2026 Fall Projects

Overview

The Virtual Math Research Circle (VMRC) Fall/Winter 2026 term will bring motivated high-school students together with university research mentors for six weeks of collaborative mathematical research.

The term will begin the week of October 26, 2026 and conclude by December 13, 2026, with no regular program meetings during Thanksgiving week. Additional research projects will be posted as mentors and topics are finalized.

Tuition: $1,200 per student for the six-week Fall/Winter 2026 term.

Registration for Fall/Winter 2026 is now open.

Register for Fall/Winter 2026

Fall/Winter 2026 Research Projects

Exploring Waddington’s Epigenetic Landscape: Modeling Stem Cell Differentiation

Research Proposal — Virtual Math Research Circle

Term
Fall/Winter 2026: October 26, 2026 – December 13, 2026
Six instructional weeks with a Thanksgiving week break
Mentor
Kaylie Green
PhD Candidate
Department of Biomathematics
Florida State University

Project Title
Exploring Waddington’s Epigenetic Landscape: Modeling Stem Cell Differentiation
Topic Area
Mathematics in Biology and Probability Theory

Background
High school calculus is not required, but students should have a basic understanding of algebra, graphing, and functions. Calculus concepts such as derivatives may be introduced as an extension for advanced students interested in analyzing stability in Waddington’s landscape.

Students should also have a foundational understanding of cell biology, including the cell cycle, stem cell types, and differentiation. A background in algebra and probability will be useful for modeling cell fate decisions, while pre-calculus concepts such as exponential and logistic growth may be helpful but are not required.

Basic data analysis and computational skills using tools such as Excel, MATLAB, Desmos, or related graphing and computational software may enhance the project experience, but guidance will be provided.

Abstract
This six-week interdisciplinary research project explores stem cell differentiation through the lens of Waddington’s epigenetic landscape, integrating ideas from biology, mathematics, probability, and computational modeling.

Students will investigate how stem cells transition into specialized cell types, using the epigenetic landscape as a conceptual and mathematical framework for understanding gene regulation and cell fate decisions. Through mathematical modeling and computational experiments, students will analyze how changes in environmental signals, regulatory mechanisms, and model parameters can influence differentiation pathways.

The project begins with an introduction to stem cell biology, including pluripotency, self-renewal, and differentiation. Students will then explore mathematical representations of Waddington’s landscape using potential functions, probability models, phase diagrams, and computational simulations.

As the term progresses, students will transition into individual mini-projects in which they investigate a more focused research question within the broader theme. Possible directions may include stochastic models of cell fate, parameter sensitivity, optimization, disease modeling, regenerative medicine, or analysis of preprocessed gene-expression data.

Returning students may build upon work begun during Summer 2026, while new students will be guided through the foundational material needed to develop their own research direction. The term will conclude with research-style presentations of the students’ individual or collaborative findings.

Possible Extension
Students may develop stochastic models of cell fate decisions using probability trees, Markov-style models, or simulation-based approaches to represent differentiation pathways.

Another direction is to investigate parameter sensitivity and optimization by modifying mathematical landscape models and studying how external factors alter the location, depth, or stability of possible cell states.

Students interested in computational biology may investigate preprocessed gene-expression data and explore how patterns in gene activity correspond to different stages or pathways of differentiation.

Particularly motivated students may also continue developing their work after the term toward a more formal research project, poster, or future presentation opportunity.

Outline/Timeline
Week 1 — Stem Cells and Waddington’s Landscape.
Students will study stem cells, differentiation, pluripotency, self-renewal, and the biological motivation for Waddington’s epigenetic landscape. The group will begin connecting biological questions with mathematical representations.

Week 2 — Mathematical Foundations of the Landscape.
Students will explore potential functions, probability models, graph-based representations, and qualitative ideas from dynamical systems. Computational tools will be introduced for visualizing landscapes and cell-fate trajectories.

Week 3 — Computational Modeling and Parameter Exploration.
Students will construct and modify mathematical or computational models of differentiation. They will investigate how changing parameters affects landscape geometry, stability, transition probabilities, and predicted cell outcomes.

Week 4 — Developing Individual Research Questions.
Students will identify a focused mini-project within the broader research theme. Possible directions include stochastic cell-fate models, parameter sensitivity, regenerative medicine, disease modeling, optimization, or computational analysis of biological data. Students will formulate research questions and begin independent investigation.

Thanksgiving Week — Program Break.
No regular VMRC meetings are planned. Students may continue independent work if desired, but no new material will be required during the break.

Week 5 — Independent Research and Analysis.
Students will develop their mini-projects with guidance from the mentor, refine their mathematical or computational models, analyze results, and create appropriate figures and visualizations.

Week 6 — Research Synthesis and Final Presentation.
Students will interpret their findings, discuss limitations and future directions, and prepare a research-style final presentation or poster summarizing their question, methods, results, and conclusions.

References
  1. Murray, J. D. Mathematical Biology, 3rd ed., Springer, 2002.
  2. Waddington, C. H. Organisers and Genes. Cambridge University Press, 1940.
  3. Waddington, C. H., and Editors. Computational Biology of Stem Cells. Springer, 2006.

