Abstract The stick number of a knot is the minimum number of segments needed to build a polygonal version of the knot. Despite its elementary definition and relevance to physical knots, the stick number is poorly understood: for most knots we only know bounds on the stick number. We adopt a Monte Carlo approach to finding better bounds, producing very large ensembles of random polygons in tight confinement

Abstract We conjecture a formula supported by computations for the valuation of Kac polynomials of a quiver, which only depends on the number of loops at each vertex. We prove a convergence property of renormalized Kac polynomials of quivers when increasing the number of arrows: they converge in the ring of power series, with a linear rate of convergence. Then, we propose a conjecture concerning the

Abstract Let ϕ:X→Y be a (possibly ramified) cover, with X and Y of strictly positive genus. We develop tools to identify the Prym variety of ϕ, up to isogeny, as the Jacobian of a quotient curve C of the Galois closure of a composition X→Y→P 1 of ϕ with a well-chosen map Y→P 1 that identifies branch points of ϕ. To our knowledge, this method recovers all previously obtained descriptions of Prym varieties

Abstract In 1993, Broué, Malle and Michel initiated the study of spetses on the Greek island bearing the same name. These are mysterious objects attached to non-real Weyl groups. In algebraic topology, a p-compact group X is a space which is a homotopy-theoretic p-local analogue of a compact Lie group. A connected p-compact group X is determined by its root datum which in turn determines its Weyl group

Abstract We introduce a Sinkhorn-type algorithm for producing quantum permutation matrices encoding symmetries of graphs. Our algorithm generates square matrices whose entries are orthogonal projections onto one-dimensional subspaces satisfying a set of linear relations. We use it for experiments on the representation theory of the quantum permutation group and quantum subgroups of it. We apply it

Abstract We formulate, using heuristic reasoning, conjectures for the range of the number of primes in intervals of length y around x, where y≪( log x)2. In particular, we conjecture that the maximum grows surprisingly slowly as y ranges from log x to ( log x)2. We will exhibit the available data, showing that it somewhat supports our conjectures, though not so well that there may not be room for some

Abstract We extend the heuristic introduced by Conrey, Farmer, Keating, Rubinstein, and Snaith in order to formulate conjectures for the (k,l)-moments of L-functions of elliptic curves twisted by cubic characters. We also apply the work of Keating and Snaith on the (k,l)-moments of characteristic polynomials of unitary matrices to extend our conjecture to k,l∈C such that Re(k),Re(l), and Re(k+l)>−1

Abstract Let [n]:={1,2,…,n} and let a k-set denote a set of cardinality k. A family of sets is union-closed (UC) if the union of any two sets in the family is also in the family. Frankl’s conjecture states that for any nonempty UC family F⊆2[n] such that F≠{∅}, there exists an element i∈[n] that is contained in at least half the sets of F, where 2[n] denotes the power set on [n]. The 3-sets conjecture

Abstract The quintic threefold X is the most studied Calabi-Yau 3-fold in the mathematics literature. In this article, using Čech-to-derived spectral sequences, we investigate the mod 2 and integral cohomology groups of a real Lagrangian L⌣R, obtained as the fixed locus of an anti-symplectic involution in the mirror to X. We show that L⌣R is the disjoint union of a 3-sphere and a rational homology

Abstract We extend the computations from our previous work to determine the maximum number of rational points on a curve over F3 and F4 with fixed gonality and small genus. We find, for example, that there is no curve of genus 5 and gonality 6 over a finite field. We propose two conjectures based on our data. First, an optimal curve of genus g has gonality at most ⌊g+32⌋. Second, an optimal curve of

Abstract In this article, we describe all the hyperbolic 24-cell 4-manifolds with exactly one cusp. There are four of these manifolds up to isometry. These manifolds are the first examples of one-cusped hyperbolic 4-manifolds of minimum volume.

Abstract Given a finite covering of graphs f:Y→X, it is not always the case that H1(Y;C) is spanned by lifts of primitive elements of π1(X). In this article, we study graphs for which this is not the case, and we give here the simplest known nontrivial examples of covers with this property, with covering degree as small as 128. Our first step is focusing our attention on the special class of graph

Abstract Permutations that can be sorted greedily by one or more stacks having various constraints have been studied by a number of authors. A pop-stack is a greedy stack that must empty all entries whenever popped. Permutations in the image of the pop-stack operator are said to be pop-stacked. Asinowki, Banderier, Billey, Hackl, and Linusson recently investigated these permutations and calculated

Abstract I propose a formalization challenge. The text below is a slightly edited version of the blog post Xena Project – Liquid Tensor Experiment, and I have kept its informal style.

