Related papers: A short introduction to Monstrous Moonshine
It is well known that every finite simple group can be generated by two elements and this leads to a wide range of problems that have been the focus of intensive research in recent years. In this survey article we discuss some of the…
In this survey article we explain the intricate links between Conway-Coxeter friezes and cluster combinatorics. More precisely, we provide a formula, relying solely on the shape of the frieze, describing how each individual entry in the…
This lecture note is intended to be a brief introduction to a recent development on the interplay between the ultradiscrete (or tropical) soliton systems and the combinatorial representation theory. We will concentrate on the simplest cases…
The phenomenology of MOND (flat rotation curves of galaxies, baryonic Tully-Fisher relation, etc.) is a basic set of phenomena relevant to galaxy dynamics and dark matter distribution at galaxy scales. Still unexplained today, it enjoys a…
The electrostatics of 2D system with complicated inner boundary is studied. The object which we call "monster" is built by an iterative process of multiple conformal mapping of the circle exterior. The procedure leads to the figures built…
In their study of a quartic integral, Boros and Moll discovered a special class of Jacobi polynomials, which we call the Boros-Moll polynomials. Kauers and Paule proved the conjecture of Moll that these polynomials are log-concave. In this…
A short, fairly self-contained proof is given of the Poincar\'e Conjecture. In the previous version there was an error on Page 8. This gap has now been filled.
The motivation for this work was to construct a nontrivial knot with trivial Jones polynomial. Although that open problem has not yielded, the methods are useful for other problems in the theory of knot polynomials. The subject of the…
The well known Andrews-Curtis Conjecture [2] is still open. In this paper, we establish its finite version by describing precisely the connected components of the Andrews-Curtis graphs of finite groups. This finite version has independent…
We give an introduction to the theory of wonderful G-varieties, with many examples when G is simple of type F4. We present results and open problems about these varieties: on their classification, on their isotropy groups, on morphisms…
In 1967 Moser proved the existence of a normal form for real analytic perturbations of vector fields possessing a reducible Diophantine invariant quasi-periodic torus. In this paper we present a proof of existence of this normal form based…
This paper introduces the notion of twisted toric manifolds which is a generalization of one of symplectic toric manifolds, and proves the weak Delzant type classification theorem for them. The computation methods for their fundamental…
This article explores the different formation scenarios of the Kronian moons system in the context of a highly dissipative Saturn, with the objective of identifying the most likely of these scenarios. First, we review the diversity of…
We analyse the possibility of defining complex valued Knot invariants associated with infinite dimensional unitary representations of $SL(2,R)$ and the Lorentz Group taking as starting point the Kontsevich Integral and the notion of…
Given a countable set X (usually taken to be the natural numbers or the integers), an infinite permutation \pi of X is a linear ordering of X. This paper investigates the combinatorial complexity of the infinite permutation on the natural…
Our understanding of space and time is probed to its depths by black holes. These objects, which appear as a natural consequence of general relativity, provide a powerful analytical tool able to examine macroscopic and microscopic…
In our current cosmological model, the main constituents of the Universe are dark matter and dark energy, whose nature is unknown, and for which there is no place in the standard model of particle physics. How to include dark matter and…
This is an expository article invited for the ``Commentary'' section of PNAS in connection with Y.-Z. Huang's article, ``Vertex operator algebras, the Verlinde conjecture, and modular tensor categories,'' appearing in the same issue of…
We introduce the notion of a Morse sequence, which provides a simple and effective approach to discrete Morse theory. A Morse sequence is a sequence composed solely of two elementary operations, that is, expansions (the inverse of a…
In this article we study the (cohomological) Hodge conjecture for singular varieties. We prove the conjecture for simple normal crossing varieties that can be embedded in a family where the Mumford-Tate group remains constant. We show how…