Summer Workshop for Intrepid Mathematicians

SWIM 2024

Summer 20247 talks · via ZoomArchived

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.

Organizers

Felix GottiMIT Marly GottiApple Harold PoloUC Irvine

Schedule

Click a talk to read its abstract

August 05Monday

On the Ascent of Atomicity to Monoid Algebras

Felix Gotti · MIT

Abstract

Given a submonoid \(M\) of a torsion-free abelian group and a commutative ring \(R\), the monoid algebra of \(M\) over \(R\), denoted by \(R[M]\), is the commutative ring consisting of all polynomial expressions with coefficients in \(R\) and exponents in \(M\), with addition and multiplication defined as for polynomials. When \(R\) is an integral domain, \(R[M]\) is also an integral domain. A commutative monoid (resp., an integral domain) is called atomic provided that each nonunit (resp., nonzero nonunit) factors into atoms (i.e., irreducibles). In this talk, we will discuss some progress on the ascent of atomicity carried out in recent years by PRIMES and CrowdMath participants.

August 08Thursday

On Finitary Power Monoids of Linearly Orderable Monoids

Jiya Dani, Leo Hong · MIT PRIMES

Abstract

Let \(M\) be a linearly ordered monoid (i.e., a commutative monoid along with a total order relation compatible with its operation). The (finitary) power monoid \(P(M)\) of \(M\) is the commutative monoid consisting of all nonempty finite subsets of \(M\) under the Minkowski sum (also called sumset), that is, \(S+T := \{s+t : s \in S \text{ and } t \in T\}\) for any nonempty finite subsets \(S\) and \(T\) of \(M\). In this talk (which is based on our current PRIMES research project), we will discuss our recent findings on factorizations in power monoids of linearly ordered monoids.

August 12Monday

Comparative Analysis of Machine Learning Models for Thyroid Cancer Recurrence Prediction

Anay Aggarwal, Ekam Kaur, Susie Lu · MIT PRIMES

Abstract

In this presentation, we will explore the effectiveness of various machine learning algorithms in predicting the recurrence of Differentiated Thyroid Cancer (DTC). We have analyzed models such as Artificial Neural Networks, K-Nearest Neighbors, Support Vector Machines, Logistic Regression, Extreme Gradient Boosting, and Random Forests using a robust dataset from the UCI Machine Learning Repository. By comparing their performance, our goal is to identify the most promising algorithm for enhancing personalized treatment strategies and improving outcomes in thyroid cancer care. This work is part of our 2024 PRIMES project.

August 15Thursday

A Goldbach Theorem for Group Semidomains

Eddy Li, Advaith Mopuri, Charles Zhang · MIT PRIMES

Abstract

A semidomain is an integral domain that does not require additive inverses. Given a semidomain \(S\) and a torsion-free group \(G\), the group semidomain \(S[G]\) consists of all polynomials with coefficients in \(S\) and exponents in \(G\). Similarly, the group series semidomain \(S[[G]]\) consists of all power series with coefficients in \(S\) and exponents in \(G\). In our talk, we will discuss a variant of the Goldbach conjecture over group semidomains \(S[G]\) and a variant of the weak Goldbach conjecture over group series semidomains \(S[[G]]\) for semidomains and groups satisfying certain properties. The results discussed in this talk is part of our 2024 PRIMES research project.

August 19Monday

The Davenport Constant and Automorphically Equivalent Elements

Arjun Agarwal, Rachel Chen, Rohan Garg · MIT PRIMES

Abstract

Let \(G\) be a finite abelian group and define \(D(G)\) to be the Davenport Constant of the group. In this paper we demonstrate several bounds on the Davenport constant of the group. We also investigate whether \(D(G)\), along with other numerical invariants of the group, is sufficient to uniquely determine its structure. Our investigations lead us to a conjecture that relates the divisibility of the Davenport constant of the subgroup to the structure of the group. We also study the inverse Davenport problem– the structure of maximal 0-sequences of length \(D(G)\). The structure of these sequences motivates the study of the necessary and sufficient conditions for two elements \(x\), \(y \in G\) to be automorphic images of each other. We finally prove that there exists \(\phi \in \text{Aut}(G)\) such that \(\phi(x) = y\) if and only if \(G/<x>\) is isomorphic to \(G/<y>\). This result leads to the development of the fastest known algorithm to determine if two elements of a finite abelian group are automorphic images of each other.

August 22Thursday

On Monoids and Rings from Gauss’s Lemma

Victor Gonzalez, Ishan Panpaliya · CrowdMath

Abstract

Following the terminology introduced by Arnold and Sheldon, we say that an integral domain \(R\) is a GL-domain or has the GL-property if the product of any two primitive polynomials over \(R\) is a primitive polynomial (also over \(R\)), which means that \(R\) satisfies the statement of Gauss’s Lemma (this explains the `GL’ in the chosen terminology). The GL-property was characterized by Anderson and Quintero in pure multiplicative terms. Such a characterization allows us to consider the GL-property in the setting of commutative monoids. In this talk, we will discuss the GL-property not only in integral domains but also in the more general setting of commutative monoids. Among other preliminary results, we offer another purely multiplicative characterization of the GL-property. Also, we will discuss the ascent of the GL property to monoid algebras and, finally, we will answer a question posed by Felix Gotti and Mohammad Zafrullah on the ascent of the idf-property.

August 26Monday

On the Internal Sum of Positive Monoids

Jonathan Du, Bryan Li, Nick Zhang · MIT PRIMES

Abstract

In this talk, we will discuss arithmetic and factorization properties of the internal (finite) sum of submonoids of rank-\(1\) torsion-free abelian groups. The main properties we will see are Cohn’s notion of atomicity and the classical bounded and finite factorization properties introduced and studied in 1990 by Anderson, Anderson, and Zafrullah in the setting of integral domains, and then generalized by Halter-Koch to commutative monoids. We pay especial attention to how each of the considered properties behaves under the internal sum with a finitely generated monoid. We will also present several examples to illustrate that our primary results cannot be strengthened and, in particular, that they do not hold for submonoids of torsion-free abelian groups with rank larger than one.

Participants

33 registered

Anay AggarwalMIT PRIMES

Arjun AggarwalMIT PRIMES

Rachel ChenMIT PRIMES

Jim CoykendallClemson University

Jiya DaniMIT PRIMES

Jonathan DuMIT PRIMES

Leyanis FalconUniversity of Havana

Rohan GargMIT PRIMES

Victor GonzalezMiami Dade College

Felix GottiMIT

Marly GottiApple

Leo HongMIT PRIMES

Henry JiangMIT PRIMES

Ekam KaurMIT PRIMES

Jared KettingerClemson University

Harry KimMIT PRIMES

Jason LeeMIT PRIMES

Benjamin LiMIT

Bryan LiMIT PRIMES

Eddy LiMIT PRIMES

Evin LiangMIT PRIMES

Sophia LiaoMIT PRIMES

Susie LuMIT PRIMES

Advaith MopuriMIT PRIMES

Ishan PanpaliyaCrowdMath

Harold PoloUC Irvine

Louis QiuCrowdMath

Henrick RabinovitzMIT

Pedro RodriguezClemson University

Willy RodriguezUniversity of Toulouse

Atticus StewartCrowdMath

Charles ZhangMIT PRIMES

Nick ZhangMIT PRIMES