Spring 2023

Finding Balance in Chaos: Approximating Orthogonal Polynomials on Julia Sets

Speakers: Madison Shirazi

Abstract: Equilibrium measures describe optimal charge distributions on sets while Julia sets partially characterize the iterative behavior of polynomials. We will construct the equilibrium measures of certain Julia sets in the complex plane and then define the orthogonal polynomials relative to these equilibrium measures. Surprisingly, certain orthogonal polynomials turn out to be the iterates of the generating polynomials of the Julia sets. We will see how we can approximate these polynomials using Gram-Schmidt and study different refinements of the Gram-Schmidt algorithm.

Finding Simple Models of Complex Objects: From Regularity Lemmas to Algorithmic Fairness

Speakers: Sílvia Casacuberta Puig

Abstract: In recent years, algorithms are increasingly informing decisions that can deeply affect our lives. A major concern that arises is whether prediction algorithms are fair across different subpopulations. The notions of multiaccuracy and multicalibration were proposed by Hébert-Johnson et al. in 2018 as mathematical measures of algorithmic fairness.

Our starting point is the observation that multiaccuracy is exactly what is given by the Regularity Lemma, which is an older result in computational complexity shown by Trevisan, Tulsiani, and Vadhan in 2009 that has many important implications in different areas. By formalizing this observation, we then ask: If we start with a multicalibrated predictor instead, what versions of these implications do we obtain? Through the lenses of algorithmic fairness, we are able to cast the notion of multicalibration back into the realm of complexity theory and obtain stronger and more general versions of these theorems.

Lean into Theorem Proving

Speakers: Janna Withrow, Mark Pekala, William Hu, and Peter Chon

Abstract: The programming language Lean has ushered in a new era in formal mathematics, providing a robust platform for validating proofs with exceptional reliability. Grounded in a rigorous, type-theoretic framework, Lean empowers mathematicians to engage with avant-garde mathematical inquiries and experiment with innovative proof methodologies. In our presentation, we will illuminate the mechanics of Lean and its myriad applications, emphasizing its potential to elevate our comprehension of intricate mathematical notions. We will showcase its versatility in formalizing diverse mathematical structures by highlighting renowned theorems that have been meticulously verified using Lean. Furthermore, we will exemplify its capacity to unveil fresh insights, as illustrated by Peter Scholze's recent contributions. Lastly, we will recount our firsthand experiences with employing Lean in Math 161, highlighting its merit as a pedagogical instrument for cultivating a deeper understanding of proof theory. We hope you will join us as we delve into the unique opportunities offered by Lean to examine, authenticate, and intensify our grasp of mathematics.

Non-compact Calabi-Yau manifolds from Calabi-Yau cones

Speaker: Benjy Firester

Abstract: Calabi-Yau manifolds are complex Kähler manifolds satisfying algebraic criteria that implies the existence of a unique Ricci-flat metric (in the compact case), which is the so-called the Calabi Conjecture as proven by Yau, giving rise to the eponymous name. Yau’s proof supplies a large class of interesting geometries, including most known examples of Einstein manifolds, new tools in string theory, and many geometric and topological corollaries, however the abstract existence technique means we have very little understanding of explicit formulas for the metric. In the non-compact case, existence and uniqueness is rarely known, but we can better understand the geometry of Calabi-Yau metrics. In this talk, I will discuss recent constructions of complete non-compact Calabi-Yau manifolds with motivations to understand compact manifolds, especially in degenerate limits. 

Quaternions and abcs

Speaker: Raphael Tsiamis

Abstract: Can you solve a 3x3 equation that encodes quaternions into differential geometry?

Quaternions have three imaginary units i,j,k; they are, to complex numbers, what complex numbers are to the reals. These are the building blocks for smooth and complex manifolds. The next step are hyperkähler manifolds, resembling the quaternions; they are full of interesting properties, but difficult to construct. Kronheimer identified such structures coming from differential equations. In a simple case, studying the solutions of a 3x3 equation reveals the geometry of a hyperkähler manifold through interesting pictures.

Graduate School Panel Discussion

Please join us for a panel discussion about graduate school! We'll cover questions about the application process, how to choose a graduate program, planning courses that help with grad school, and more!

Panelists:

• Daniel Abdulah, MIT Planetary Science

• Anne Larsen, MIT Mathematics

• Lucy Liu, Harvard Applied Mathematics

• Phillip Nicol, Harvard Biostatistics

Explicit Formulas for Permutation Pattern Character Polynomials

Speaker: Jonas Iskander

Abstract: Given a permutation σ on k letters and a permutation π on n letters, one can ask how many times the ordering pattern σ occurs in substrings of π. For fixed σ, Janson, Nakamura, and Zeilberger showed that the number of pattern occurrences in a uniformly random permutation π follows a normal distribution. However, it is natural to ask how the distribution of numbers of pattern occurrences interacts with the group structure on permutations. Recently, Christian Gaetz, Laura Pierson, and Christopher Ryba introduced the family of permutation pattern character polynomials, which measure the expected value of the number of pattern occurrences times a symmetric group character. In this talk, I will describe how I extended their work to derive explicit formulas for certain permutation pattern character polynomials.

The Inhomogeneous TASEP and Evil-Avoiding Permutations

Speaker: Katherine Tung

Abstract: The asymmetric simple exclusion process (ASEP) is a family of models for interacting particles. A particular type of ASEP -- the inhomogeneous TASEP -- is only partly understood, and is related to Schubert polynomials and a family of permutations called "evil-avoiding." In this talk, I will discuss the inhomogeneous TASEP, pattern-avoiding permutations, and some related research I did last summer at the Duluth REU.