Current Projects

Mentor: Srishti Singh

Area: Combinatorics

Overview of the topic: If you have an infinite amount of $2 and $3 coins, what is the largest amount for which you canNOT give me exact change? This problem of finding amount that cannot be obtained using only coins of specified denominations is called the Frobenius problem. The amounts for which you CAN give exact change form an object called a Numerical Semigroup. In 1978, Wilf conjectured that every numerical semigroup satisfies a certain inequality. During the semester, we will learn about numerical semigroups and try to make progress towards Wilf's conjecture.  

Pre-requisites: has taken (and is confident in) MATH3000, and working with matrices 

Text: NA

Mentor: Christian Hirni

Area: Numerical Analysis

Overview of the topic: Learn introductory or advanced methods in the field. 

Pre-requisites: Knows differential equations or knows calculus. 

Text: NA

Mentor: Nathan Bushman

Area: History of Mathematics 

Overview of the topic: We'll read Dunham's book, which covers the stories and proofs of some important theorems in the history of math. Time permitting, we may branch out to other sources as well. 

Pre-requisites: Minimal: calculus 1 and/or linear algebra would be sufficient. Some familiarity with proof techniques (math 3000) would be helpful, but is not required. 

Text: Journey Through Genius by William Dunham

Mentor: Erik Amézquita 

Area: Applied Mathematics 

Overview of the topic: Plenty of real-world data comes encoded as networks: collections of vertices and edges. For instance, the edges could be individual faculty and edges indicate certain interaction between them. How "connected" is the graph? How "vital" is an individual node? If we could add a new edge, where would it be the most "impactful"? Are there "typical" motifs repeated over and over again? How do we mathematically define "connected", "vital", "impactful", and "typical" in the first place?

The general aim is to understand and use concepts from network analysis and to then apply them to network data from various academia-inspired contexts. We will review network analysis concepts from Estrada's The Structure of Complex Networks and Newman's Networks to analyze contributions of nodes and edges to the overall structure of the graph. The ultimate goal is to mathematically model the network of academic collaboration between faculty members of the Interdisciplinary Plant Group (IPG) at MU. This would allow us to understand better current collaborations and suggest key collaborations to foster in the future.

Pre-requisites: The student should feel comfortable with some linear algebra and calculus at the bare minimum. Some knowledge on discrete mathematics or real analysis would be helpful albeit not required. Basic coding skills (e.g. feel comfortable using `numpy` in python) would be very helpful but not required. The exact path of the research project will vary depending on the student's strengths. The student should be open to discuss their knowledge with biologists and be receptive of biology input when it comes to data analysis and interpretation.

Texts: 

E. Estrada (2011) The Structure of Complex Networks: Theory and Applications. Oxford University Press.

M. Newman (2010) Networks: An Introduction. Oxford University Press.

Mentor: Brent Koogler 

Area: Applied Mathematics 

Overview of the topic: We will review the construction and training of neural networks. With these tools, we will attempt to solve regression problems. There will be an emphasis on reading modern papers. 

Pre-requisites: An understanding of multivariate calculus, linear algebra, and Python programming experience. (These can be reviewed.) 

Texts: Reference book: https://www.deeplearningbook.org/. Initial literature: https://arxiv.org/abs/2307.08934.