We use the order complex corresponding to a symmetric matrix (defined by Giusti et al in 2015). In this note, we use it to define a class of models of random graphs, and show some surprising experimental results, showing sharp phase transitions.

We present a software suite for the analysis and optimization of ideal convex polyhedra in hyperbolic 3-space ℍ^3. Using Rivin's variational characterization of ideal polyhedra, we develop efficient algorithms for checking combinatorial realizability and finding volume-maximizing configurations. Our systematic computational study reveals two striking phenomena: (1) maximal volume ideal polyhedra consistently exhibit dihedral angles that are rational multiples of π – a property with no obvious explanation from the optimization formulation; and (2) the distribution of volumes for random configurations is well-approximated by a Beta distribution, with mean normalized volume converging to approximately ln 2 ≈ 0.69 as the vertex count increases. We provide complete data for small vertex counts, including vertex positions, triangulations, and verified rational angle structures. An interactive implementation is publicly available.

Estimating the variance of prices of financial assets is a long-standing difficult problem, since estimating the second moment is a hard problem. Here, we describe some more robust methods, primarily for the family of Student T distributions (which includes the Gaussian distribution).

It is a known empirical observation that the "speed'' of trading in the equity markets is maximal at open and close, and minimal in the middle of the day. This is important both for its own sake and for correctly backtesting quantitative strategies. In this note we analyze this phenomenon, and give a model.We will perform a similar analysis for the "volatility smile", which has features quite different from those of the volume smile.

Betti curves of symmetric matrices were introduced in [3] as a new class of matrix invariants that depend only on the relative ordering of matrix entries. These invariants are computed using persistent homology, and can be used to detect underlying structure in biological data that may otherwise be obscured by monotone nonlinearities. Here we prove three theorems that characterize the Betti curves of rank 1 symmetric matrices. We then illustrate how these Betti curve signatures arise in natural data obtained from calcium imaging of neural activity in zebrafish.

I introduce a simple Bayesian methodology to analyze citation metrics across fields of Mathematics. I collect and analyze the MathSciNet http://www.ams.org/mathscinet} profiles of Full Professors of Mathematics at all 131 R1, research oriented US universities. The data recorded was citations, field, time since first publication, and gender. We perform basic analysis and provide a ranking of US math departments, based on age corrected and field adjusted citations. This is a (brief) follow-up to the recent paper of J.D.Paik and the author, where non-Bayesian techniques were

An opinion piece by Abigail Thompson in the Notices of the American Mathematical Society has engendered a lot of discussion, including three open letters with over 1400 signatures. We analyze the professional profiles of signatories of these three letters, and, in particular, their citation records. We find that when restricting to R1 math professors, the means of their citations and citations per year are ordered $\mu(A) < \mu(B) < \mu(C)$. The significance of these findings are validated using a one-sided permutation test.

We introduce a methodology to analyze citation metrics across fields of Mathematics. We use this methodology to collect and analyze the MathSciNet profiles of Full Professors of Mathematics at all 131 R1, research oriented US universities. The data recorded was citations, field, and time since first publication. We perform basic analysis and provide a ranking of US math departments, based on age corrected and field adjusted citations.

In this article we examine some macroeconomic data over the last several decades, and see if we can we can find anything interesting. In particular, we investigate how macroeconomic data affect equity prices.

We investigate the statistical behavior of the eigenvalues and diameter of random Cayley graphs of SL2[Z/pZ] as the prime number p goes to infinity. We prove a density theorem for the number of exceptional eigenvalues of random Cayley graphs, i. e., the eigenvalues with absolute value bigger than the optimal spectral bound. Our numerical results suggest that random Cayley graphs of SL2[Z/pZ] and the explicit LPS Ramanujan projective graphs of P1(Z/pZ) have optimal spectral gap and diameter as the prime number p goes to infinity.

In this paper we study the manifolds in the census of small 3-manifolds as available in SnapPy. We compare our results with the statistics of random 3-manifolds obtained using the Dunfield Thurston and Rivin models.

We discuss some (numerical and theoretical) results about the coefficients and zeros of Tutte (dichromatic) polynomial of graphs of bounded degree whose size increases. We also discuss related results for Bollobás-Riordan polynomials.

We describe the design of the CCI30 cryptocurrency index.

In this study, we conduct a preliminary investigation of coalition action against ISIS, which will provide a foundation for further analysis. This work has two goals. One is to understand when the tide turned against ISIS, and the other (and lesser) goal is to see and understand how to model the war on Terror mathematically. By examining the data we show that ISIS expansion was significantly and categorically retarded by mid-2015, and its back (as a nation-state, but not as a terrorist organization) was broken by mid-2016.

We give some historical antecedents for the current geopolitical situation.

The Sharpe ratio is the most widely used risk metric in the quantitative finance community - amazingly, essentially everyone gets it wrong. In this note, we will make a quixotic effort to rectify the situation.

We take a look the changes of different asset prices over variable periods, using both traditional and spectral methods, and discover universality phenomena which hold (in some cases) across asset classes.

We show that for any n >= 2, two elements selected uniformly at random from a symmetrized Euclidean ball of radius X in SLn(Z) will generate a thin free group with probability tending to 1 as X -> infinity. This is done by showing that the two elements will form a ping-pong pair, when acting on a suitable space, with probability tending to 1. On the other hand, we give an upper bound < 1 for the probability that two such elements will form a ping-pong pair in the usual Euclidean ball model in the case where n > 2.

We study random knots, which we define as a triple of random periodic functions (where a random function is a random trigonometric series, \[f(\theta) = \sum_{k=1}^\infty a_k \cos (k \theta) +b_k (\sin k \theta),\] with $a_k, b_k$ are independent gaussian random variables with mean $0$ and variance $\sigma(k)^2$ - our results will depend on the functional dependence of $\sigma$ on $k.$ In particular, we show that if $\sigma(k) = k^\alpha,$ with $\alpha < -3/2,$ then the probability of getting a knot type which admits a projection with $N$ crossings, decays at least as fast as $1/N.$ The constant $3/2$ is significant, because having $\alpha < -3/2$ is exactly the condition for $f(\theta)$ to be a $C^1$ function, so our class is precisely the class of random \emph{tame} knots. We also find some suprising experimental observations on the zeros of Alexander polynomials of random knots (with slowly and non-decaying coefficients), and even more surprising observations on their coefficients. Our observations persist in other models of random knots, making it likely that the results are universal.