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Abhinav Kumar

Publications and source records attributed to Abhinav Kumar.

126 records · Page 7Linked to original sources
arXiv2015

Optimal simplices and codes in projective spaces

We find many tight codes in compact spaces, i.e., optimal codes whose optimality follows from linear programming bounds. In particular, we show the existence (and abundance) of several hitherto unknown families of simplices in quaternionic projective spaces and the octonionic projective plane. The most noteworthy cases are 15-point simplices in HP^2 and 27-point simplices in OP^2, both of which are the largest simplices and the smallest 2-designs possible in their respective spaces. These codes are all universally optimal, by a theorem of Cohn and Kumar. We also show the existence of several positive-dimensional families of simplices in the Grassmannians of subspaces of R^n with n <= 8; close numerical approximations to these families had been found by Conway, Hardin, and Sloane, but no proof of existence was known. Our existence proofs are computer-assisted, and the main tool is a variant of the Newton-Kantorovich theorem. This effective implicit function theorem shows, in favorable conditions, that every approximate solution to a set of polynomial equations has a nearby exact solution. Finally, we also exhibit a few explicit codes, including a configuration of 39 points in OP^2 that form a maximal system of mutually unbiased bases. This is the last tight code in OP^2 whose existence had been previously conjectured but not resolved.

math.MG↗
arXiv2015

Examples of abelian surfaces with everywhere good reduction

We describe several explicit examples of simple abelian surfaces over real quadratic fields with real multiplication and everywhere good reduction. These examples provide evidence for the Eichler-Shimura conjecture for Hilbert modular forms over a real quadratic field. Several of the examples also support a conjecture of Brumer and Kramer on abelian varieties associated to Siegel modular forms with paramodular level structures.

math.NT↗
arXiv2015

Multiplicative excellent families of elliptic surfaces of type E_7 or E_8

We describe explicit multiplicative excellent families of rational elliptic surfaces with Galois group isomorphic to the Weyl group of the root lattices E_7 or E_8. The Weierstrass coefficients of each family are related by an invertible polynomial transformation to the generators of the multiplicative invariant ring of the associated Weyl group, given by the fundamental characters of the corresponding Lie group. As an application, we give examples of elliptic surfaces with multiplicative reduction and all sections defined over Q for most of the entries of fiber configurations and Mordell-Weil lattices in [Oguiso-Shioda '91], as well as examples of explicit polynomials with Galois group W(E_7) or W(E_8).

math.AG↗
arXiv2015

K3 surfaces and equations for Hilbert modular surfaces

We outline a method to compute rational models for the Hilbert modular surfaces Y_{-}(D), which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in Q(sqrt{D}), via moduli spaces of elliptic K3 surfaces with a Shioda-Inose structure. In particular, we compute equations for all thirty fundamental discriminants D with 1 < D < 100, and analyze rational points and curves on these Hilbert modular surfaces, producing examples of genus-2 curves over Q whose Jacobians have real multiplication over Q.

math.NT↗
arXiv2014

Optimal WiFi Sensing via Dynamic Programming

The problem of finding an optimal sensing schedule for a mobile device that encounters an intermittent WiFi access opportunity is considered. At any given time, the WiFi is in any of the two modes, ON or OFF, and the mobile's incentive is to connect to the WiFi in the ON mode as soon as possible, while spending as little sensing energy. We introduce a dynamic programming framework which enables the characterization of an explicit solution for several models, particularly when the OFF periods are exponentially distributed. While the problem for non-exponential OFF periods is ill-posed in general, a usual workaround in literature is to make the mobile device aware if one ON period is completely missed. In this restricted setting, using the DP framework, the deterministic nature of the optimal sensing policy is established, and value iterations are shown to converge to the optimal solution. Finally, we address the blind situation where the distributions of ON and OFF periods are unknown. A continuous bandit based learning algorithm that has vanishing regret (loss compared to the optimal strategy with the knowledge of distributions) is presented, and comparisons with the optimal schemes are provided for exponential ON and OFF times.

cs.IT↗
arXiv2014

Elliptic fibrations on a generic Jacobian Kummer surface

We describe all the elliptic fibrations with section on the Kummer surface X of the Jacobian of a very general curve C of genus 2 over an algebraically closed field of characteristic 0, modulo the automorphism group of X and the symmetric group on the Weierstrass points of C. In particular, we compute elliptic parameters and Weierstrass equations for the 25 different fibrations and analyze the reducible fibers and Mordell-Weil lattices. This answers completely a question posed by Kuwata and Shioda in 2008.

math.AG↗
arXiv2014

Using Elimination Theory to construct Rigid Matrices

The rigidity of a matrix A for target rank r is the minimum number of entries of A that must be changed to ensure that the rank of the altered matrix is at most r. Since its introduction by Valiant (1977), rigidity and similar rank-robustness functions of matrices have found numerous applications in circuit complexity, communication complexity, and learning complexity. Almost all nxn matrices over an infinite field have a rigidity of (n-r)^2. It is a long-standing open question to construct infinite families of explicit matrices even with superlinear rigidity when r = Omega(n). In this paper, we construct an infinite family of complex matrices with the largest possible, i.e., (n-r)^2, rigidity. The entries of an n x n matrix in this family are distinct primitive roots of unity of orders roughly exp(n^2 log n). To the best of our knowledge, this is the first family of concrete (but not entirely explicit) matrices having maximal rigidity and a succinct algebraic description. Our construction is based on elimination theory of polynomial ideals. In particular, we use results on the existence of polynomials in elimination ideals with effective degree upper bounds (effective Nullstellensatz). Using elementary algebraic geometry, we prove that the dimension of the affine variety of matrices of rigidity at most k is exactly n^2-(n-r)^2+k. Finally, we use elimination theory to examine whether the rigidity function is semi-continuous.