Abstract In this article, we study the Mahler measures of more than 500 families of reciprocal polynomials defining genus 2 and genus 3 curves. We numerically find relations between the Mahler measures of these polynomials with special values of L-functions. We also numerically discover more than 100 identities between Mahler measures involving different families of polynomials defining genus 2 and

Abstract In this article, we investigate the powerful nilpotency class of powerfully nilpotent groups of standard nilpotency class 2. We outline the process of collecting data using the computer algebra system GAP, formulating a conjecture based on the data, and finally we prove the conjecture. In particular, we prove that for a powerfully nilpotent group of nilpotency class 2 and order pn, where p

Abstract If k[x1,…,xn]/I=R=∑i≥0Ri, k a field, is a standard graded algebra, then the Hilbert series of R is the formal power series ∑i≥0dimkRiti. It is known already since Macaulay which power series are Hilbert series of graded algebras. A much harder question is which series are Hilbert series if we fix the number of generators of I and their degrees, say for ideals I=(f1,…,fr), degfi=di,i=1,…,r

Abstract In this paper, we give a complete description of the farthest point map, as defined on the regular octahedron with its intrinsic flat cone metric. As a consequence we show that every well-defined orbit of the map converges to the 1-skeleton of the barycentric subdivision of the triangulation.

Abstract We compute the Sn -equivariant rational homology of the tropical moduli spaces Δ 2 , n for n ≤ 8 using a cellular chain complex for symmetric Δ-complexes in Sage.

We study arithmetic functions $\Phi(x;d,a)$, called prime running functions, whose value at $x$ sums the gaps between primes $p_k \equiv a\ (\text{mod}\ d)$ below $x$ and the next following prime $p_{k+1}$, up to $x$. (The following prime $p_{k+1}$ may be in any residue class $(\text{mod}\ d)$.) We empirically observe systematic biases of order $x / \log x$ in $\Phi(x;d,a) - \Phi(x;d,b)$ for different

Liedtke (2008) has introduced group functors $K$ and $\tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work we relate $K$ and $\tilde K$ to a group functor $\tau$ arising in the construction of the non-abelian exterior square of a group

Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this article, we study counting problems that result from focusing on properties of the square torus one by one. After drawing insights from experimental evidence, we consider the implications between

Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this paper, we study counting problems that result from focusing on properties of the square torus one by one. After drawing insights from experimental evidence, we consider the implications between

We present a fast algorithm that takes as input an elliptic curve defined over $\mathbb Q$ and an integer $d$ and returns all the number fields $K$ of degree $d'$ dividing $d$ such that $E(K)_{tors}$ contains $E(F)_{tors}$ as a proper subgroup, for all $F \varsubsetneq K$. We ran this algorithm on all elliptic curves of conductor less than 400.000 (a total of 2.483.649 curves) and all $d \leq 23$ and

We used the MACH2 supercomputer to study coefficients in the q-series expansion of (1−q)(1−q2)…(1−qn), for all n≤75000. As a result, we were able to conjecture some periodic properties associated w...

We present new computational results for symplectic monodromy groups of hypergeometric differential equations. In particular, we compute the arithmetic closure of each group, sometimes justifying a...

In this work, we present an algorithm running over SAGE, which allows users to deal with group actions on Riemann surfaces up to topological equivalence. Our algorithm allows us to study the equisy...

We study the translation surfaces obtained by considering the unfoldings of the surfaces of Platonic solids. We show that they are all lattice surfaces and we compute the topology of the associated Teichmuller curves. Using an algorithm that can be used generally to compute Teichmuller curves of translation covers of primitive lattice surfaces, we show that the Teichmuller curve of the unfolded dodecahedron

We recognize certain special hypergeometric motives, related to and inspired by the discoveries of Ramanujan more than a century ago, as arising from Asai $L$-functions of Hilbert modular forms.

