GILAD MOSKOWITZ

MATHEMATICIAN, PROGRAMMER, AVID FAN OF TABLE-TOP GAMES

About Me



I completed my undergraduate education at UCLA where I got my B.S. in Chemistry and B.S. in Mathematics. I then took a small break from education to work and travel the world. I returned to school to pursue an M.S. in Applied Mathematics with a Concentration in Communication Systems at SDSU. My primary mathematic interests are number theory, abstract algebra, and combinatorics. I also enjoy applying machine learning and other algorithmic approaches to attempt to solve difficult math problems. For more information on my educational and mathematical background you can check out my resume.

In my free time, I enjoy playing a variety of tabletop games, such as Settlers of Catan (with the Cities and Knights expansion, of course), chess, and D&D. I also love playing basketball and some video games.

My Mathematical Interests

 Most of my mathematical research involves the study of numerical semigroups
\begin{defn} A numerical semigroup is a subsemigroup of $(\!ℕ_0, +\!).$ \end{defn}
We often write a numerical semigroup as $S = \langle\!n_1, n_2, ..., n_k \!\rangle$ for $n_j \in ℕ$ and $n_j < n_{j+1}.$ The elements of $S$ are then the non-negative integers that can be written as linear combinations of the generators of $S$.
\begin{defn} A semigroup is a set, $S,$ with an associative binary operator, $+.$ \end{defn}
and their properties. Particularly, the properties that I study are the factorizations,
\begin{defn} Let $S = \langle n_1, n_2, \dots, n_k \rangle$ be a numerical semigroup. We say that $\mathbf{p} = (p_1, p_2, ..., p_k)$ is a \emph{factorization} of $n \in S$ if \[n = p_1n_1 + p_2n_2 + ... + p_kn_k.\] We say that this factorization has \emph{length} $|$ $\mathbf{p}| = p_1 + p_2 + ... + p_k$. \end{defn} \begin{defn} The \emph{factorization set} of an integer $n \in S$ is \[\mathsf{Z}_S(n) = \{(p_1, p_2, ..., p_k) : n = p_1n_1 + p_2n_2 + ... + p_kn_k\}.\] When it is clear which semigroup we are referring to, we simply write $\mathsf{Z}_S(n)$. \end{defn}
length sets,
\begin{defn} The \emph{length set} of an integer $n \in S$ is \[\mathsf{L}_S(n) = \{|\mathbf{p}| : \mathbf{p} = (p_1, p_2, ..., p_k) \in \mathsf{Z}_S(n) \}.\] When there is no ambiguity, we simply write $\mathsf{L}(n)$. \end{defn}
and gaps.
\begin{defn} Let $S$ be a numerical semigroup, then we define the gaps of $S$ as the set \[\{i : i \in ℕ_0 \mbox{ and } i \notin S\}.\] \end{defn}
I have worked several projects that were centered around the factorization set and length set of a numerical semigroup.
 Currently, I have shifted my focus to examine the gaps of a numerical semigroup. This project is centered on the game Sylver Coinage, and attempts to tackle the problem with the lens of a numerical semigroup. I am actively building towards a competition where the competitors submit bots to play the game Sylver Coinage, and the winner is the bot with the best overall score. This project has required me to develop serveral pieces of code for working with numerical semigroups in Javascript and Python1.
 Beyond numerical semigroups, I have also explored other areas of mathematics, including toying around with factorization algorithms, using machine learning to try and predict properties of the integers, and a few cryptography related projects. Additionally, I truly enjoy teaching mathematics and am in the process of writting up notes to introduce students to proof based mathematics. I hope one day to put together an introductory Algebra textbook for elementry school students, but I know that this will be a long and timely endeavor.

World Travels

 I love to travel and hope to one day explore the entire world. I believe that learning about the culture and values of different countries is extremely important and enjoyable. I have been very fortunate in my life to be able to travel to at least some parts of the world, but I have a lot more to explore. My next big travel destination will be Japan, where I hope to find a nice small apartment to rent for a little while (a month or two) and really experience the life there.

Taken at Stonehenge during one of my trips to the UK.
Taken at Osteria Francescana in Modena, Italy.
Taken in Jerusalem, Israel. This slide is known as "The Monster", a classic landmark in Israel.
Taken at the Lourve in Paris, France.
Taken while trekking through Zion National Park in Utah, USA.