I am a PhD student at the Einstein Institute of Mathematics and the Federmann Center for the Study of Rationality at the Hebrew University of Jerusalem, Israel. My advisor is Prof. Nati Linial.
I am interested in high-dimensional combinatorics, especially random high-dimensional permutations and designs. I also enjoy thinking about random (hyper) graphs and (hyper) graph processes.
I am the librarian at the Federmann Center for the Study of Rationality. Please send submissions to the Center's Discussion Paper series to my email, below.
Here is my curriculum vitae.
Email: menahem.simkin followed by @ followed by mail.huji.ac.il
Office: Ross 35, Givat Ram campus of the Hebrew University.
My Favorite Open Problem
An order-$n$ Latin square
is an $n \times n$ matrix in which every column and every row contains all the values from $[n]$. This is equivalent to an $n \times n \times n$ $(0,1)$-array in which every row, column, and "shaft" contains a single $1$. Let $A$ be a random $n \times n \times n$ $(0,1)$-array in which the $n^3$ entries are independent random variables that equal $1$ with probability $p$. What is the threshold function $p(n)$ above which $A$ contains a Latin square with high probability?
I have TAed the following courses, all at Hebrew University:
- Spring 2019: Linear Algebra 2: Second undergraduate course in linear algebra.
- Spring 2018: Linear Algebra 1: First year undergraduate course in linear algebra.
- Fall2019, Fall 2018, Fall 2017, and Fall 2016: Mathematical Tools in Computer Science: Graduate course for CS students. Topics include:
- Probability (emphasizing the probabilistic method).
- Linear algebra: Spectral theorems and singular value decomposition for real matrices.
- Markov chains.
- Linear programming.
- Spring 2017 and Fall 2015: Topics in Analysis for Computer Science Students: Second year undergraduate course for CS students. Topics include:
- Norms, inner products, Banach and Hilbert spaces.
- Notions of convergence for function sequences.
- Fourier series.
- Spring 2016 and Spring 2015: Infinitesimal Calculus 2 for Computer Science Students: Second course in undergraduate calculus.
Theses and Notes
Finding structure with randomness. My PhD thesis, supervised by Prof. Nati Linial.
Methods for analyzing random designs. Notes for a lecture I gave at the Israel Institute for Advanced Studies in March, 2018. The highlight is a proof, using Matthew Kwan's beautiful method, that almost every Steiner triple system on $n$ vertices contains at least $\left(1 - o(1) \right)n^2/24$ Pasch configurations. As far as I know this result is new, even if not ground-breaking.
Construction and enumeration with the probabilistic method. Notes for a lecture I gave at the HUJI math department's student seminar in Spring 2017. I gave two examples of probabilistic tools in combinatorics: Erdős's lower bound on the diagonal Ramsey numbers, and an application of the Linial-Luria entropy method to bound from above the number of order-$n$ Latin squares.
Random-turn and Richman games. My MSc thesis, supervised by Prof. Gil Kalai. In which the following question is considered: Many games (e.g. Tic-Tac-Toe, chess) employ a mechanism where players alternate taking turns. How are games affected if, instead of simple alternation, each turn is assigned randomly as we go along? What if before each turn there is an auction for the privilege of taking the next turn?
Self-similar structures near the edges of strained elastic sheets. My Amirim (undergraduate honors) thesis, supervised by Prof. Raz Kupferman. Tear a plastic bag. Go ahead! Do it! Now examine the newly-created edge. Most likely you'll see wave-like patterns. In this project I numerically tested a proposed mathematical explanation for the phenomenon.
Matchings and Latin squares. Slides for a talk given at the Rationality Center's Caesarea retreat (2018). The lecture is intended for a broad academic audience. I motivate the view of Latin squares as high-dimensional matchings, and explain the high-dimensional Erdős-Szekeres theorem.
Noam Yonat learns about triangle decompositions.
With Shanee in the French Alps.
Carrying Noam up Har HaTayasim.
Aqaba... seems more colorful in real life.