In this paper we discuss minimizing a generalized frame potential - F(p,m,n) - for vectors in CPn. We discuss the minimal frame potential as well as the space of solutions for the minimization prob...

We derive a refined conjecture for the variance of Gaussian primes across sectors, with a power saving error term, by applying the L-functions Ratios Conjecture. We observe a bifurcation point in the main term, consistent with the Random Matrix Theory (RMT) heuristic previously proposed by Rudnick and Waxman. Our model also identifies a second bifurcation point, undetected by the RMT model, that emerges

Geodesic nets on Riemannian manifolds form a natural class of stationary objects generalizing geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean plane or the round $2$-sphere. In the first half of this paper we survey some results and open questions (old and new) about geodesic nets on Riemannian manifolds

This is a collection of notes on embedding problems for 3-manifolds. The main question explored is “which 3-manifolds embed smoothly in S?” The terrain of exploration is the Burton/Martelli/Matveev/Petronio census of triangulated prime closed 3-manifolds built from 11 or less tetrahedra. There are 13766 manifolds in the census, of which 13400 are orientable. Of the 13400 orientable manifolds, only

This paper provides an algorithm enumerating superspecial trigonal curves of genus $5$ over finite fields. Executing the algorithm over a computer algebra system Magma, we enumerate them over finite fields $\mathbb{F}_{p^a}$ for any natural number $a$ if $p \leq 7$ and for odd $a$ if $p \leq 13$.

We study orthogonal polynomials with respect to self-similar measures, focusing on the class of infinite Bernoulli convolutions, which are defined by iterated function systems with overlaps, especi...

This short note resolves the most important part of the PS braid conjecture while introducing the first known examples of knots and links with odd torsion of order 9, 27, 81, and 25 in their even Khovanov homology.

We present eigenvalue data and pictures of eigenfunctions of the classic and quadratic snowflake fractal and of quadratic filled Julia sets. Furthermore, we approximate the area and box-counting dimension of selected Julia sets to compare the eigenvalue counting function with the Weyl term.

Alt's problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial equations which was first solved numerically using homotopy continuation by Wampler, Morgan, and Sommese in 1992. Since there is still not a proof that all solutions

Recently, Hirschhorn investigated vanishing coefficients of the arithmetic progressions in the following two q-series expansions ∑n=0∞a(n)qn:=∏n=1∞(1+q5n−4)(1+q5n−1)(1−q10n−9)3(1−q10n−1)3,∑n=0∞b(n)qn:=∏n=1∞(1+q5n−3)(1+q5n−2)(1−q10n−7)3(1−q10n−3)3. He proved that for n≥0,a(5n+2)=a(5n+4)=b(5n+1)=b(5n+4)=0. In this paper, we further study these two q-series expansions and obtain the generating functions

This paper presents some configuration theorems concerning rectangles inscribed in four lines.

The problem of determining $DS_n$, the complex numbers that occur as an eigenvalue of an $n$-by-$n$ doubly stochastic matrix, has been a target of study for some time. The Perfect-Mirsky region, $PM_n$, is contained in $DS_n$, and is known to be exactly $DS_n$ for $n \leq 4$, but strictly contained within $DS_n$ for $n = 5$. Here, we present a Boundary Conjecture that asserts that the boundary of $DS_n$

AbstractWe relate Reiner, Tenner, and Yong’s coincidental down-degree expectations (CDE) property of posets to the minuscule doppelganger pairs studied by Hamaker, Patrias, Pechenik, and Williams. ...

AbstractGay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. This p...

We construct a one (real) dimensional deformation of an acylindrical hyperbolic 3-manifold, compute some associated arithmetic invariants and exhibit the skinning map along the deformation locus. T...

In differential geometry of surfaces the Dirac operator appears intrinsically as a tool to address the immersion problem as well as in an extrinsic flavour (that comes with spin transformations to comformally transfrom immersions) and the two are naturally related. In this paper we consider a corresponding pair of discrete Dirac operators, the latter on discrete surfaces with polygonal faces and normals

In this note, given a pair $(\mathfrak{g}, \lambda)$, where $\mathfrak{g}$ is a complex semisimple Lie algebra and $\lambda \in \mathfrak{h}^*$ is a dominant integral weight of $\mathfrak{g}$, where $\mathfrak{h} \subset \mathfrak{g}$ is the real span of the coroots inside a fixed Cartan subalgebra, we associate an $SU(2)$ and Weyl equivariant smooth map $f: X \to (P^m(\mathbb{C}))^n$, where $X \subset

A monic, irreducible polynomial in one variable having integer coefficients and all real roots deserves particular interest if its roots lie in an interval of length 4 whose end-points are not inte...

