רון אהרוני

Mathematical interests

  Currently I am mostly interested in matchings in hypergraphs, a field that has more conjectures than insights. A topological method I developed with P. Haxell (based on a topological proof of Hall's theorem) sheds some light on the topic, but less so than we expected when it was discovered.

In the past I was interested in infinite matching theory. One achievement there was the solution of the "infinite marriage problem", namely the extension of Hall's theorem to infinite graphs. It turned out that Hall's theorem is valid in the infinite case, if cardinalities are measured in the graph. Here is the precise formulation: if there is no marriage of the men then there exists a set A of men that is smaller in the graph than the set B of women connected to it, namely there exists an injection consisting of graph edges from B into A, while there is no such injection in the opposite direction. A few years ago, Eli Berger and I settled the last major problem in the field, extending this result to general graphs, not necessarily bipartite, namely extending Menger's theorem to the infinite case.

 

Poetry

Following a short bout of writing poetry, which was probably triggered by the birth of my first child, I have been interested in the question of what does poetry do to its reader and to its writer. More specifically - what is the effect of the various poetical stratagems on the reader. How do they induce the poetic impression, and what is this impression. When a publisher ordered from me a book on mathematics and beauty, I used the opportunity to write on the similar ways beauty is generated in mathematics and poetry. I won't divulge the secret here. The book appeared eventually in 2008 in "Hakibutz Hameuchad". A large part of it was devoted to the above mentioned problem, of how poetry works. Nobody noticed. It was read solely as a book about the beauty of mathematics. I hope that one day some people interested in poetry will overcome the intimidation of the mathematical part and will read it.

 

Jokes and poetry

Another book I have written, not published as yet, is on mechanisms common to jokes and poetry. Surprisingly, these two fields share the same arsenal of techniques. This fact has its roots in a deeper level structure shared by both. In particular, the book gives an answer to the old questions of what is a joke and what is a poem. The common definition of "joke" speaks of the collision of two viewpoints, or change of viewpoint, or meeting of two distinct frames of concepts. My answer is that the real underlying process is some detachment between a symbol and its meaning, and that this process is also typical of poems. Where do the ways of the joke and the poem split, this is the main topic of the book.

 

Facial expressions

My interest in this field started in my student days. One of our professors used to react to our attempted answers with an expression of disgust: pulling down the edges of the lower lip, accompanied by a slight protrusion of the tongue, and some low vocal utterance. I understood that this is just an imitation of spitting food out. This led me to analyze other common facial expression, only to discover quite soon that I was preceded by Darwin, by about 100 years. In his beautiful book "The expression of emotions in man and animals" he gave ingenious explanations for most of the expressions, finding that their origin is always functional, namely that they evolved from motions that had some purpose. My contribution to the field - an understanding of the origin of one of the most mysterious phenomena in this field, the so-called "principle of antithesis", namely that opposite states of minds induce opposite expressions.

Philosophy

Philosophy is possibly the most enigmatic of human intellectual endeavors. What does it study? Does it have an object in reality? And if not, how come its sentences seem to be meaningful? How is it possible that 2500 years of research have not lead to any progress on the big problems?

For me, the question of what is philosophy is not philosophical at all. To answer it, one has just to inspect philosophers and see what they do. In a book published in Magness Press in 2010, "The cat that is not there", I try to give an answer to this question, as well as clarify the riddles surrounding philosophy. My answer is that philosophers study human thought, without fulfilling an elementary requirement: separating the concepts used for the investigation and the concepts that are being investigated. The result is that the philosopher is like a person defining a number as "the number hereby defined, plus 1", and then calling to help him out of the paradox generated. The "big" philosophical problems - skepticism, Determinism-Free will and the Mind-Body problem, are all born directly from this flawed conceptual structure, and they owe it their very existence. Hence rectifying the error causes them to disappear. Other philosophical problems have also a component related to reality, and hence the effect of separating inspector and inspected on them is less drastic - they do not disappear, but they lose their philosophical flavor.

 

Mathematical education

I reached this field almost by chance, but now I am deeply involved. It is a truly fascinating subject, in which I feel mostly as a student, learning from the teachers whom I am supposed to guide. They usually know, intuitively, how to approach children, a knowledge that cannot be found in the academy. A book of mine on the subject, "Arithmetic for Parents", appeared in Schocken press, in 2004.