Mentor: Erik Amézquita 

Area: Applied math 

Overview: Plenty of real-world data comes encoded as networks: collections of vertices and edges. For instance, the edges could be individual genes and edges indicate certain interaction between them. More often than not, complex traits such as plant height or flowering time are controlled not by one but by multiple genes and multiple interactions. Providing an adequate mathematical description of such a vast array of vertices and edges can be daunting. 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 some concepts of network analysis and to then apply them to network data from various plant biology-inspired contexts. I am particularly interested in reading the work of De La Cruz Cabrera et al (2022). In this paper, they provide ways to analyze contributions of individual nodes and edges to the overall structure of the graph. To the best of my knowledge, these specific network analysis techniques have not been applied yet to any plant biology context. The ultimate goal would be to code ---preferably in python--- the proposed analysis techniques by De La Cruz Cabrera et al and apply them to both social and yeast network data.

This project can be extended for future semesters, where we can be involved in active discussion with plant biologists to truly understand the significance of our results and fuel future directions. Funding opportunities are available through the BIPS grant of the DPST. More details on goals and funding can be read here:

https://ejamezquita.github.io/docs/undergrad_network_analysis.pdf 

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. 

Text: 

• O. De la Cruz Cabrera, J. Jin, S. Noschese, L. Reichel (2022) Communication in complex networks. Applied Numerical Mathematics 172: 186--205. DOI: 10.1016/j.apnum.2021.10.005

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

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

Mentor: Stephen Landsittel 

Area: Point set topology (and possibly applications to number theory) 

Overview: Discuss fundamental properties of spaces such as the T axioms, analyze basic results regarding closure, continuity, compactness, and products. Further exploration and applications in topological vector spaces and other structures in number theory may be covered depending on the student's background and progress. Starting on chapter 1 of the book 'Topology' by Munkres, is the "default" plan, but this might be adjusted depending on the student's background (see next answer). 

In the past I have had to modify my plan somewhat quickly, so if there is more interest, I would also be happy to instead form a plan involving a crash course in Modules, or commutative rings. 

Pre-requisites: Not necessarily any at all, other than a first course in proofs, and understanding of basic set theory (i.e. something equivalent to math3000). If the student already knows some topology (and or algebra) then we can start further along in topology and focus on deeper results and applications.

Text: "Topology", Munkres (I will find a pdf if the student does not have this text) 

Mentor: Saptarshi Banerjee 

Area: Commutative Algebra 

Overview: In this semester we will focus on developing a strong foundation of Commutative Algebra. For that I propose going through the first four chapters of Atiyah Macdonald, however if needed we may use other textbooks as supplementary material. We will try to solve few excercise problems from each chapter and we will finish with proving Hilbert's Nullstellensatz, one of the key results in commutative algebra. 

Pre-requisites: Good understanding of Ring theory. Materials taught in Math 4510/4720 would suffice. 

Text: Commutative Algebra by Atiyah Macdonald 

Mentor: Brent Koogler 

Area: Applied Math 

Overview: Cover the fundamentals of linear state space modeling and corresponding theoretical results. 

Pre-requisites: Proof writing, knowledge of calculus, and an interest in applied work. 

Text: Linear State-Space Control Systems (Williams and Lawrence) 

Mentor: Arun Suresh

Area: Applied math/analysis 

Overview: This project will (more or less) revolve around understanding the effectiveness of (deep) neural networks by viewing them as universal approximators. Depending on the student's inclination, the project can be made as theoretical or practical as needed. The program could potentially include an independent coding project, if time permits. 

Pre-requisites: It is preferred that the student is comfortable with some programming language (eg. Python, Julia, etc.) as some of the exercise problems would involve a coding/visualization component. Comfort with linear algebra and experience with basic real analysis would be beneficial.

Text: One of the following (with the rest being optional/supplemental material, sections from which will be assigned as needed)

• Neural networks and deep learning - Micheal Neilsen.

• An introduction to statistical learning - James, Witten, et. al.

•  Stanford's online deep learning tutorial/lecture notes - Andrew Ng, et. al. 

Mentor: Luis Flores 

Area: Applied Mathematics/Algebra 

Overview: In linear algebra, we learn about vector spaces and "ideal" subsets of these vector spaces. These subsets are called basis and are defined through the concepts of span and linear independence. Basis, however "ideal" they may be, are very restrictive. If we drop the linear independence requirement, the result is called a frame. Frames are useful for a myriad of reasons including but not limited to signal decomposition, data compression, phase retrieval, and robust encodings of data. Our goal for the semester will be to study finite frames along with their many applications. 

Pre-requisites: Math 3000, Linear Algebra (We will be using this a lot), some analysis would be beneficial but not necessary (We might encounter some sequences and supremums/infimums). 

Text: Frames for Undergraduates by Han, Kornelson, Larson, and Weber, and Finite Frames: Theory and Applications by Casazza and Kutyniok. 

Mentor: Eliot Bitting 

Area: Applied Topology 

Overview: Work through the online TDA book by Tamal Dey and Yusu Wang 

Pre-requisites: Math 3000, some linear algebra 

Text: Computational Topology for Data Analysis by Tamal Dey and Yusu Wang