We confirm, for the primes up to 3000, the conjecture of Bourgain, Gamburd, and Sarnak on strong approximation for the Markoff surface $x^2+y^2+z^2 = 3xyz$ modulo primes. For primes congruent to 3 modulo 4, we find data suggesting that some natural graphs constructed from this equation are asymptotically Ramanujan. For primes congruent to 1 modulo 4, the data suggest a weaker spectral gap. In both

We present a property satisfied by a large variety of complex continued fraction algorithms (the "finite building property") and use it to explore the structure of bijectivity domains for natural extensions of Gauss maps. Specifically, we show that these domains can each be given as a finite union of Cartesian products. In one complex coordinate, the sets come from explicit manipulation of the continued

When calculating the index of a minimal surface, the set of smooth functions on a domain with compact support is the standard setting to describe admissible variations. We show that the set of admissible variations can be widened in a geometrically meaningful manner by considering the difference of area functional, leading to a more general notion of index. This allows us to produce explicit examples

The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height (the maximum absolute value of the coefficients) of the polynomial. We improve the known bounds for this problem and related ones. Then we report on extensive experiments

(2020). Correction. Experimental Mathematics: Vol. 29, No. 3, pp. 360-360.

Let $E$ be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer $A(E)$, that we call the {\it Serre's constant associated to $E$}, that gives necessary conditions to conclude that $\rho_{E,m}$, the mod m Galois representation associated to $E$, is non-surjective. In particular, if there exists a prime factor $p$ of

We consider the integer Chebyshev problem, that of minimizing the supremum norm over polynomials with integer coefficients on the interval $[0,1]$. We implement algorithms from semi-infinite programming and a branch and bound algorithm to improve on previous methods for finding integer Chebyshev polynomials of degree $n$. We give 16 new integer Chebyshev polynomials of degrees in the range 147 to 244

We construct an explicit example of a genus $2$ curve $C$ over a number field $K$ such that the adelic Galois representation arising from the action of $\operatorname{Gal}(\overline{K}/K)$ on the Jacobian of $C$ has image $\operatorname{GSp}_4(\widehat{\mathbb Z})$.

We define oldforms and newforms for Drinfeld cusp forms of level $t$ and conjecture that their direct sum is the whole space of cusp forms. Moreover we describe explicitly the matrix $U$ associated to the action of the Atkin operator $\mathbf{U}_t$ on cusp forms of level $t$ and use it to compute tables of slopes of eigenforms. Building on such data, we formulate conjectures on bounds for slopes, on

We propose an approach for showing rationality of an algebraic variety $X$. We try to cover $X$ by rational curves of certain type and count how many curves pass through a generic point. If the answer is $1$, then we can sometimes reduce the question of rationality of $X$ to the question of rationality of a closed subvariety of $X$. This approach is applied to the case of the so-called Ueno-Campana

In a convex mosaic in $\mathbb{R} ^d$ we denote the average number of vertices of a cell by $\bar v$ and the average number of cells meeting at a node by $\bar n$. Except for the $d=2$ planar case, there is no known formula prohibiting points in any range of the $[\bar n, \bar v]$ plane (except for the unphysical $\bar n, \bar v < d+1$ strips). Nevertheless, in $d=3$ dimensions if we plot the 28 points

A Lie (super)algebra with a non-degenerate invariant symmetric bilinear form $B$ is called a nis-(super)algebra. The double extension $\mathfrak{g}$ of a nis-(super)algebra $\mathfrak{a}$ is the result of simultaneous adding to $\mathfrak{a}$ a central element and a derivation so that $\mathfrak{g}$ is a nis-algebra. Loop algebras with values in simple complex Lie algebras are most known among the

The Kontsevich deformation quantization combines Poisson dynamics, noncommutative geometry, number theory, and calculus of oriented graphs. To manage the algebra and differential calculus of series...