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Summer Workshop for Intrepid Mathematicians
SWIM 2025
A series of accessible talks given primarily by high-school students, undergraduates, and early-career graduate students. Speakers present their research, develop their talks with feedback from the organizers, and strengthen their expository skills. The workshop ran during the last six weeks of summer, right before the new academic year.
Announcement
CrowdMath 2025 — Fall Term has begun
CrowdMath is a large-scale online collaborative research project where high-school and college students work together on open problems in real time, sharing ideas and building on each other’s work. Learn about past projects →
Organizers
Schedule
Click a talk to read its abstract
August 04Monday
Learning Probability Distributions with Variational Autoencoders: Theory, Geometry, and Applications
Willy Rodriguez · Torus
Learning Probability Distributions with Variational Autoencoders: Theory, Geometry, and Applications
Willy Rodriguez · Torus
Abstract
Variational Autoencoders (VAEs) are a class of deep generative models that combine principles from variational inference, latent variable modeling, and neural networks. In this talk, we explore VAEs through a mathematical lens, presenting the theoretical foundations that underpin their architecture: variational inference, the Evidence Lower Bound (ELBO), and the geometry of the latent space. We will examine the role of Kullback–Leibler divergence (KL-divergence) in regularizing the latent representation, and introduce the reparametrization trick that makes gradient-based optimization feasible. Beyond the theory, we will discuss how VAEs can be leveraged for tasks such as representation learning and data classification, with illustrative examples. Throughout the presentation, we aim to highlight the interplay between probabilistic modeling, optimization, and geometry that makes VAEs a powerful tool in both theory and practice.
August 05Tuesday
The Goldbach Property in Semidomains
Victor Gonzalez · CrowdMath
The Goldbach Property in Semidomains
Victor Gonzalez · CrowdMath
Abstract
In this talk, we will introduce the notion of commutative semirings, which are, roughly speaking, commutative rings without the requirement that every element has an additive inverse. Our central algebraic objects are the homomorphic images of the polynomial semiring \(\mathbb{N}_0[x]\) we obtain when we evaluate polynomials at rational numbers: we call such semirings rational cyclic semidomains. We discuss some analogs of the statement of the Goldbach conjecture in connection with rational cyclic semidomains and their localizations.
August 07Thursday
On the Additive Structure of Cyclic Semirings: The Size of Their Sets of Atoms and Strong Atoms
Anna Deng, Jason Zeng · MIT CMI
On the Additive Structure of Cyclic Semirings: The Size of Their Sets of Atoms and Strong Atoms
Anna Deng, Jason Zeng · MIT CMI
Abstract
For an algebraic number \(a\), we let \(M_a\) denote the underlying additive monoid of the subsemiring \(\mathbb{N}_0[a]\) of the field of complex numbers, which is a cyclic semiring. A non-zero element of \(M_a\) is called an atom if it cannot be written as the sum of two non-invertible elements, while a given atom of \(M_a\) is called a strong atom if whenever we add any finite number of copies of the given atom, the element we obtain can be written as a sum of atoms in a unique way (the obvious one). In this talk, we will discuss some aspects related to the atomicity of the monoids \(M_a\) for algebraic numbers \(a\). We put special emphasis on the problem of determining the pairs \((m,n)\) of integers for which there exists an algebraic number \(a\) such that \(m\) is the number of strong atoms of \(M_a\) and \(n\) is the number of atoms of \(M_a\).
August 12Tuesday
On the Multiplicative Structure of Cyclic Semirings
Jiya Dani, Bryan Li, Arav Paladiya · MIT CMI
On the Multiplicative Structure of Cyclic Semirings
Jiya Dani, Bryan Li, Arav Paladiya · MIT CMI
Abstract
For an algebraic number \(a\), the set \(\mathbb{N}_0[a] := \{ p(a) : p(x) \in \mathbb{N}_0[x] \}\) is a subsemiring of the field of complex numbers, which is called the cyclic semidomain/semiring generated by \(a\). In this talk, we will discuss the group of units of the semidomain \(\mathbb{N}_0[a]\) as well as some aspects of its factorization behavior, including the classical unique factorization (UF) property (which is an abstraction of the Fundamental Theorem of Arithmetic) and the finite factorization property (which is a natural generalization of the UF property that was introduced and first studied by Anderson, Anderson, and Zafrullah back in 1990).
August 14Thursday
Transformer-Based Embeddings for Topic Coherence
Alex Ding, Tarun Rapaka · PRIMES
Transformer-Based Embeddings for Topic Coherence
Alex Ding, Tarun Rapaka · PRIMES
Abstract
Text is messy and hard to work with directly. Our plan is to turn a large collection of documents into simple math objects that a computer can handle. First, we split the text into tokens (the words or pieces of words) and remove stop words like “the” and “and.” Then we map the remaining text to vectors, which are just lists of numbers. Once everything is numbers, we can measure similarity, group similar items, and summarize groups as topics. We will show this pipeline in news articles and biomedical abstracts. After that, we will ask whether we really need very large models or if smaller ones already give clear, useful topics.
August 15Friday
On the MCD and MCD-Finite Properties
Darren Han, Hengrui Liang · PRIMES
On the MCD and MCD-Finite Properties
Darren Han, Hengrui Liang · PRIMES
Abstract
Let \(M\) be a commutative monoid. A common divisor \(d\) of a nonempty subset \(S\) of \(M\) is called a maximal common divisor (MCD) if the only elements of \(M\) that are common divisors of the set \(\{s/d : s \in S\}\) are the units of \(M\). The monoid \(M\) is called an MCD monoid if every nonempty finite subset of \(M\) has at least an MCD, while \(M\) is called an MCD-finite monoid if every nonempty finite subset of \(M\) has only finitely many MCDs (up to associates). In our talk we will discuss the MCD and MCD-finite properties in the setting of monoid algebras and power monoids of torsion-free commutative monoids.
