IGL Projects

A sequence in [0,1] is uniformly distributed if for all continuous functions F, the averages of F at the points in the sequence converge to the integral of F on [0,1]. We know how to construct lots of such sequences. A suitably randomly chosen sequence will work. But many very simple sequences work too. For example, for any irrational number r, the fractional parts of the arithmetic sequence (kr : 1,2,3,…) are uniformly distributed. It is not so easy to see, but we also know there are MANY r > 1 such that the fractional parts of the kth powers of r are uniformly distributed. But we do not know of any concrete example of r with this property! It has long been conjectured hat r = 3/2 will work. Can we figure out if this is true or false? On the other hand, there are values of r such that the fractional parts of the kth powers of r are not uniformly distributed. For example, r = 1/2 + 5^{1/2}/2. Why is this? Can we characterize this behavior? In contrast, very different behavior goes on if we do not use all the whole numbers in our sequences. Suppose we fix an increasing sequence (n(k) : k=1,2,3,…). Can we describe, or even just find, values of r such that the fractional parts of the sequence (n(k) r : k=1,2,3,…) are also uniformly distributed? In direct contrast to uniform distribution, we do not understand very well the rigidity of sequences like the ones above. That is, fix an increasing sequence (n(k) : k=1,2,3,…). What are the values of r such that the fractional parts of n(k)r go to 0 as k goes to infinity? Are there any? How does it depend on how fast n(k) is increasing? For example, are there any irrational numbers r such that the fractional parts of (exp(kr) : k=1,2,3,…) go to 0 as k goes to infinity? What about something a little faster like replacing k by a polynomial function of k? As part of this project, we will explore many interesting facts and conjectures related to uniform distribution and rigidity. These involve an interplay of number theory, harmonic analysis, and dynamical systems. We will seek to understand what we know, and better understand what we do not know, about this interplay. We will not only use existing techniques and results, but we will also see what we can do with numerical experiments to explore the behavior of sequences modulo one, especially where we do not have theorems that adequately explain the behavior of the sequences.

For more details see here.

Faculty Member: Joseph Rosenblatt

Difficulty: Intermediate

Team Meetings: Once or twice per week

Prerequisites: Calculus, linear algebra. We will do some numerical experiments using coding with software that the participants know, or that we can easily learn.

More information on Uniform Distribution and Rigidity

It has been known for some time that the solution profile of dispersive equations posed on periodic domains depends heavily on the algebraic properties of time, a phenomenon called Talbot effect in the literature. In this project we will set up numerical experiments to study this behavior for model linear equations such as the Schrodinger equation posed on the two-dimensional sphere. Specifically, we will numerically test the optimality of the limited results known for the fractal dimension of the graphs of solutions to certain linear equations.

Faculty Member: Burak Erdogan

Difficulty: Intermediate

Team Meetings: Biweekly

Prerequisites: Experience with python is a must. Some experience with differential equations would be useful but it is not crucial.

More information on Talbot effect on the 2-sphere

The math department has a rare collection of models illustrating mathematical surfaces and other geometric concepts. The models, which date back to 1900, are mostly 3-D plaster models but some are wood and some consist of metal frames with silk string threading. They are all currently housed in cabinets spread throughout Altgeld Hall but will be displayed properly when the Altgeld Hall renovations are completed. This IGL project is part of the effort to prepare the models for this transition and to enhance their impact. In Fall 2022 we selected a small set of the models and focussed on their mathematical significance; this semester we will expand our scope, either by including more models, or by investigating other aspects of the same models, or both.

Faculty Member: Steve Bradlow and Karen Mortensen

Difficulty: Intermediate

Team Meetings: Weekly

Prerequisites: N/A

More information on Math Models

In the last IGL project, we came close to identify the research task of creating a design of a quantum machine which allows us to detect which permutation is implemented in a certain quantum machine. Depending on the audience we will start with a crash course on background and previous finding. The new task is a) create a template for the classical computation needed for the detection problem, and b) discover similar designs coming from the theory of queries, and c) write a paper. Those students wanting to continue with Roy's project are very welcome to join, and will be accommodated.

