# Summer 2022 Projects

The 2022 UC Davis Math REU will have between four and six research projects (see below), representing various research areas of the UC Davis math department. Students will be directly mentored by UC Davis faculty and postdoctoral researchers, and will have the opportunity to interact with UC Davis.  Students will be expected to write a final report and give an oral presentation to their peers at the end of the program.

### Project 1: Rotating fluid dynamics experiments and partial differential equations

The Earth is a rotating frame of reference over which the fluid of the atmosphere and ocean move.   Rotation organizes planetary motion by driving large eastward winds, and cyclones.   But the effect of rotation was thought to be most weakly experienced in the deep tropics.   In recent mathematical models, myself and my collaborator, Prof. Matt Igel of the atmospheric sciences group, have developed models for the tropical atmosphere which recognize the important roll that rotation plays in organizing tropical thunderstorms.

In collaboration with Prof. Michael Toney of the Chemistry department, we have assembled and implemented a rotating fluid tank and a pair of small vortex cannons which are intended to mimic thunderstorms in the tropical atmosphere.  The vortices that we have generated travel in circular arc in the rotating tank, and they interact with one another to create flows which are on larger scales than the original vortices.

In this REU project, students will participate in running the experiment.  They will generate fluid vortices using the vortex cannons.  They will take data with the Raspberry Pi camera and process that data with Matlab or similar software.

In the process, the students will learn about the partial differential equations of fluid dynamics, line and toroidal vortices, and the theory of rotating fluid dynamics, both at the equator and in the middle latitudes.

### Project 2: Pseudorandom graphs from sum-and-product games

The following puzzle is from a 1979 column by Martin Gardner in Scientific American: Two numbers (not necessarily different) are chosen from the range of positive integers greater than 1 and not greater than 20. Only the sum of the two numbers is given to mathematician S. Only the product of the two is given to mathematician P.

On the telephone S says to P, "I see no way you can determine my sum." An hour later P calls him back to say, "I know your sum." Later S calls P again to report, "Now I know your product." What are the two numbers? The variation discussed by Sallows in a 1995 Intelligencer column increases the number 20 to 100 and the sequence of "I don’t know"s substantially. The analysis of this puzzle entails a deterministic graph process in which the bipartite graphs G(N) all have the possible sums and the possible products as vertices and at step N in the process there is an edge between a sum S and product P if S = A + B and P = AB and 2 ≤ A ≤ B ≤ N. A puzzle solution is a maximum length branch in a non-tree component. The project would be to compare this process to a random graph process such as the one studied by ErdÅ‘s and Rényi. Deterministic structures involving both sums and products of integers often turn out to behave similarly to random objects.

Possible questions would be asymptotic properties of degree distributions, component size distributions, cycle length distributions and the longest leaf branch (hence the number of "Don’t Know"s in a puzzle solution). Computer tests would be one option.

### Project 3: Lagrangian skeleta for Stein traces

Advisor: Roger Casals and Orsola Capovilla-Searle

This is a project on knots in 3-dimensional space, in the area of mathematics known as geometry and topology. In particular, we will be studying how surfaces can be embedded into space and how their boundaries might be knotted. From a research perspective, we will first be learning Kirby calculus, which is a diagrammatic technique that allows us to manipulate knots and surfaces in spaces of dimensions 3 and 4 by drawing curves in surfaces. Once we have mastered the smooth aspects of this diagrammatic technique, we will venture into contact and symplectic geometry, where the knots will be representing wavefronts of light and the surfaces will be caustics. We will explore the diagrammatic calculus in this modified setting and use it to construct new Lagrangian skeleta for a certain family of 4-dimensional spaces, known as Stein traces. These new results are the goal of the project but the journey, and what you can learn from it, is as important as its end. The fundamental mathematical skills that are required for this project are a lot of enthusiasm and eagerness to learn how to think visually about abstract mathematical concepts, as well as manipulating lots of pictures. We will also have a primer and preliminary readings on knot theory in case this is your first hike into the realm of topology, so you can learn how to draw and visualize knots and surfaces efficiently. The pictures that we will be discussing are quite similar to the one of a surface with 3-holes and many (colorful) curves wrapping around it.

### Project 4: Quantum error correction in novel quantum geometries

The concept of an error-correcting code in combinatorics and computer science is closely related to the concept of a sphere packing in geometry.   In any metric space X, one can consider minimum-distance sets (subsets S such that minimum distance between any two points in S is at least some number t), and the nearly equivalent concept of sphere packings of some radius s.   Either way, one of the fundamental questions is simply how to pack in as much as possible.  If X is a Euclidean space, then the question becomes the classic sphere-packing problem; if X is a Hamming space, then the question becomes standard error correction with binary codes.   There is a well-developed classical theory of both examples and packing bounds for these two choices of the metric space X, and in many other cases.

There is now an analogous theory of quantum error correction in quantum Hamming spaces, which is important for quantum computing.  Many of the ideas from classical error correction have quantum counterparts; for instance, linear codes in a Hamming space become additive codes in a quantum Hamming space.   Already in quantum Hamming spaces, there are many new phenomena and open problems.  Moving beyond that special case, in this project we will look at error correction and minimum-distance subspaces in the general setting of finite quantum metric spaces, by analogy with packing problems in general classical metric spaces.   Despite the word "quantum", this project is entirely rooted in algebra and combinatorics.  For instance, a finite quantum metric space is defined as an algebra filtration on the vector space of n x n complex matrices.   It is particularly fun to look at quantum metric spaces that are constructed from linear representations of Lie groups, and yet these very natural quantum geometries have been explored very little, leaving us with nearly everything to discover, aside from the special case of quantum Hamming spaces.