arXiv subjects

F. Janda

Publications and source records attributed to F. Janda.

4 recordsLinked to original sources

arXiv2020

A comparison of group testing architectures for COVID-19 testing

An important component of every country's COVID-19 response is fast and efficient testing - to identify and isolate cases, as well as for early detection of local hotspots. For many countries, producing a sufficient number of tests has been a serious limiting factor in their efforts to control COVID-19 infections. Group testing is a well-established mathematical tool, which can provide a substantial and inexpensive expansion of testing capacity. In this note, we compare several popular group testing schemes in the context of qPCR testing for COVID-19. We find that in practical settings, for identification of individuals with COVID-19, Dorfman testing is the best choice at prevalences up to 30%, while for estimation of COVID-19 prevalence rates in the total population, Gibbs-Gower testing is the best choice at prevalences up to 30% given a fixed and relatively small number of tests. For instance, at a prevalence of up to 2%, Dorfman testing gives an efficiency gain of 3.5--8; at 1% prevalence, Gibbs-Gower testing gives an efficiency gain of 18, even when capping the pool size at a feasible number . This note is intended as a helpful handbook for labs implementing group testing methods.

stat.ME↗

arXiv2018

Double ramification cycles with target varieties

Let X be a nonsingular projective algebraic variety, and let S be a line bundle on X. Let A = (a_1,..., a_n) be a vector of integers. Consider a map f from a pointed curve (C,x_1,...,x_n) to X satisfying the following condition: the line bundle f*(S) has a meromorphic section with zeroes and poles exactly at the marked points x_i with orders prescribed by the integers a_i. A compactification of the space of maps based upon the above condition is given by the moduli space of stable maps to rubber over X. The main result of the paper is an explicit formula (in tautological classes) for the push-forward of the virtual fundamental class of the moduli space of stable maps to rubber over X via the forgetful morphism to the moduli space of stable maps to X. In case X is a point, the result here specializes to Pixton's formula for the double ramification cycle. Applications of the new formula, viewed as calculating double ramification cycles with target X, are given.

math.AG↗

arXiv2016

Double ramification cycles on the moduli spaces of curves

Curves of genus g which admit a map to CP1 with specified ramification profile mu over 0 and nu over infinity define a double ramification cycle DR_g(mu,nu) on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves. The cycle DR_g(mu,nu) for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for DR_g(mu,nu) in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain's formula in the compact type case. When mu and nu are both empty, the formula for double ramification cycles expresses the top Chern class lambda_g of the Hodge bundle of the moduli space of stable genus g curves as a push-forward of tautological classes supported on the divisor of nonseparating nodes. Applications to Hodge integral calculations are given.

math.AG↗

arXiv2015

The hypergeometric functions of the Faber-Zagier and Pixton relations

The relations in the tautological ring of the moduli space M_g of nonsingular curves conjectured by Faber-Zagier in 2000 and extended to the moduli space of stable pointed curves by Pixton in 2012 are based upon two hypergeometric series A and B. The question of the geometric origins of these series has been solved in at least two ways (via the Frobenius structures associated to 3-spin curves and to CP1). The series A and B also appear in the study of descendent integration on the moduli spaces of open and closed curves. We survey here the various occurrences of A and B starting from their appearance in the asymptotic expansion of the Airy function (calculated by Stokes in the 19th century). Several open questions are proposed.

math.AG↗