Gedeon, Proudfoot, and Young first conjectured that the graded Sn -representations H∙(Conf(n,Rd);Q) are strongly equivariantly log concave for even d. Later work of Matherne, Miyata, Proudfoot, and...

We give experimental support for a conjecture of Louder and Wilton saying that words of imprimitivity rank greater than two yield hyperbolic one-relator groups.

Dr Jonathan Spreer Associate Professor Address: School of Mathematics and Statistics F07 The University of Sydney NSW 2006 Australia Office: Carslaw Building, Room 623 Email:…

For an isolated hypersurface singularity f:(\CC^n,0_\rightarrow(\CC,0), we study the Łojasiewicz exponent ł(f) of f. We formulate three conjectures concerning: (i) attainment of ł(f) on coordinate polar curves, (ii) topological invariance of ł(f), and (iii) a local Newton-diagram formula for ł(f) in the Kushnirenko non-degenerate case. We compute ł(f) for all singularities of modality at most 2 in Arnold’s list and for several further families, obtaining systematic evidence in favor of these conjectures. We also find to what extent the collection of numerical invariants {modality⁡(f),ł(f),\mu(f),corank⁡(f),stabord⁡(f)} determines the stable equivalence class of f for the aforementioned singularities.

We establish a higher-dimensional irrationality criterion for periods which are presented as Mellin integrals depending on many parameters. The criterion is stated as an upper bound on the multi-variate transfinite diameter of the image of the domain of integration under the Mellin arguments. Most of the paper is devoted to studying notions of transfinite diameter relative to very general multivariate Vandermonde matrices. As a proof of principle, we illustrate how this approach works with detailed computations in the case of a 5-parameter family of integrals for \zeta(2) on \mathcal{M}_{0,5}, the moduli space of curves of genus 0 with 5 marked points. This yields a “higher-dimensional” proof of the irrationality of \zeta(2), based on an upper bound for a certain kind of transfinite diameter associated to \mathcal{M}_{0,5}.

Several authors have shown that there exist nonsolvable Bieberbach groups of dimension 15. In this paper, we show that this is, in fact, a minimal dimension for such groups.

We tell the story of how schemes were formalized in three different ways in the Lean theorem prover.

Gorenstein formats present the equations of regular canonical, Calabi–Yau and Fano varieties embedded by subcanonical divisors. We present a new algorithm for the enumeration of these formats based on orbifold Riemann–Roch and knapsack packing-type algorithms. We apply this to extend the known lists of threefolds of general type beyond the well-known classes of complete intersections and also to find classes of Calabi–Yau threefolds with canonical singularities.

We classify four-dimensional quasismooth weighted hypersurfaces with small canonical class and verify a conjecture of Johnson and Kollár on infinite series of quasismooth hypersurfaces with anticanonical hyperplane section in the case of fourfolds. By considering the quotient singularities that arise, we classify those weighted hypersurfaces that are canonical, Calabi–Yau and Fano fourfolds. We also consider other classes of hypersurfaces, including Fano hypersurfaces of index greater than 1 in dimensions 3 and 4.

We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of \pi, log(2), or zeta values. In order to perform these simplifications, we view the series as specializations of generating series. For these generating series, we derive integral representations in terms of root-valued iterated integrals or directly in terms of cyclotomic harmonic polylogarithms. Using substitutions, we express the root-valued iterated integrals as cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive expressions in terms of several constants. We provide an algorithimic machinery to prove identities which so far could only be proved using classical hypergeometric approaches. These methods are implemented in the computer algebra package HarmonicSums.

We give the first examples of smooth Fano and Calabi–Yau varieties violating the (narrow) canonical strip hypothesis, which concerns the location of the roots of Hilbert polynomials of polarized varieties. They are given by moduli spaces of rank 2 bundles with fixed odd-degree determinant on curves of sufficiently high genus, hence our Fano examples have Picard rank 1, index 2, are rational, and have moduli. The hypotheses also fail for several other closely related varieties.

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 manifolds. Assuming certain conjectures on curve counting, we show that the previously open cases X_{4,6} and X_{5,6} are both rational. Our conjectures are evidenced by computer experiments. In an unexpected twist, existence of lattices D_6, E_8, and \Lambda_{10} turns out to be crucial.

We discuss the space VPS(Q, H) of ideals with Hilbert function H=(1,n,n,...) that are apolar to a full rank quadric Q. We prove that its components of saturated ideals are closely related to the locus of Gorenstein algebras and to the Slip component in border apolarity. We also point out an important error in ["The variety of polar simplices", Ranestad, K., Schreyer, F.-O. (2013)] and provide the necessary corrections.

The Surface Evolver is a computer program that minimizes the energy of a surface subject to constraints. The surface is represented as a simplicial complex. The energy can include surface tension, gravity and other forms. Constraints can be geometrical constraints on vertex positions or constraints on integrated quantities such as body volumes. The minimization is done by evolving the surface down the energy gradient. This paper describes the mathematical model used and the operations available to interactively modify the surface.

