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Summer Workshop for Intrepid Mathematicians
SWIM 2026
A series of accessible research talks presented primarily by high school students, undergraduate students, and early-career graduate students. Speakers develop their presentations with feedback from the organizers, providing an opportunity to strengthen both their communication and expository skills. The workshop will run from August 3 to August 27. Talks will be held every Tuesday and Thursday at 6:00 p.m. ET, with the opening talk taking place on Monday, August 3. To join, please use the following Zoom link: https://mit.zoom.us/j/93323362619.
Organizers
Schedule
Click a talk to read its abstract
August 03Monday
LLMs, Matheo, and the Mathematical Research Workflow
Harold Polo · Clemson University
LLMs, Matheo, and the Mathematical Research Workflow
Harold Polo · Clemson University
Abstract
Large language models can accelerate mathematical research, but only when paired with careful verification. Through demonstrations of the custom GPT Matheo, this talk presents practical methods for using AI to improve mathematical writing, explore ideas, and support research while maintaining the standards of mathematical rigor.
August 04Tuesday
When the Computer Says QED: A Friendly Introduction to the Lean Theorem Prover
Marly Gotti · iMathLab Co-Founder
When the Computer Says QED: A Friendly Introduction to the Lean Theorem Prover
Marly Gotti · iMathLab Co-Founder
Abstract
LLMs changed the world almost overnight. They leapt from chatbots that answered questions to agents capable of writing and debugging software, searching vast bodies of literature, coordinating complex workflows, and completing in hours tasks that once demanded days or weeks. Their impact has spread so quickly that, only a few days ago, a seventy-year-old friend asked me to install Gemini on her phone.
Lean remains unimpressed.
One system daydreams; the other refuses to be fooled. It demands that every dream be built, piece by piece, from thousands upon thousands of tiny, verifiable truths.
This is a talk about the future. 🚀
August 11Tuesday
When the Machine and Paper Disagree: Who is Right?
Jonathan Liu, Jason Yang, Alan Yao · MIT CMI
When the Machine and Paper Disagree: Who is Right?
Jonathan Liu, Jason Yang, Alan Yao · MIT CMI
Abstract
Can a computer catch mathematical errors that humans miss? In this talk, we explore this question through the Frobenius problem for numerical semigroups and the use of the Lean proof assistant. We develop a computational framework based on Apéry sets to calculate Frobenius numbers and related invariants, and use it alongside GAP to check results from the mathematical literature. Our investigation verifies numerous published computations while also uncovering several discrepancies, illustrating how formal verification and computer-assisted mathematics can complement traditional mathematical reasoning and peer review.
Building on these themes of semigroups, computation, and mathematical verification, the final part of the talk explores a complementary question about factorization, where the interplay between addition and multiplication reveals striking algebraic rigidity. What happens when unique factorization is required under both addition and multiplication? In this final part, we talk about the Bi-UFS Positive Conjecture for algebraic monogenic semidomains of the form \(\mathbb{N}_0[\alpha]\), where \(\alpha\) is a positive algebraic number. We prove that \(\mathbb{N}_0[\alpha]\) can have unique factorization in both its additive and multiplicative structures if and only if \(\mathbb{N}_0[\alpha] = \mathbb{N}_0\). Our approach combines structural restrictions on additive atoms with multiplicative divisibility arguments: in the quadratic case, these reduce the problem to two exceptional candidates, while in higher degrees, constraints on the minimal polynomial lead to contradictions with multiplicative unique factorization. These results illustrate how the interaction between additive and multiplicative factorization can impose strong rigidity on algebraic semidomains.
Building on these themes of semigroups, computation, and mathematical verification, the final part of the talk explores a complementary question about factorization, where the interplay between addition and multiplication reveals striking algebraic rigidity. What happens when unique factorization is required under both addition and multiplication? In this final part, we talk about the Bi-UFS Positive Conjecture for algebraic monogenic semidomains of the form \(\mathbb{N}_0[\alpha]\), where \(\alpha\) is a positive algebraic number. We prove that \(\mathbb{N}_0[\alpha]\) can have unique factorization in both its additive and multiplicative structures if and only if \(\mathbb{N}_0[\alpha] = \mathbb{N}_0\). Our approach combines structural restrictions on additive atoms with multiplicative divisibility arguments: in the quadratic case, these reduce the problem to two exceptional candidates, while in higher degrees, constraints on the minimal polynomial lead to contradictions with multiplicative unique factorization. These results illustrate how the interaction between additive and multiplicative factorization can impose strong rigidity on algebraic semidomains.
