finite geometry

Love in Projective Planes, Chinese Valentine’s Day & Phonotactics
A part of the video.

A few days ago China celebrated one of many Chinese Valentine’s days: the 20th of May. Why is this a special day in China? Chinese has around 30 to 36 phonemes which is plenty, but Chinese phonotactics dictate that you can only make around 1200 syllables out of them, for instance, see this video. English has more than 8000 possible syllables. Additionally, Chinese (unlike many other languages) uses one syllables per morpheme. This distinguished from languages such as Japanese which have very few syllables, but usually use two or three syllables for one morpheme. Often Chinese avoids any resulting confusion by various means, for instance, by combining at least two morphemes/syllables. Yet you find around 1500 one syllable words in Mandarin. By the pigeonhole principle, we find at least two words which sound the same!

(more…)

Classifying Cameron-Liebler Sets/Boolean Degree 1 Functions

This week I put two new preprint on the arXiv. Both are on a similar theme, so I will discuss them together. One is with Morgan Rodgers on regular sets of lines in rank 3 polar spaces. The other one is solves a problem which I have been thinking about very regularly since November 2017: The classification of Boolean degree {1} functions (or Cameron-Liebler classes) of {k}-spaces in an {n}-dimensional vector space {V} over the field with {q} elements for {n} large enough (and {q} and {k \geq 2} fixed).

Not only did I (and many other researcher) try to solve this problem for many years, it also turns out that the solution has a very short and concise proof. So for now I am very happy about it. [And please do not find a mistake. Any mistake must be embarrissingly simple.] The problem itself (for {k=2}) goes back to a paper by Cameron and Liebler in 1982, so it is also a reasonably old problem.

(more…)

Constructing Cospectral Graphs

Last week I had a cold and could not do much thinking. So I spent my time making TikZ pictures for an upcoming talk of mine. This talk is on my recent work with Akihiro Munemasa on constructing cospectral strongly regular graphs. I think that the pictures are nice for a blog post, so here we go.

1. The Spectrum of a Graph

Two graphs {\Gamma} and {\overline{\Gamma}} are called cospectral if their adjacency matrices {A} and {\overline{A}} have the same eigenvalues. This is the same as saying that there is an orthogonal matrix {Q} with {\overline{A} = Q^T A Q}. Any permutation matrix is a valid choice for {Q}, but this is not very interesting as then {\Gamma} and {\overline{\Gamma}} are isomorphic. Chris Godsil and Brendan McKay described one of the easiest interesting choices for {Q} in 1982. For a graph {\Gamma} on {v} vertices, a simplified version of their matrix is

\displaystyle Q = \begin{pmatrix} \frac{1}{m} J_{2m} - I_{2m} & 0 \\ 0 & I_{v-2m} \end{pmatrix}.

(more…)

Translating Terminology: Equitable Partitions and Related Concepts

Permutation Groups, Analysis of Boolean Functions, Finite Geometry, Coding Theory and Algebraic Graph Theory

Important mathematical concepts get reinvented many times. In my recent work with Yuval Filmus we explored objects that are called (in random ordering) Boolean degree 1 functions, Cameron-Liebler line classes, equitable partitions, completely regular strength 0 codes with covering radius 1, intriguing sets, perfect colorings, sets with dual width 1, tactical decompositions or tight sets — all depending on the context and who you ask. While the article with Yuval explains to some extent how these notions connect, a research article does not seem to be the right format to explain all concepts in sufficient detail. This post tries to amend this. It also prepares a future post which will elaborate my research with Yuval in more detail. (more…)

The chromatic number of the Kneser graphs and the q-Kneser graphs

This year is László Lovász‘s 70th birthday and I went to his birthday conference, so a blog post about one of his famous results seems to be appropriate: the chromatic number of the Kneser graph.

Let \Gamma be a graph with a vertex set V and an adjacency relation \sim. A coloring of \Gamma is a map from V from V to a set of colors C such that no pair of adjacent vertices gets mapped to the same element of C. The chromatic number of \Gamma is the smallest number of colors needed to color \Gamma. One of the nicest graphs to investigate is the Kneser graph: Here V consists of all subsets of \{ 1, 2, \ldots, n \} of size k and two vertices are adjacent if the corresponding k-sets are disjoint. For k=2 and n=5, this is the famous Petersen graph:

400px-Kneser_graph_KG(5,2).svg (more…)

Erdős-Ko-Rado Theorems for Spherical Buildings

Today’s topic combines three of my favorite subjects: Erdős-Ko-Rado theorems (EKR theorems), finite buildings and spectral techniques. All of these topics deserve their own books (and have them, here some examples which I read: Erdos-Ko-Rado Theorems: Algebraic Approaches by Chris Godsil and Karen Meagher, Spectra of Graphs by Andries E. Brouwer and Willem H. Haemers, The Structure of Spherical Buildings by Richard M. Weiss), so I will only touch these topics slightly.

My main aim is to present a variation of the EKR theorem which is motivated by questions about spherical buildings. The variation was recently formulated by Klaus Metsch, Bernhard Mühlherr, and me. If you already know spherical buildings, then you might prefer to read the introduction of our paper instead. (more…)