cs.CC↗
arXiv2013

K3 surfaces associated with curves of genus two

It is known (work of Galluzzi, Lombardo, Dolgachev and Naruki) that there is a unique K3 surface X which corresponds to a genus 2 curve C such that X has a Shioda-Inose structure with quotient birational to the Kummer surface of the Jacobian of C. In this paper we give an explicit realization of X as an elliptic surface over P^1 with specified singular fibers of type II^* and III^*. We describe how the Weierstrass coefficients are related to the Igusa-Clebsch invariants of C.

math.AG↗
arXiv2012

Point configurations that are asymmetric yet balanced

A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in R^3, and his classification is equivalent to the converse for R^3. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.

math.MG↗
arXiv2012

Rigidity of spherical codes

A packing of spherical caps on the surface of a sphere (that is, a spherical code) is called rigid or jammed if it is isolated within the space of packings. In other words, aside from applying a global isometry, the packing cannot be deformed. In this paper, we systematically study the rigidity of spherical codes, particularly kissing configurations. One surprise is that the kissing configuration of the Coxeter-Todd lattice is not jammed, despite being locally jammed (each individual cap is held in place if its neighbors are fixed); in this respect, the Coxeter-Todd lattice is analogous to the face-centered cubic lattice in three dimensions. By contrast, we find that many other packings have jammed kissing configurations, including the Barnes-Wall lattice and all of the best kissing configurations known in four through twelve dimensions. Jamming seems to become much less common for large kissing configurations in higher dimensions, and in particular it fails for the best kissing configurations known in 25 through 31 dimensions. Motivated by this phenomenon, we find new kissing configurations in these dimensions, which improve on the records set in 1982 by the laminated lattices.

math.MG↗
arXiv2010

Ground states and formal duality relations in the Gaussian core model

We study dimensional trends in ground states for soft-matter systems. Specifically, using a high-dimensional version of Parrinello-Rahman dynamics, we investigate the behavior of the Gaussian core model in up to eight dimensions. The results include unexpected geometric structures, with surprising anisotropy as well as formal duality relations. These duality relations suggest that the Gaussian core model possesses unexplored symmetries, and they have implications for a broad range of soft-core potentials.

cond-mat.stat-mech↗
arXiv2009

Algorithmic design of self-assembling structures

We study inverse statistical mechanics: how can one design a potential function so as to produce a specified ground state? In this paper, we show that unexpectedly simple potential functions suffice for certain symmetrical configurations, and we apply techniques from coding and information theory to provide mathematical proof that the ground state has been achieved. These potential functions are required to be decreasing and convex, which rules out the use of potential wells. Furthermore, we give an algorithm for constructing a potential function with a desired ground state.

cond-mat.stat-mech↗
arXiv2008

Counterintuitive ground states in soft-core models

It is well known that statistical mechanics systems exhibit subtle behavior in high dimensions. In this paper, we show that certain natural soft-core models, such as the Gaussian core model, have unexpectedly complex ground states even in relatively low dimensions. Specifically, we disprove a conjecture of Torquato and Stillinger, who predicted that dilute ground states of the Gaussian core model in dimensions 2 through 8 would be Bravais lattices. We show that in dimensions 5 and 7, there are in fact lower-energy non-Bravais lattices. (The nearest three-dimensional analog is the hexagonal close-packing, but it has higher energy than the face-centered cubic lattice.) We believe these phenomena are in fact quite widespread, and we relate them to decorrelation in high dimensions.

cond-mat.stat-mech↗
arXiv2008

The D_4 root system is not universally optimal

We prove that the D_4 root system (equivalently, the set of vertices of the regular 24-cell) is not a universally optimal spherical code. We further conjecture that there is no universally optimal spherical code of 24 points in S^3, based on numerical computations suggesting that every 5-design consisting of 24 points in S^3 is in a 3-parameter family (which we describe explicitly, based on a construction due to Sali) of deformations of the D_4 root system.

math.MG↗
arXiv2007

Uniqueness of the (22,891,1/4) spherical code

We use techniques of Bannai and Sloane to give a new proof that there is a unique (22,891,1/4) spherical code; this result is implicit in a recent paper by Cuypers. We also correct a minor error in the uniqueness proof given by Bannai and Sloane for the (23,4600,1/3) spherical code.

math.MG↗
arXiv2006

Universally optimal distribution of points on spheres

We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m distances between distinct points in it and it is a spherical (2m-1)-design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the E_8 and Leech lattices. We also prove the same result for the vertices of the 600-cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact two-point homogeneous spaces, and we conclude with an extension to Euclidean space.

math.MG↗
arXiv2004

The densest lattice in twenty-four dimensions

In this research announcement we outline the methods used in our recent proof that the Leech lattice is the unique densest lattice in R^24. Complete details will appear elsewhere, but here we illustrate our techniques by applying them to the case of lattice packings in R^2, and we discuss the obstacles that arise in higher dimensions.

math.MG↗