The Illinois Mathematics Lab (previously known as Illinois Geometry Lab) provides a framework for faculty and graduate students to engage local undergraduates in research, administering large-scale research programs each fall and spring semester and smaller programs during the summer months.

The Fall 2023 semester IML projects are:

## Linear Poisson structures in dimension 4

Poisson brackets naturally arises in the context of Hamiltonian mechanics to describe the Hamiltonian equations of motions. In this project we look at linear Poisson brackets in dimension 4. The goal of this project is to understand, on a formal level, the underlying foliation, infinitesimal automorphisms and deformations by means of the associated cohomology groups, i.e. formal Poisson cohomology, for some of the linear Poisson brackets. |

**Faculty Member:**Florian Zeiser**Difficulty:**Intermediate**Team Meetings:**Weekly**Prerequisites:**Math 416 (Abstract Linear Algebra) or equivalent and Calculus I-III or equivalent.

## Automated proofs of partition congruences

Partition functions are a basic object in combinatorics and number theory. A beautiful fact is that their values often satisfy congruence properties, and these have been studied for a long time by a large number of people. There are various ways to prove such properties; some are elementary and others use some deep machinery. The goal of this project is to use a computer algebra system to give automated proofs of a large number of such congruences and to discover families of new congruences. To do will require learning some of the theory of “modular functions,” surveying existing literature, and a good deal of computer experimentation. |

**Faculty Member:**Scott Ahlgren**Difficulty:**Intermediate**Team Meetings:**Weekly**Prerequisites:**A knowledge of basic number theory (e.g. Math 453) will be extremely helpful. Familiarity with Mathematica (or a willingness to learn) and a willingness to experiment with the computer.

## Transforms of Functions – Theory and Applications

Transforms have a long and rich history. But they are also objects of active research and applications today. A good, old example of a transform is to take a logarithm: x → log(x). This converts multiplication to addition, a very important feature of the logarithm function that made it so useful in the early days of scientific calculations. It turns out that many of the transforms of functions f → T(f) that are most important today, like the Fourier transforms and Laplace transforms, have analogous properties. These transforms convert convolutions into products, they convert differentiation into multiplication, and so on… These features and others are why these transforms are useful, why understanding them is important, and why computing them quickly and accurately is a worthwhile goal in modern theoretical and applied mathematics.In this project, we will look at how these and other important transforms (like wavelet transforms) are defined and how they are computed. We will learn about how these transforms get applied in mathematics, science, and engineering. The goal of this project will be for everyone, students and mentors alike, to understand very well at least one type of transform of functions, find good applications of it, and see how to illustrate computationally and graphically how the transform works. By doing this, we expect that we will all discover new and exciting ideas in mathematics, and how to apply them. |

**Faculty Member:**Joseph Rosenblatt**Difficulty:**Intermediate**Team Meetings:**A couple of times a week, at least once in person**Prerequisites:**Basic calculus and differential equations. While knowledge of coding/software is not essential, being able to use Mathematica, Matlab, and/or C++/Python would be helpful.

## Win/loss Sequences in Sports Leagues

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 issues such as the predictability of games,the effect of home advantage on game outcomes, and the parity and competitive balance among teams in a given sports league. 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, and NHL, with the primary focus being on the MLB.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. This project is similar in sprit to, but independent from, a previous (Spring 2023)project of the same title; familiarity with the earlier project is not expected. Knowledge of any of the sports mentioned is not required. However, if you are very knowledgeable in one of the mentioned sports and its recent history, please mention this in your application. |

**Faculty Member:**AJ Hildebrand**Difficulty:**Intermediate**Team Meetings:**once or twice per week**Prerequisites:**There are no hard course prerequisites; any needed 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 *may* be helpful for certain aspects of the project, but is not required.

## Scale-free First Passage Percolation

First passage percolation is a mathematical model used to study the spread of rumor or information among nodes in a random network. In the long-range model, a node passes the information to another node after a random time with a mean proportional to some power A of the distance, independent of each other. For larger A, the travel time for information will get larger. This project considers the scale-free version, where each node u is assigned a non-negative random weight W(u). One can think of W(u) as something quantifying the node u’s importance. When the weight variables have unbounded support, nodes with very high weight will be present. These form the hubs of the network and play a crucial role in the functionality of the network. In the new model, the time needed for information to pass from node u to node v is scaled by W(u)*W(v). We want to study the effect of the tail-exponent of W and power-exponent A on the passage time between two far-away nodes and, using simulation, understand the phase transition when the base graph is a one-dimensional lattice. If time permits, we will use simulation to understand the distributional limit for the passage time and generalize to higher dimensional lattices. |

**Faculty Member:**Partha Dey**Difficulty:**Advanced**Team Meetings:**Weekly**Prerequisites:**Probability, Linear algebra, Optimization, Python and/or C++

