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Summer Workshop for Intrepid Mathematicians
SWIM 2023
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
Schedule
Click a talk to read its abstract
August 02Wednesday
Numerical-valued polynomials
Harold Polo · UC Irvine
Numerical-valued polynomials
Harold Polo · UC Irvine
Abstract
A numerical monoid is a cofinite additive submonoid of the nonnegative integers, while a numerical-valued polynomial is a polynomial with rational coefficients that maps a numerical monoid \(N\) into \(\mathbb{N}\). In this talk, we will discuss the algebraic and factorization properties of numerical-valued polynomials.
August 09Wednesday
Goldbach conjecture for Laurent polynomials with positive integer coefficients
Sophia Liao · MIT PRIMES
Goldbach conjecture for Laurent polynomials with positive integer coefficients
Sophia Liao · MIT PRIMES
Abstract
The Goldbach conjecture posits that every even natural number can be written as the sum of two primes. While it has neither been proven true or false, variations of this problem have been solved: In particular, in polynomial extensions of certain rings and fields, it has been shown that elements can be written as the sum of two irreducible polynomials. In this talk, we discuss a proof of this modified Goldbach conjecture over Laurent polynomials of positive coefficients.
August 16Wednesday
Divisibility and a weak ascending chain condition on principal ideals
Bangzheng Li · MIT
Divisibility and a weak ascending chain condition on principal ideals
Bangzheng Li · MIT
Abstract
An integral domain \(R\) is atomic if each nonzero nonunit of \(R\) factors into irreducibles. In addition, an integral domain \(R\) satisfies the ascending chain condition on principal ideals (ACCP) if every increasing sequence of principal ideals (under inclusion) becomes constant from one point on. Although every integral domain satisfying ACCP is atomic, examples of atomic domains that do not satisfy ACCP are notoriously hard to construct. The first of such examples was constructed by A. Grams back in 1974. In this talk we will discuss the class of atomic domains that do not satisfy ACCP. We will generalize Grams’ construction and talk about the notion of weak-ACCP, a condition weaker than ACCP that was recently introduced in a joint paper with Felix Gotti.
August 23Wednesday
On a weaker notion of the finite factorization property
Henry Jiang, Shihan Kanungo, Harry Kim · MIT PRIMES
On a weaker notion of the finite factorization property
Henry Jiang, Shihan Kanungo, Harry Kim · MIT PRIMES
Abstract
Following Geroldinger and Zhong, we say that an atomic monoid \(M\) is a length-finite factorization monoid if each \(b\) in \(M\) has only finitely many factorizations of any prescribed length. An additive submonoid consisting of nonnegative real numbers is called a positive monoid. Factorizations in positive monoids have been actively studied in recent years. The main purpose of this talk is to give a better understanding of the non-unique factorization phenomenon in positive monoids through the lens of the length-finite factorization property. We will discuss a large class of positive monoids which satisfy the length-finite factorization property. Then we will compare the length-finite factorization property to the bounded and the finite factorization properties, which are two properties that have been systematically investigated for more than thirty years.
August 30Wednesday
Recent open problems in factorization theory
Felix Gotti · MIT
Recent open problems in factorization theory
Felix Gotti · MIT
Abstract
In this talk we will discuss about ten recent open problems in atomicity and factorization theory. This list of open problems is designed for the participants of the summer workshop.
Participants
23 registered
Juliet BringasUniversity of Paris
Ernesto FerrerUniversity of Havana
Agastya GoelMIT PRIMES
Víctor GonzalezMiami Dade College
Felix GottiMIT
Marly GottiApple
Henry JiangMIT PRIMES
Aditya KamurUniversity of Florida
Shihan KanungoMIT PRIMES
Harry KimMIT PRIMES
Benjamin LiMIT
Sophia LiaoMIT PRIMES
Shi LuUniversity of Florida
Jason MaoMIT PRIMES
Skyler MaoMIT PRIMES
Harold PoloUC Irvine
Henrick RabinovitzMIT PRIMES
Pedro RodriguezUniversity of Havana
Miguel SecillanoUniversity of Florida
Easton SingerHarvard University
Marcos TiradorUniversity of Havana
Joseph VulakhMIT PRIMES
Tiger ZhangMIT PRIMES