August 19Tuesday
The Valuation Property on the Additive Structure of Cyclic Semirings
Timothy Chen, Tony Lu, Alan Yao · PRIMES
The Valuation Property on the Additive Structure of Cyclic Semirings
Timothy Chen, Tony Lu, Alan Yao · PRIMES
Abstract
Let \(\mathbb{N}_0[x]\) be the semiring consisting of polynomials with nonnegative integer coefficients. For each \(\alpha \in \mathbb{C}\), \(\mathbb{N}_0[\alpha]\) denotes the cyclic semiring generated by \(\alpha\), the image of the semiring homomorphism \(\mathbb{N}_0[x] \to \mathbb{C}\) by evaluation at \(\alpha\). We provide an algebraic characterization of the \(\alpha\) for which the underlying additive monoid \(M_\alpha\) has the valuation property and relate it to other well-studied conditions. For instance, any antimatter \(M_\alpha\) decomposes as the product of finitely many valuation monoids. As a consequence, we show that the valuation monoids are abundant: the set of \(\alpha\) yielding valuation \(M_\alpha\) is dense in \((0,1)\).
August 21Thursday
AI-Based Histopathology for Lung Cancer Diagnosis
Omar Graia, Katrina Liu, Audrey Wang · PRIMES
AI-Based Histopathology for Lung Cancer Diagnosis
Omar Graia, Katrina Liu, Audrey Wang · PRIMES
Abstract
Histopathology is central to lung cancer diagnosis, but traditional manual assessments are time-consuming and prone to variability. In this presentation, we explore how artificial intelligence can support pathologists by analyzing lung tissue images. Using the LungHist700 dataset and validating on LC25000, we compare approaches ranging from logistic regression and random forests to deep convolutional neural networks. We examine four themes: cross-dataset robustness, classification of adenocarcinoma subtypes, the role of multi-scale learning, and insights from explainable AI methods such as SHAP. Our results highlight both the promise and limitations of current models, showing how magnification, feature design, and interpretability affect performance. We conclude with lessons on balancing accuracy, generalizability, and transparency in AI-based histopathology.
August 26Tuesday
Chip-Firing on Infinite k-ary trees
Selena Ge, Dohun Kim, et al. · MIT PRIMES Step
Chip-Firing on Infinite k-ary trees
Selena Ge, Dohun Kim, et al. · MIT PRIMES Step
Abstract
We use an infinite k-ary tree with a self-loop at the root as our underlying graph and consider a chip-firing process starting with \(N\) chips at the root. One of our goals is to describe the stable configurations and calculate the number of fires for each vertex and the total number of fires. We also study a sequence of the number of root fires for a given \(k\) as a function of \(N\) and study its properties. We do the same for the total number of fires.
August 28Thursday
Prime Element Stability in Ring Extensions
Bofan Liu, Seabert Mao, Michael Zhao · PRIMES
Prime Element Stability in Ring Extensions
Bofan Liu, Seabert Mao, Michael Zhao · PRIMES
Abstract
The behavior of prime elements under ring extensions is a fundamental question in commutative algebra. Given an extension of domains \(R \subseteq T\) and a prime element \(p\) of \(R\), we identify conditions under which \(p\) remains prime in intermediate rings. Assuming that \(p\) is prime in \(T\), we prove that this holds whenever \(T\) is an integral overring of a one dimensional ring \(R\). Furthermore, we show that if \(p\) is coprime to the conductor of the extension \(R \subseteq T\), then \(p\) remains prime in \(T\) and all intermediate rings. Finally, with the help of a result on prime behavior in minimal extensions, we prove that this prime stability holds for any extension satisfying the FCP condition.
August 29Friday
On the Atomic Density of Polynomials with Integer Coefficients
Neil Kolekar, Maiya Qiu, Richard Wang · PRIMES
On the Atomic Density of Polynomials with Integer Coefficients
Neil Kolekar, Maiya Qiu, Richard Wang · PRIMES
Abstract
In this talk, we discuss the asymptotic density of irreducible elements in the context of polynomial semidomains. In particular, we show that, in a precise asymptotic sense, almost every polynomial in \(\mathbb{N}_0[x]\) is irreducible.
Participants
40 registered
Leo AbediMathRoots
Anay AggarwalMIT PRIMES
Grant BlitzMIT PRIMES
Olivia ChenMIT CMI
Timothy ChenMIT PRIMES
Jim CoykendalClemson University
Jiya DaniMIT CMI
Anna DengMIT CMI
Alex DingMIT PRIMES
Jonathan DuMIT PRIMES
Leyanis FalconClemson University
Victor GonzalezMIT
Felix GottiMIT
Marly GottiArcus Biosciences
Omar GraiaMIT PRIMES
Darren HangMIT PRIMES
Leo HongMIT PRIMES
Jared KettingerClemson University
Harry KimMIT
Neil KolekarMIT PRIMES
Aarush KulkarniCrowdMath
Bryan LiMIT CMI
Hengrui LiangMIT PRIMES
Bofan LiuMIT PRIMES
Katrina LiuMIT PRIMES
Guanjie LuMIT PRIMES
Tony LuMIT PRIMES
Seabert MaoMIT PRIMES
Advaith MopuriMIT PRIMES
Arav PaladiyaMIT CMI
Tanish ParidaPRIMES Step
Harold PoloUCI Irvine
Maiya QiuMIT PRIMES
Tarun RapakaMIT PRIMES
Pedro RodriguezClemson University
Willy RodriguezTorus
Audrey WangMIT PRIMES
Jessica WuMIT CMI
Alan YaoMIT PRIMES
Jason ZengMIT CMI