## Mapping out the quantum channel zoo

Mapping out the quantum channel zoo – creating a website hosting a database of quantum channels and their mathematical and information-theoretic properties.Long description:Quantum systems, such as photons or electrons, can be used to store and process information in a completely novel way. This is studied in the field of quantum information theory. A special form of correlation called entanglement gives rise to protocols like quantum teleportation and quantum computation that cannot be achieved using classical systems. However, quantum systems are delicate and tend to be affected greatly by noise from the environment in the form of radiation, heat, etc. We model this noise as a “quantum channel”, a special class of linear maps on vector spaces of operators. The big question is then: how can we protect correlations such as entanglement from a certain type of noise, in order to do useful things with quantum systems and quantum computers? In information theory, this question is answered by treating the quantum channel as a noisy communication link through which we aim to transmit information. Answering the above question amounts to determining the so-called “capacity” of the corresponding quantum channel. Channel capacities express the fundamental information-processing capabilities of the channel, depending on the type of information that we want to transmit. We have formulas that express quantum channel capacities as an optimization of entropic quantities. However, solving these optimization problems is typically hard unless the channel has some nice mathematical property. A large number of such mathematical properties are known, and researchers have found examples of channels with these properties and figured out what they imply for channel capacities.The goal of this project is to map out this “quantum channel zoo” by creating a website hosting a dynamic wiki-style database of the known quantum channels and their mathematical and information-theoretic properties. The idea is to create a valuable resource for scientists researching quantum channels. In the course of the project you will learn basics of quantum information theory, quantum channels, and their mathematical characterizations and properties. Prerequisites are a good understanding of linear algebra (Math 416 or equivalent). Some familiarity/experience with web design is beneficial, but not necessary to work on this project. |

**Faculty Member:**Felix Leditzky**Difficulty:**Intermediate**Team Meetings:**Weekly**Prerequisites:**Math 416 or equivalent preferred (but not necessary): * some basic knowledge in quantum theory. Basic web design skills, scientific computing skills in MATLAB/python

## Real superalgebras

Supersymmetry in physics postulates a symmetry exchanging two species of particles: bosons and fermions. The mathematics of supersymmetry carries a Z/2-grading corresponding to these two species. Hence, a supervector space is a Z/2-graded vector space, a superalgebra is a Z/2-graded algebra, etc. The rules for manipulating these Z/2-graded objects involve subtle plus and minus signs related to fermionic statistics in physics. The goal of this project will be to formulate definitions and prove basic results for inner products and real structures on supervector spaces. Here, an inner product is a bilinear map with properties, and a real structure on a complex vector space is taken to be a complex anti-linear map with properties. This project is really just ordinary linear algebra with new minus signs: no prior physics knowledge will be assumed. |

**Faculty Member:**Dan Berwick-Evans**Difficulty:**Intermediate**Team Meetings:**Biweekly**Prerequisites:**Math 416 necessary, Math 417 or Math 427 recommended.

## List coloring with requests for planar graphs

A proper graph coloring is an assignment of a color to each vertex of a graph so that no two adjacent vertices receive the same color. Thomassen proved that if G is a planar graph in which each vertex receives a list of five possible colors, then G has a proper coloring in which each vertex receives a color from its list. In this project, we aim to strengthen this result. Suppose G is a planar graph in which each vertex has a list of five possible colors. Suppose further that some of the vertices of G have a preferred color. We will ask, can we give G a proper coloring using these color lists so that at least 10% of these coloring preferences are satisfied? If 10% is not possible, what about 1%? If 1% is not possible, what about 0.000001%? In other words, we aim to find some (likely very small) constant c > 0 such that for every planar graph G, every assignment of lists of size 5 to the vertices, and every set of coloring preferences, we can find a proper coloring on G using these lists that satisfies c% of the coloring preferences. |

**Faculty Member:**Peter Bradshaw**Difficulty:**Intermediate**Team Meetings:**Weekly**Prerequisites:**Math 412 (Graph theory) is necessary. Math 413 (Combinatorics) is preferred but not necessary.

## Finding the Math Department’s Deep Structure

What is our department good at? The question is not as simple as it seems: the research group composition changes fast, and their impact on the global scale might be smaller of larger than it seems. The goal of this project is two-fold: on one hand, to detect the intrinsic research clusters (i.e. groups of people working in close areas, talking to the same communities, publishing in the same journals) within the department. On the other hand, to see how significant these clusters are in the context of their respective fields: which are strong, which are growing. This is a continuation of the Spring ’22 IGL project on finding the department’s deep structure. As most of the data are already in place, the team can concentrate on extracting the resulting clustering(s) and aggregating them (here some theory of averaging on CAT(0) spaces, using Sturm’s algorithm will be used.)

**Faculty Member:**Yuliy Baryshnikov**Difficulty:**Intermediate**Team Meetings:**Weekly**Prerequisites:**Interest in data analysis, some confidence with R, Python

**Applications for Fall 2023 projects are now closed**. See the Upcoming Projects page for Spring 2024 application information. To be notified when the applications for future semesters will be open, join our mailing list here.

Illinois Mathematics Lab research is supported by the Department of Mathematics at the University of Illinois at Urbana-Champaign, and by the generous private donors.