Can AI Recognize Incomplete Shapes? Exploring Co-presheaf Transformers

Research Proposal — Virtual Math Research Circle

Term
Fall/Winter 2026: October 26, 2026 – December 13, 2026
Six instructional weeks with a Thanksgiving week break
Mentor
Dr. Ibrahem AlJabea
Professional in Residence in Artificial Intelligence
Division of Computer Science and Engineering
Louisiana State University

Project Title
Can AI Recognize Incomplete Shapes? Exploring Co-presheaf Transformers
Topic Area
Artificial Intelligence, Computer Vision, Graph Neural Networks, Transformers, and Topology

Background
This project introduces high-school students to artificial intelligence through a familiar question: can a computer recognize a shape when parts are missing? Students should understand basic geometry, coordinates, and percentages. No previous knowledge of graph neural networks or transformers is required.

Using Python and instructor-provided notebooks, students will learn about image classification, training and testing data, and prediction accuracy. They will explore how co-presheaf transformers learn transformations that pass information between image-patch representations.

Abstract
People can often recognize a triangle or square even when part of its outline is hidden. This project investigates whether a co-presheaf transformer can recognize incomplete shapes and how its performance compares with a regular transformer.

Students will automatically generate images of triangles, squares, and non-square rectangles with different sizes and orientations. They will make the task more challenging by hiding image regions or adding noise. Both models will be trained and evaluated using separate datasets, with identical test examples used for comparison.

The central question is whether the co-presheaf transformer’s learned information-transfer mechanisms improve recognition when visual information is missing. Students will compare accuracy across different corruption levels, examine mistakes, and present their findings through plots and a research poster.

Real-world images may contain noise, hidden objects, or missing regions. Simple shapes give students a clear, controlled way to investigate how these problems affect AI predictions. The project teaches students to design fair experiments and evaluate whether a more advanced model actually provides an advantage.

Possible Extension
Students may extend the study to text by comparing how the models classify short reviews when words are missing or misspelled.

They will also investigate how deleting important words, such as not, can change the meaning and the correct classification.

Additional research directions may include examining which types of visual corruption are most difficult for each model, comparing performance across different shape classes, or investigating whether particular missing image regions affect predictions more strongly than others.

Outline/Timeline
This Fall/Winter version expands the project across six instructional weeks so that students can move from foundational ideas through controlled experimentation, comparative analysis, and research presentation.

Week 1 — Artificial Intelligence, Images, and the Research Question.
Students will begin with the central question of whether an artificial-intelligence system can recognize a geometric shape when some of the visual information is missing. They will review basic ideas in image classification, training data, testing data, prediction accuracy, and fair experimental comparison. Students will also become familiar with the Python notebooks and computational environment used throughout the project.

Week 2 — Building the Shape Dataset and Creating Incomplete Images.
Students will generate images of triangles, squares, and non-square rectangles with varying sizes, positions, and orientations. They will develop controlled ways to make the classification task more difficult by hiding image regions, removing parts of shapes, or introducing noise. The group will discuss how to separate training and testing examples and how to make model comparisons fair.

Week 3 — Transformers, Graph-Based Representations, and Co-presheaf Ideas.
Students will develop an intuitive understanding of transformers and image-patch representations. They will be introduced to the idea that information can be represented across connected components and transferred between representations. Through the instructor-provided notebooks, students will explore how a regular transformer and a co-presheaf transformer process the shape-classification problem.

Week 4 — Training Models and Designing the Comparison.
Students will train and evaluate both models using controlled datasets. Identical test examples will be used when appropriate so that model performance can be compared directly. Students will measure prediction accuracy and begin systematically varying the amount and type of missing visual information.

Thanksgiving Week — Program Break.
No regular VMRC meetings are planned. Students may continue independent exploration if desired, but no new material will be required during the break.

Week 5 — Robustness Analysis, Error Analysis, and Interpretation.
Students will compare the models across different corruption levels and examine examples that are classified correctly or incorrectly. They will create plots showing how accuracy changes as visual information is removed or noise is increased. The group will investigate whether the co-presheaf transformer appears to provide an advantage under particular types or levels of corruption and will discuss the limitations of the experiment.

Week 6 — Research Synthesis and Final Presentation.
Students will synthesize their experimental results, identify their main conclusions, and prepare figures that communicate the comparison clearly. The project will conclude with a research-style poster or presentation describing the research question, experimental design, models, results, limitations, and possible future directions.

How Math Explains Medicine: Modeling Drug Movement in the Body

Research Proposal — Virtual Math Research Circle

Term
Fall/Winter 2026: October 26, 2026 – December 13, 2026
Six instructional weeks with a Thanksgiving week break
Mentor
Roshana Sandadini Pitiyegedara
Ph.D. Candidate in Applied Mathematics
Department of Mathematics and Statistics
Washington State University

Project Title
How Math Explains Medicine: Modeling Drug Movement in the Body
Topic Area
Mathematical Modeling, Differential Equations, and Pharmacokinetics

Background
This project requires only basic algebra and a willingness to explore new ideas. No prior knowledge of differential equations or pharmacokinetics is required, as all necessary concepts will be introduced during the project.