We consider the growth rate of the Mahler measure in discrete dynamical systems with the Laurent property, and in cluster algebras, and compare this with other measures of growth. In particular, we formulate the conjecture that the growth rate of the logarithmic Mahler measure coincides with the algebraic entropy, which is defined in terms of degree growth. Evidence for this conjecture is provided by exact and numerical calculations of the Mahler measure for a family of Laurent polynomials generated by rank 2 cluster algebras, for a recurrence of third order related to the Markoff numbers, and for the Somos-4 recurrence. Also, for the sequence of Laurent polynomials associated with the Kronecker quiver (the cluster algebra of affine type \tilde{A}_1 we prove a precise formula for the leading order asymptotics of the logarithmic Mahler measure, which grows linearly.

Much progress has been made in classifying when the weak Lefschetz property holds for A=F[x,y,z]/I where char(F)=0 and I=(xd1,yd2,zd3,xa1ya2za3) is a monomial almost complete intersection. We conne...

We investigate the average value of the Frobenius trace at p over elliptic curves in a fixed conductor range with given rank. Plotting this average as p varies over the primes yields a striking oscillating pattern, the details of which vary with the rank. Based on this observation, we perform various data-scientific experiments with the goal of classifying elliptic curves according to their ranks.

In this article we study adjoint hypersurfaces of geometric objects obtained by intersecting simple polytopes with few facets in P^5 with the Grassmannian Gr(2,4). These generalize the positive Grassmannian, which is the intersection of Gr(2,4) with the simplex. We show that if the resulting object has five facets, it is a positive geometry and the adjoint hypersurface is unique. For the case of six facets we show that the adjoint hypersurface is not necessarily unique and give an upper bound on the dimension of the family of adjoints. We illustrate our results with a range of examples. In particular, we show that even if the adjoint is not unique, a positive hexahedron can still be a positive geometry.

We provide conjectural necessary and (separately) sufficient conditions for the Hilbert scheme of points of a given length to have the maximum dimension tangent space at a point. The sufficient condition is claimed for 3D and reduces the original problem to a problem in convex geometry. Proving either of the two conjectural statements will in particular resolve a long-standing conjecture by Briançon and Iarrobino back in the ’70s for the case of the powers of the maximal ideal. Furthermore, for specific classes of lengths, we conjecturally classify points satisfying the conjectural sufficient conditions. This in particular (conjecturally) provides many new explicit families of examples of maximum dimension tangent space at a point of the Hilbert schemes of points of lengths strictly between two consecutive tetrahedral numbers {3+k \choose 3}.

We have developed a reinforcement learning agent that often finds a minimal sequence of unknotting crossing changes for a knot diagram with up to 200 crossings, hence giving an upper bound on the unknotting number. We have used this to determine the unknotting number of 57k knots. We took diagrams of connected sums of such knots with oppositely signed signatures, where the summands were overlaid. The agent has found examples where several of the crossing changes in an unknotting collection of crossings result in hyperbolic knots. Based on this, we have shown that, given knots K and K' that satisfy some mild assumptions, there is a diagram of their connected sum and u(K)+u(K') unknotting crossings such that changing any one of them results in a prime knot. As a by-product, we have obtained a dataset of 2.6 million distinct hard unknot diagrams; most of them under 35 crossings. Assuming the additivity of the unknotting number, we have determined the unknotting number of 43 at most 12-crossing knots for which the unknotting number is unknown.

A set of 2n points in P^{n-1} is self-dual if it is invariant under the Gale transform. Motivated by Mukai’s work on canonical curves, Petrakiev showed that a general self-dual set of 14 points in P^6 arises as the intersection of the Grassmannian Gr(2,6) in its Plücker embedding in P^14 with a linear space of dimension 6. In this paper, we focus on the inverse problem of recovering such a linear space associated to a general self-dual set of points. We use numerical homotopy continuation to approach the problem and implement an algorithm in Julia to solve it. Along the way, we also implement the forward problem of slicing Grassmannians and use it to experimentally study the real solutions to this problem.

I work part-time as universitair docent (assistant professor) at Utrecht University, and part time at the Lean FRO. My main interests lie in arithmetic geometry and formalization of mathematics. From…

We study the problem of recovering a collection of n numbers from the evaluation of m power sums. This yields a system of polynomial equations, which can be underconstrained (m < n), square (m = n), or overconstrained (m > n). Fibers and images of power sum maps are explored in all three regimes, and in settings that range from complex and projective to real and positive. This involves surprising deviations from the Bézout bound, and the recovery of vectors from length measurements by p-norms.

We consider problems concerning the largest degrees of irreducible characters of symmetric groups, and the multiplicities of character degrees of symmetric groups. Using evidence from computer experiments, we posit several new conjectures or extensions of previous conjectures, and prove a number of results. One of these is that, if n >= 21, then there are at least eight irreducible characters of S_n, all of which have the same degree, and which have irreducible restriction to A_n. We explore similar questions about unipotent degrees of GL_n(q). We also make some remarks about how the experiments here shed light on posited algorithms for finding the largest irreducible character degree of S_n.