Schedule, Notes/Handouts & Videos

In this tutorial, participants will be introduced to the Lean proof assistant as both a programming environment and a platform for formal mathematical reasoning. Students will learn how to implement number-theoretic and combinatorial functions, such as the Fibonacci and Catalan sequences. A distinctive feature of Lean is that programs and proofs coexist in the same framework. Beyond computing values, participants will learn how to formally state mathematical claims about their definitions and develop proofs that are fully verified by the system. Through guided, interactive exercises, the tutorial will demonstrate how computation and proof can inform and reinforce one another.

This tutorial introduces an experimental approach to mathematics, with a focus on computation and data analysis. Drawing inspiration from the sciences, participants will run “experiments”, using Python libraries and Colab notebooks to gather data. They will then use Machine Learning to observe patterns in the data and thus “discover” theorems. The emphasis is on exploration and discovery: experimenting with functions, generating datasets, and using ML techniques to posit conjectures.

Artificial intelligence has renewed interest in a fundamental question: What mathematical principles give rise to learning and intelligent behavior? This talk presents two complementary mathematical perspectives on intelligence inspired by research conducted at the Institute for Pure and Applied Mathematics (IPAM). The first explores how spectral methods from dynamical systems use eigenvalues to reveal hidden structure in complex systems, leading to interpretable approaches for learning and prediction. The second examines adaptive reservoir computing through a nonlocal mathematical framework, where interactions occur across neighborhoods rather than only at individual points, allowing network connections to evolve through homeostatic mechanisms such as pruning and impairment to study how structure influences memory and computation. Together, these projects illustrate how modern mathematics provides new ways to understand intelligent behavior alongside traditional machine learning. The talk will emphasize the underlying mathematical ideas, computational experiments, and open research questions, with the goal of introducing undergraduate students to active areas of research at the intersection of dynamical systems, scientific computing, and artificial intelligence.

In evolution, hybridization occurs when two distinct species merge genetically to create a new species. Recent advances in phylogenetic analyses of DNA have revealed that hybridization has played an important role in the evolutionary history of many species. Reconstructing evolutionary relationships is already challenging because of the many processes that shape species' histories, and the presence of hybridization  adds a layer of complexity. In this talk, we explore some of the challenges of inferring evolutionary relationships in the presence of hybridization under the Network Multispecies Coalescent model (NMSC).  Specifically, we discuss which aspects of evolutionary history can be reliably recovered from data (are identifiable) and also present recent advances in statistically consistent inference methods. Additionally, we  present some of the consequences of model misspecification when the true evolutionary process violates the assumptions of an inference model.

Braids are entangled strings in space, fixed at the endpoints, never doubling back. Mathematically, they can be abstracted as infinitely stretchable, deformable curves, and thus a topological object. There is also an elegant method of discretizing them to give an algebraic definition. Due to braids' clear physical interpretation, images and diagrams have long played a role in both topological and algebraic proofs. How can we temper the intuitive understanding granted by this visual imagery with mathematical rigor? Modern mathematical tools, such as the Lean interactive theorem prover, help us in this quest. While often viewed as mere bookkeepers, proof-verification tools can grant us opportunity and license to explore -- and visualize -- new theorems, proofs, and conjectures, all with safe, enforced boundaries on correctness and rigor.     

This tutorial introduces an experimental approach to mathematics, with a focus on computation and data analysis. Drawing inspiration from the sciences, participants will run “experiments”, using Python libraries and Colab notebooks to gather data. They will then use Machine Learning to observe patterns in the data and thus “discover” theorems. The emphasis is on exploration and discovery: experimenting with functions, generating datasets, and using ML techniques to posit conjectures.  

This tutorial introduces an experimental approach to mathematics, with a focus on computation and data analysis. Drawing inspiration from the sciences, participants will run “experiments”, using Python libraries and Colab notebooks to gather data. They will then use Machine Learning to observe patterns in the data and thus “discover” theorems. The emphasis is on exploration and discovery: experimenting with functions, generating datasets, and using ML techniques to posit conjectures.

Many problems in algebraic combinatorics have geometric objects hidden in the background, and bringing this geometry to the foreground can reveal structure that is difficult to see from formulas alone. We study a refinement of the q,t-Catalan numbers introduced by Xin and Zhang in 2022 and 2023 using tools from polyhedral geometry. These refined polynomials depend on a vector of parameters k and recover the classical q,t-Catalan numbers when every entry of k is equal to one.

We begin with an introduction to polyhedral cones and their connection to multivariable generating functions. We then construct cones arising from the constraints that define k-Dyck paths and show how their area and bounce statistics can be encoded geometrically. This viewpoint provides a new interpretation of Xin and Zhang’s generating functions and recovers their q,t-symmetry results for parameter vectors of length three and for the four-parameter family (k, k, k, k). It also leads to extensions, including the family (k, k+m, k+m, k+m), and offers insight into related generalizations and open problems concerning refined q,t-Catalan numbers. No prior experience with polyhedral geometry will be assumed. The talk is based on joint work with Matthias Beck, Mitsuki Hanada, Max Hlavacek, John Lentfer, and Katie Waddle.

Algebraic geometry studies the geometric objects that arise as solutions to polynomial equations. Two guiding questions I will discuss are: how can we classify these objects? How does their geometry influence the number of rational solutions to the polynomial equations that define them? These questions lead to the subjects of moduli theory and arithmetic geometry, respectively.

The Dirichlet problem for the real elliptic Monge-Ampère equation in two dimensions arises in many important applications including optimal transport, machine learning, and geometry. In this talk, I will discuss the performance of two numerical methods for the solution of this problem. The first method is an augmented Lagrangian based method that reformulates the problem as a saddle point problem. The second method is a relaxation algorithm using a least squares formulation. This work is part of an ongoing undergraduate research project, and I will present preliminary numerical experiments in two dimensions comparing the two methods with respect to accuracy, convergence behavior, and computational cost.

Work by J.C. Meza and Isaic Cruse, UC Merced.