Faculty Member: Marius Junge

Difficulty: Advanced

Team Meetings: Weekly

Prerequisites: Quantum circuits and queries, machine learning. Linear Algebra, probability helps.

More information on Quantum Circuits and Quantum Machine Learning II

We consider the simplest rumor propagation model, where an agent passes the rumor to one of its neighbors (chosen uniformly at random) according to its Poisson clock, independent of each other. For the base graph, we take the spread-out version of the d-dimensional lattice box of side length n. Given a positive integer L, in the L spread-out version, all vertices within distance L are connected to each other. When L = n*d, the graph becomes the complete graph, whereas, for L=1, we get the usual lattice graph. We aim to understand the first-passage time needed for the rumor to propagate from one corner to the other and how it depends on L for large n in d = 1 or 2 dimensions. In general, for L=1, the time needed is of order n, whereas for L=n*d, the time needed is of order log(n)/n. We want to find the threshold on L as a function of n and d when the time needed changes from large to small. If time permits, we will use simulation to understand the distributional limit for the passage time.

Faculty Member: Partha Dey

Difficulty: Intermediate

Team Meetings: Weekly

Prerequisites: Math 416 and 461. Some knowledge of algorithms is required for simulation. Python programming language will be used for simulation.

More information on Rumor propagation on a spread-out lattice graph

Sequences of wins and losses of sports teams over the course of a season provide a rich source of real-world "pseudo-random" sequences to explore from a variety of angles. In this project we will focus on phenomena such as streakiness, momentum, and over/underperformance that may be observed in these sequences. We will develop mathematical tools to quantify phenomena of this type and then apply these tools to data from professional sports leagues such as the MLB, NBA, NHL, NFL. This will allow one to answer, in a mathematically sound manner, questions such as the following: Which is the "streakiest" team in the history of the MLB? How do the major sports leagues compare in terms of their streakiness? How does streakiness in win/loss sequences of sports teams compare to streakiness in randomly generated sequences? This project has a theoretical component, involving reading background literature, but most of the work will be on the coding side. All participants should be proficient with Python and ideally should have some experience with large scale coding projects in Python. Knowledge of any of the sports mentioned is not required. However, if you are very knowledgeable in a particular sport and its recent history, please mention this in your application.

Faculty Member: AJ Hildebrand

Difficulty: Intermediate

Team Meetings: Once or twice per week

Prerequisites: Aside from completion of the calculus sequence, there are no hard course prerequisites. The necessary background in probability and statistics can be acquired during the course of the project. A high level of proficiency with Python is essential; if you have a Github site, please mention it in your application. Experience with webscraping tools such as Selenium *may* be helpful for certain aspects of the project, but is not required.

More information on Win/loss sequences in sports leagues

This will be a continuation of our Fall 2022 work in which we looked at quantum error correction and operator quantum error correction. This term our focus will be on the stability properties between correctable channels and their induced noncommutative graphs.

Faculty Member: Roy Araiza

Difficulty: Advanced

Team Meetings: The team will meet with faculty weekly

Prerequisites: Students should have taken Linear algebra, Analysis, Quantum mechanics or a first course in quantum information theory. More importantly, this project is only open to students who participated in the Fall 2022 IGL, or who are otherwise already working with Prof. Araiza.

More information on Quantum channels and error correction II

We are interested in several datasets associated with the conflict in Ukraine. Some elementary knowledge of statistics is needed. Students are all required to know python, and should include a link to their github repo in the application, if possible.

Faculty Member: Rich Sowers

Difficulty:

Team Meetings: The team will meet with faculty weekly. Part of the project will be remote.

Prerequisites: Students should have taken CS101 or equivalent and have experience with Python.

More information on Data From Ukraine