For the existence of an n-vertex polyhedral map of type {p, p}, it is known that n must be {\small $\geq (p-1)^2$} and equality holds if and only if K is weakly neighbourly. In 2002, Brehm et al. saw that there is a unique polyhedral map of type {\small $\{5, 5\}$} on 16 vertices. In 1990, Brehm constructed a polyhedral map of type {\small $\{6, 6\}$} with 26 vertices. In this article, we prove that there do not exist any polyhedral maps of type {\small $\{6, 6\}$} on 25 vertices. As a consequence, we show that the minimum number of edges in polyhedral maps of Euler characteristic -25 is {\small $>$} 75.

We study the Frobenius problem: Given relatively prime positive integers {\small $a_{1} , \dots , a_{d}$}, find the largest value of {\small $t$} (the Frobenius number) such that {\small $ \sum_{k=1}^d m_{k} a_{k} = t $} has no solution in nonnegative integers {\small $ m_{ 1 } , \dots , m_{ d } $}. Based on empirical data, we conjecture that except for some special cases, the Frobenius number can be bounded from above by {\small $ \sqrt{a_{1}a_{2}a_{3}}^{5/4} - a_1 - a_2 - a_3 $}.

Morse theory is a fundamental tool for investigating the topology of smooth manifolds. This tool has been extended to discrete structures by Forman, which allows combinatorial analysis and direct computation. This theory relies on discrete gradient vector fields, whose critical elements describe the topology of the structure. The purpose of this work is to construct optimal discrete gradient vector fields, where optimality means having the minimum number of critical elements. The problem is equivalently stated in terms of maximal hyperforests of hypergraphs. Deduced from this theoretical result, a algorithm constructing almost optimal discrete gradient fields is provided. The optimal parts of the algorithm are proved, and the part of exponential complexity is replaced by heuristics. Although reaching optimality is MAX-SNP hard, the experiments on odd topological models are almost always optimal.

We discuss the relationship between polygonal knot energies and smooth knot energies, concentrating on ropelength. We show that a smooth knot can be inscribed in a polygonal knot in such a way that the ropelength values are close. For a given knot type, we show that polygonal ropelength minima exist and that the minimal polygonal ropelengths converge to the minimal ropelength of the smooth knot type. A subsequence of these polygons converges to a smooth ropelength minimum. Thus, ropelength minimizations performed on polygonal knots do, in fact, approximate ropelength minimizations for smooth knots.

A Heron triangle is a triangle having the property that the lengths of its sides as well as its area are positive integers. Let {\small ${\cal P}$} be a fixed set of primes; let S denote the set of integers divisible only by primes in {\small ${\cal P}$}. We prove that there are only finitely many Heron triangles whose sides {\small $a,b,c\in S$} and are reduced, that is {\small $\gcd (a,b,c)=1$}. If {\small ${\cal P}$} contains only one prime {\small $\equiv 1\; (\bmod \; 4)$}, then all these triangles can be effectively determined. In case {\small ${\cal P}=\{2,3,5,7,11\}$}, all such triangles are explicitly given.

In Zagier and Kramarz, the authors computed the critical value of the L-series of the family of elliptic curves {\small $x^3+y^3=m$} and they pointed out some numerical phenomena concerning the frequency of curves with a positive rank and the frequency of occurrences of the Tate-Shafarevich groups {\small$\TSg$} in the rank 0 case (assuming the Birch and Swinnerton-Dyer conjecture). In this paper, we give a similar study for the family of elliptic curves associated to simplest cubic fields. These curves have a nonzero rank and we discuss the density of curves of rank 3 that occurs. We also remark on a possible positive density of nontrivial Tate-Shafarevitch groups in the rank 1 case. Finally, we give examples of curves of rank 3 and 5 for which the group {\small $\TSg$} is nontrivial.

We consider the "volume conjecture,'' which states that an asymptotic limit of Kashaev's invariant (or, the colored Jones type invariant) of knot {\small $\mathcal{K}$} gives the hyperbolic volume of the complement of knot {\small $\mathcal{K}$}. In the first part, we analytically study an asymptotic behavior of the invariant for the torus knot, and propose identities concerning an asymptotic expansion of q-series which reduces to the invariant with q being the N-th root of unity. This is a generalization of an identity recently studied by Zagier. In the second part, we show that "volume conjecture'' is numerically supported for hyperbolic knots and links (knots up to 6-crossing, Whitehead link, and Borromean rings).

Let {\small $\gamma$} be a Jordan curve in {\small $\sph{2}$}, considered as the ideal boundary of {\small $\hyp{3}$}. Under certain circumstances, it is known that for any {\small $c \in (-1,1)$}, there is a disc of constant mean curvature c embedded in {\small $\hyp{3}$} with {\small $\gamma$} as its ideal boundary. Using analysis and numerical experiments, we examine whether or not these surfaces in fact foliate {\small $\hyp{3}$}, and to what extent the known conditions on the curve can be relaxed.

In this paper, we propose a method to obtain the syzygies of the Dirac complex defined on abstract vector variables. We propose a generalized theory of differential forms which acts as a de Rham-like sequence for the Dirac complex and we show that closure in this complex is equivalent to the syzygies for the Dirac complex.

We discuss Dror Bar-Natan's experimental data on cohomology groups of all prime knots with 11 or fewer crossings.

We look at numerical computations of the pluricomplex Green function g with two poles of equal weight for the bidisk. The results we obtain strongly suggest that Coman's conjecture holds in this setting, that is that g equals the Lempert function. We also prove this in a special case.

Furthermore, we show that Coman's conjecture fails in the case of two poles of different weight in the unit ball of $\C2$.