There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Sphere Packings, Lattices and Groups

Computational geometry

Note: Professor Pach's number: [045]; Also in: Facsimile edition: World Classics in Mathematics Series, vol. 28, China Science Press, Beijing, 2006. Mimeographed from 100 Research Problems in Discrete Geometry (with W. O. J. Moser). Reference DCG-BOOK-2008-001 URL: http://www.math.nyu.edu/~pach/research_problems.html Record created on 2008-11-18, modified on 2017-05-12

Research Problems in Discrete Geometry

Bounding multiplicative energy by the sumset

We prove almost tight bounds on the number of incidences between points and k-dimensional varieties of bounded degree in R d . Our main tools are the polynomial ham sandwich theorem and induction on both the dimension and the number of points.

/pdf/an-incidence-theorem-in-higher-dimensions-4nfg79j4fp.pdf

An Incidence Theorem in Higher Dimensions

Keywords: Ramsey property ; ordered tournaments Note: Professor Pach's number: [143] Reference DCG-ARTICLE-2001-001doi:10.1007/s004930100016 Record created on 2008-11-14, modified on 2017-05-12

/pdf/ramsey-type-theorems-with-forbidden-subgraphs-1bf0l2xopt.pdf

Ramsey-type theorems with forbidden subgraphs

We prove almost tight bounds on incidences between points and $k$-dimensional varieties of bounded degree in $\R^d$. Our main tools are the Polynomial Ham Sandwich Theorem and induction on both the dimension and the number of points.

An incidence theorem in higher dimensions

A new lower bound on $\max \{|A+A|,|A\cdot A|\}$ is given, where $A$ is a finite set of complex numbers.

/pdf/on-the-number-of-sums-and-products-3q7shqtchs.pdf

József Solymosi

Papers

Bounding multiplicative energy by the sumset

An Incidence Theorem in Higher Dimensions

Ramsey-type theorems with forbidden subgraphs

An incidence theorem in higher dimensions

On the Number of Sums and Products