Max Lieblich

Moduli of complexes on a proper morphism

Given a proper morphism X -> S, we show that a large class of objects in the derived category of X naturally form an Artin stack locally of finite presentation over S. This class includes S-flat coherent sheaves and, more generally, contains the collection of all S-flat objects which can appear in the heart of a reasonable sheaf of t-structures on X. In this sense, this is the Mother of all Moduli Spaces (of sheaves). The proof proceeds by studying the finite presentation properties, deformation theory, and Grothendieck existence theorem for objects in the derived category, and then applying Artins representability theorem.

Moduli of twisted sheaves

We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of surfaces, we show (under a mild hypothesis on the twisting class) that the spaces are asympotically geometrically irreducible, normal, generically smooth, and l.c.i. over the base. We also develop general tools necessary for these results: the theory of associated points and purity of sheaves on Artin stacks, twisted Bogomolov inequalities, semistability and boundedness results, and basic results on twisted Quot-schemes on a surface.

Twisted sheaves and the period-index problem

We use twisted sheaves and their moduli spaces to study the Brauer group of a scheme. In particular, we (1) show how twisted methods can be efficiently used to re-prove the basic facts about the Brauer group and cohomological Brauer group (including Gabber’s theorem that they coincide for a separated union of two affine s chemes), (2) give a new proof of de Jong’s period-index theorem for surfaces over algebraically closed fields, and (3) prove an analogous result for surfaces over finite fields. We also include a reduction of all period-index problems for Brauer groups of function fields over algebraically closed fields to characteristic 0, which (among other things) extends de Jong’s result to include classes of period divisible by the characteristic of the base field. Finally, we use the theory developed here to give counterexamples to a standard type of local-to-global conjecture for geometrically rational varieties over the function field of the projective plane.

Remarks on the stack of coherent algebras

We consider the stack of coherent algebras with proper support, a moduli problem generalizing Alexeev and Knutsons stack of branchvarieties to the case of an Artin stack. The main results are proofs of the existence of Quot and Hom spaces in greater generality than is currently known and several applications to Alexeev and Knutsons original construction: a proof that the stack of branchvarieties is always algebraic, that limits of one-dimensional families always exist, and that the connected components of the stack of branchvarieties are proper over the base under certain hypotheses on the ambient stack.

Period and index in the Brauer group of an arithmetic surface

Abstract In this paper we introduce two new ways to split ramification of Brauer classes on surfaces using stacks. Each splitting method gives rise to a new moduli space of twisted stacky vector bundles. By studying the structure of these spaces we prove new results on the standard period-index conjecture. The first yields new bounds on the period-index relation for classes on curves over higher local fields, while the second can be used to relate the Hasse principle for forms of moduli spaces of stable vector bundles on pointed curves over global fields to the period-index problem for Brauer groups of arithmetic surfaces. We include an appendix by Daniel Krashen showing that the local period-index bounds are sharp.

Nagata compactification for algebraic spaces

We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian approximation for such algebraic spaces, generalizing earlier results in the case of schemes.

Automation of Workplace Lifting Hazard Assessment for Musculoskeletal Injury Prevention

ObjectivesExisting methods for practically evaluating musculoskeletal exposures such as posture and repetition in workplace settings have limitations. We aimed to automate the estimation of parameters in the revised United States National Institute for Occupational Safety and Health (NIOSH) lifting equation, a standard manual observational tool used to evaluate back injury risk related to lifting in workplace settings, using depth camera (Microsoft Kinect) and skeleton algorithm technology.MethodsA large dataset (approximately 22,000 frames, derived from six subjects) of simultaneous lifting and other motions recorded in a laboratory setting using the Kinect (Microsoft Corporation, Redmond, Washington, United States) and a standard optical motion capture system (Qualysis, Qualysis Motion Capture Systems, Qualysis AB, Sweden) was assembled. Error-correction regression models were developed to improve the accuracy of NIOSH lifting equation parameters estimated from the Kinect skeleton. Kinect-Qualysis errors were modelled using gradient boosted regression trees with a Huber loss function. Models were trained on data from all but one subject and tested on the excluded subject. Finally, models were tested on three lifting trials performed by subjects not involved in the generation of the model-building dataset.ResultsError-correction appears to produce estimates for NIOSH lifting equation parameters that are more accurate than those derived from the Microsoft Kinect algorithm alone. Our error-correction models substantially decreased the variance of parameter errors. In general, the Kinect underestimated parameters, and modelling reduced this bias, particularly for more biased estimates. Use of the raw Kinect skeleton model tended to result in falsely high safe recommended weight limits of loads, whereas error-corrected models gave more conservative, protective estimates.ConclusionsOur results suggest that it may be possible to produce reasonable estimates of posture and temporal elements of tasks such as task frequency in an automated fashion, although these findings should be confirmed in a larger study. Further work is needed to incorporate force assessments and address workplace feasibility challenges. We anticipate that this approach could ultimately be used to perform large-scale musculoskeletal exposure assessment not only for research but also to provide real-time feedback to workers and employers during work method improvement activities and employee training.

A case-crossover study of heat exposure and injury risk in outdoor agricultural workers

Background Recent research suggests that heat exposure may increase the risk of traumatic injuries. Published heat-related epidemiological studies have relied upon exposure data from individual weather stations. Objective To evaluate the association between heat exposure and traumatic injuries in outdoor agricultural workers exposed to ambient heat and internal heat generated by physical activity using modeled ambient exposure data. Methods A case-crossover study using time-stratified referent selection among 12,213 outdoor agricultural workers with new Washington State Fund workers’ compensation traumatic injury claims between 2000 and 2012 was conducted. Maximum daily Humidex exposures, derived from modeled meteorological data, were assigned to latitudes and longitudes of injury locations on injury and referent dates. Conditional logistic regression was used to estimate odds ratios of injury for a priori daily maximum Humidex categories. Results The mean of within-stratum (injury day and corresponding referent days) standard deviations of daily maximum Humidex was 4.8. The traumatic injury odds ratio was 1.14 (95% confidence interval 1.06, 1.22), 1.15 (95% confidence interval 1.06, 1.25), and 1.10 (95% confidence interval 1.01, 1.20) for daily maximum Humidex of 25–29, 30–33, and ≥34, respectively, compared to < 25, adjusted for self-reported duration of employment. Stronger associations were observed during cherry harvest duties in the June and July time period, compared to all duties over the entire study period. Conclusions Agricultural workers laboring in warm conditions are at risk for heat-related traumatic injuries. Combined heat-related illness and injury prevention efforts should be considered in high-risk populations exposed to warm ambient conditions in the setting of physical exertion.

Deformation Theory and Rational Points on Rationally Connected Varieties

We give an account of some recent work on the existence of rational points on varieties over function fields, starting with basic material on deformation theory and the bend-and-break theorem. We emphasize the connection with the geometry of moduli spaces and include a sketch of the irreducibility of M g as a model. All details are relegated to the references.

On the Ubiquity of Twisted Sheaves

We describe some recent work on the uses of twisted sheaves in algebra, arithmetic, and geometry. In particular, we touch on the role of twisted sheaves in: 1. The geometry of the period-index problem for the Brauer group 2. The connection between finiteness of the u-invariant and Colliot-Thelene’s conjecture on 0-cycles 3. The link between the Tate conjecture for K3 surfaces and finiteness of the set of isomorphism classes of K3 surfaces over a finite field 4. The geometry of rational curves on the moduli spaces of supersingular K3 surfaces