We propose a framework for unifying the $\mathrm{sl}(N)$ Khovanov--Rozansky homology (for all $N$) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory that categorifies the HOMFLY polynomial. Moreover, this theory should have an additional formal structure of a family of differentials. Roughly speaking, the triply graded theory by itself captures the large-$N$ behavior of the $\mathrm{sl}(N)$ homology, and differentials capture nonstable behavior for small $N$, including knot Floer homology. The differentials themselves should come from another variant of $\mathrm{sl}(N)$ homology, namely the deformations of it studied by Gornik, building on work of Lee.

While we do not give a mathematical definition of the triply graded theory, the rich formal structure we propose is powerful enough to make many nontrivial predictions about the existing knot homologies that can then be checked directly. We include many examples in which we can exhibit a likely candidate for the triply graded theory, and these demonstrate the internal consistency of our axioms. We conclude with a detailed study of torus knots, developing a picture that gives new predictions even for the original $\mathrm{sl}(2)$ Khovanov homology.

Let $S_{\epsilon}$ denote the set of Euclidean triangles whose two small angles are within $\epsilon$ radians of $\frac{\pi}{6}$ and $\frac{\pi}{3}$ respectively. In this paper we prove two complementary theorems: (1) For any $\epsilon>0$ there exists a triangle in $S_{\epsilon}$ that has no periodic billiard path of combinatorial length less than $1/\epsilon$. (2) Every triangle in $S_{1/400}$ has a periodic billiard path.

A prime number $p$ is called elite if only finitely many Fermat numbers $2^{2^n}+1$ are quadratic residues modulo $p$. Previously, only fourteen elite primes were known explicitly, all of them smaller than $35$ million. Using computers, we searched all primes less than $10^9$ for other elite primes and discovered $p=159\,318\,017$ and $p=446\,960\,641$ as the fifteenth and sixteenth elite primes. Moreover, with another approach we found $26$ other elite primes larger than a billion, the largest of which has $1172$ decimal digits. Finally, we derive some conjectures about elite primes from the results of our computations.

We present a case study in experimental yet rigorous mathematics by describing an algorithm, fully implemented in both Mathematica and Maple, that automatically conjectures, and then automatically proves, closed-form expressions extending Dyson's celebrated constant-term conjecture.

A high-performance computer program for searching a tree of words in the group generated by Andrews-Curtis transformations is presented. The program enumerates words in this group using a disk-based hash table to store words already seen. We discuss several issues that affect the performance of the program and examine the growth of the size of the Cayley graph being searched.

The Shapiro conjecture in the real Schubert calculus, while likely true for Grassmannians, fails to hold for flag manifolds, but in a very interesting way. We give a refinement of the Shapiro conjecture for flag manifolds and present massive computational experimentation in support of this refined conjecture. We also prove the conjecture in some special cases using discriminants and establish relationships between different cases of the conjecture.

We study a construction of the Mathieu group $M_{12}$ using a game reminiscent of Loyd's "15-puzzle.'' The elements of $M_{12}$ are realized as permutations on $12$ of the $13$ points of the finite projective plane of order $3$. There is a natural extension to a "pseudogroup'' $M_{13}$ acting on all $13$ points, which exhibits a limited form of sextuple transitivity. Another corollary of the construction is a metric, akin to that induced by a Cayley graph, on both $M_{12}$ and $M_{13}$. We develop these results, and extend them to the double covers and automorphism groups of $M_{12}$ and $M_{13}$, using the ternary Golay code and $12 \x 12$ Hadamard matrices. In addition, we use experimental data on the quasi-Cayley metric to gain some insight into the structure of these groups and pseudogroups.

We define and study a notion of complexity for smooth, closed and orientable 4-manifolds. This notion, based on the theory of Turaev shadows, represents the 4-dimensional analogue of Matveev's complexity of 3-manifolds. We classify complexity-0 and complexity-1 4-manifolds and provide examples of manifolds of higher complexity.

In this note I describe a computational study of the successive maxima of the relative sum-of-divisors function $\rho(n):=\sigma(n)/n$. These maxima occur at superabundant and colossally abundant numbers, and I also study the density of these numbers. The values are compared with the known maximal order $e^\gamma\loglog{n}$; theorems of Robin and Lagarias relate these data to a condition equivalent to the Riemann Hypothesis. It is thus interesting to see how close these conditions come to being violated.