August 13Thursday
On the Goldbach Property to Semidomains & On the Ascent of the Goldbach Property to Semidomains of Laurent Series
Hengrui Liang, Aarush Kulkarni · MIT CMI/CrowdMath & MIT PRIMES
On the Goldbach Property to Semidomains & On the Ascent of the Goldbach Property to Semidomains of Laurent Series
Hengrui Liang, Aarush Kulkarni · MIT CMI/CrowdMath & MIT PRIMES
Abstract
A semidomain is an integral domain in which additive inverses are no longer required. We begin by investigating Goldbach-like properties in the setting of the monogenic semidomains \(\mathbb{N}_0[\alpha]\), where \(\alpha\) is a positive algebraic number. We say that a semidomain \(S\) has the weak finitary Goldbach property if there exists an \(\ell \in \mathbb{N}\) such that every element in \(S\), except for those belonging to finitely many associate classes, can be written as a sum of at most \(\ell\) irreducibles of \(S\). In the rational case, we give a complete characterization of \(\alpha\) for which the weak finitary Goldbach property holds on \(\mathbb{N}_0[\alpha]\). Then, for general algebraic \(\alpha\), we show that the weak finitary Goldbach property imposes strong restrictions on \(\alpha\), including that \(\alpha\) must be a multiplicative unit. Finally, we show that the converse almost always holds.
Building on this investigation of Goldbach-like properties within semidomains, we next turn to a complementary question: how do such properties behave when we pass from a semidomain to larger algebraic constructions over it? To address this question, we consider additively reduced semidomains, that is, semidomains \(S\) for which \(0\) is the only invertible element of the monoid \((S,+)\). We call \(S\) Goldbach if \(2s\) is the sum of two multiplicative irreducibles for every nonzero nonunit \(s \in S\), and strong Goldbach if every nonzero nonunit of \(S\) is such a sum. We study the behavior of these properties under the ascension to semidomains of Laurent polynomials and Laurent series. In particular, for an additively reduced and additively Furstenberg semidomain \(S\), we characterize when every nonzero non-monomial Laurent series in \(S[x^{\pm 1}]\) is the sum of at most two irreducibles, settling a conjecture of Kaplan and Polo first proposed in 2023.
Building on this investigation of Goldbach-like properties within semidomains, we next turn to a complementary question: how do such properties behave when we pass from a semidomain to larger algebraic constructions over it? To address this question, we consider additively reduced semidomains, that is, semidomains \(S\) for which \(0\) is the only invertible element of the monoid \((S,+)\). We call \(S\) Goldbach if \(2s\) is the sum of two multiplicative irreducibles for every nonzero nonunit \(s \in S\), and strong Goldbach if every nonzero nonunit of \(S\) is such a sum. We study the behavior of these properties under the ascension to semidomains of Laurent polynomials and Laurent series. In particular, for an additively reduced and additively Furstenberg semidomain \(S\), we characterize when every nonzero non-monomial Laurent series in \(S[x^{\pm 1}]\) is the sum of at most two irreducibles, settling a conjecture of Kaplan and Polo first proposed in 2023.
August 18Tuesday
v-Noetherian and Krull Monoids of Algebraic Evaluations
Aaditya Bilakanti, Amrit Kandasamy · MIT CMI
v-Noetherian and Krull Monoids of Algebraic Evaluations
Aaditya Bilakanti, Amrit Kandasamy · MIT CMI
Abstract
We study the arithmetic and algebraic structure of evaluation monoids generated by positive real algebraic numbers. We show two main results. First, the monoid \(\mathbb{N}_0[\alpha_1, \dots, \alpha_n]\) is v-Noetherian (Mori) if and only if they are finitely generated. Second, cyclic monoids of the form \(\mathbb{N}_0[\alpha]\) are Krull if and only if they are finitely generated and integer divisor closed in \(\mathbb{Z}[\alpha]\). Furthermore, we present terminating algorithms to test both finite generation and the Krull property.
August 20Thursday
Cognitive Decline Prediction in OASIS-3
Tanish Parida, Gary Shen, Melody Wu · MIT PRIMES
Cognitive Decline Prediction in OASIS-3
Tanish Parida, Gary Shen, Melody Wu · MIT PRIMES
Abstract
Cognitive decline is a widespread issue in the world today, and it is often recognized only after it has begun to affect daily life. We wanted to find out if routinely available information from an older adult’s first clinical visit can identify those at risk of meaningful decline in the next two to four years. Using longitudinal data from 458 adults aged 60 and older in the OASIS-3 cohort, we compared logistic regression, random forest, and gradient-boosting models using demographic, clinical, and cognitive measures. All models outperformed chance, with a clinical-and-cognitive logistic regression model performing best, indicating baseline clinical status as especially informative. These results establish a practical non-imaging benchmark for determining whether more complex MRI-based models may offer meaningful additional value.
August 25Tuesday
Directional Curvature-Rescaled Optimization for Word2Vec Training
Aarush Kulkarni, Kris Sheng, James Wu · MIT PRIMES
August 27Thursday
A Solution to the Bi-UF Positive Conjecture
Omar Graia · CrowdMath