set of vectors in the n-dimensional linear space which allows a robust decomposition of any vector in this space as a linear combination of these vectors. Random frames are used in signal processing as a means of encoding since the loss of a fraction of coordinates does not prevent the recovery. We will discuss a question when a random frame contains a copy of a nice (almost orthogonal) basis.

Despite the probabilistic nature of this problem it reduces to a completely deterministic question of existence of approximately Hadamard matrices.  An n by n matrix with plus-minus 1 entries is called Hadamard if it acts on the space as a scaled isometry. Such matrices exist in some, but not in all dimensions. Nevertheless, we will construct plus-minus 1 matrices of every size which act as approximate scaled isometries. This construction will bring us back to probability as we will have to combine number-theoretic and probabilistic methods.

Joint work with Xiaoyu Dong.

Thu Feb 29, 2024
1:00pm - 306 Rowland Hall - Harmonic Analysis
Pavlos Kalantzopoulos - (UCI)
4:00pm to 4:50pm - RH 306 - Colloquium
Thibault Lefeuvre - (Sorbonne Université)
Isospectral connections, frame flow ergodicity, and polynomial maps between spheres

Classifying real polynomial maps between spheres is a challenging problem in real algebraic geometry. Remarkably, this question has found recent applications in two seemingly unrelated fields:

- in spectral theory, it allowed to solve Kac's celebrated isospectral problem (Can one hear the shape of a drum?) for the connection Laplacian.

- in dynamical systems, it allowed to prove ergodicity for a certain class of partially hyperbolic flows (extensions of the geodesic flow on negatively-curved manifolds).

I will explain these problems and how they all connect together. No prerequisite required -- the talk is intended for a broad audience.

Joint work with Mihajlo Cekić.