Familiarity with functions and graphs is helpful but not required. Concepts from calculus, such as rates of change, will be introduced intuitively as needed.

Students will use computational tools such as graphing software or Python to visualize solutions, compare models, and explore how changes in model parameters affect drug concentration over time.

The project is suitable for both new students and returning students from the Summer 2026 VMRC project. New students will be introduced to the mathematical foundations from the beginning, while returning students will be able to build upon their previous work with intravenous administration and compartmental models.

Abstract
What happens to a medication after it enters the body? How quickly does it reach the bloodstream, how does it move between different parts of the body, and why does its concentration eventually decrease? These questions are central to pharmacokinetics, the study of how drugs are absorbed, distributed, and eliminated.

During the Summer 2026 VMRC project, students investigated intravenous drug administration and compartmental models. The Fall/Winter project extends this mathematical framework to oral drug administration, where a new process must be considered: before a drug reaches the bloodstream, it must first be absorbed.

Students will begin with accessible models of exponential decay and intravenous dosing, then introduce an absorption compartment representing a drug in the gastrointestinal system. A simple absorption model can be written as
dAg/dt = −kaAg,
where Ag represents the amount of drug awaiting absorption and ka is the absorption-rate constant.

Students can then connect absorption to a compartment model describing movement of the drug through the body:
dA1/dt = kaAg − (k12 + k10) A1 + k21A2,
dA2/dt = k12A1 − k21A2.


Using analytical reasoning and computational experiments, students will compare oral and intravenous administration, examine how absorption changes concentration-time curves, and investigate the effects of parameters such as absorption rate, elimination rate, and transfer rates between compartments.

Returning students will be able to extend their previous research in a new direction, while new students will receive the foundational background needed to participate fully. The six-week format will allow students to move beyond basic modeling toward individual research questions, computational investigations, and research-style presentations.

Possible Extension
Students may compare intravenous and oral dosing and study how the route of administration changes the maximum drug concentration, the time at which that maximum occurs, and the overall concentration-time profile.

Another direction is to investigate how changing the absorption-rate constant affects how quickly a drug enters the bloodstream. Students may explore situations corresponding to rapid absorption, delayed absorption, or different formulations of the same medication.

Students may also study repeated dosing and ask what happens when another dose is administered before the previous dose has been fully eliminated. This can lead to questions about accumulation, dosing intervals, and steady-state behavior.

More advanced investigations may consider additional compartments, time-dependent parameters, nonlinear elimination, or the relationship between mathematical model complexity and biological realism.

Outline/Timeline
Week 1 — Drug Concentration, Exponential Decay, and Pharmacokinetic Modeling.
Students will begin by exploring how drug concentrations change over time and why differential equations provide a useful language for describing rates of change. The group will study the basic exponential-decay model
dC/dt = −kC,
and interpret the parameter k in the context of drug elimination. Students will graph solutions and connect mathematical behavior to pharmacokinetic questions.

Week 2 — Intravenous Administration and Compartment Models.
Students will examine intravenous administration, where a drug enters the bloodstream directly. They will study how one- and two-compartment models represent drug distribution and elimination and use computational tools to explore how transfer and elimination parameters influence model behavior. Returning Summer students will be able to revisit this framework efficiently, while new students will develop the background needed for the oral-dosing extension.

Week 3 — Oral Administration and Drug Absorption.
Students will introduce an additional absorption compartment representing drug that has been administered orally but has not yet entered the bloodstream. They will study the equation
dAg/dt = −kaAg,
and investigate how the absorption-rate constant changes the timing and shape of a drug concentration curve. The group will compare the mathematical structure of oral and intravenous administration.

Week 4 — Comparing Oral and Intravenous Dosing.
Students will use numerical simulations and graphs to compare oral and intravenous dosing under controlled parameter choices. They will investigate quantities such as peak concentration, time to peak concentration, rate of elimination, and the duration for which a drug remains above a selected concentration level. Students will begin developing focused research questions based on the behavior they observe.

Thanksgiving Week — Program Break.
No regular VMRC meetings are planned. Students may continue independent exploration if desired, but no new material will be required during the break.

Week 5 — Independent Research and Parameter Investigation.
Students will develop individual or small-group research investigations. Possible directions include varying absorption rates, studying repeated doses, comparing different administration routes, examining parameter sensitivity, or extending the compartment structure. Students will generate figures, interpret their simulations, and evaluate the assumptions and limitations of their mathematical models.

Week 6 — Research Synthesis and Final Presentation.
Students will organize their research question, mathematical model, computational results, and conclusions into a research-style presentation or report. They will explain what their models predict, discuss the biological and mathematical assumptions involved, identify limitations, and propose possible directions for future investigation.

References
  1. Boyce, W. E., and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 10th ed., Wiley, 2012.
  2. Edelstein-Keshet, L. Mathematical Models in Biology. SIAM, 2005.
  3. Rowland, M., and Tozer, T. N. Clinical Pharmacokinetics and Pharmacodynamics, 4th ed., Lippincott Williams & Wilkins, 2011.