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Abstract We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases, our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack $\mathcal {X}$ , which specializes to the Batyrev–Manin conjecture when $\mathcal {X}$ is a scheme and to Malle’s conjecture when $\mathcal {X}$ is the classifying stack of a finite group. 

Abstract Let $X$ be a quasi-projective variety over a number field, admitting (after passage to $\mathbb {C}$) a geometric variation of Hodge structure whose period mapping has zero-dimensional fibers. Then the integral points of $X$ are sparse: the number of such points of height $\leq B$ grows slower than any positive power of $B$. For example, homogeneous integral polynomials in a fixed number of variables and degree, with discriminant divisible only by a fixed set of primes, are sparse when considered up to integral linear substitutions. 

The COVID-19 pandemic is a global threat presenting health, economic, and social challenges that continue to escalate. Metapopulation epidemic modeling studies in the susceptible–exposed–infectious–removed (SEIR) style have played important roles in informing public health policy making to mitigate the spread of COVID-19. These models typically rely on a key assumption on the homogeneity of the population. This assumption certainly cannot be expected to hold true in real situations; various geographic, socioeconomic, and cultural environments affect the behaviors that drive the spread of COVID-19 in different communities. What’s more, variation of intracounty environments creates spatial heterogeneity of transmission in different regions. To address this issue, we develop a human mobility flow-augmented stochastic SEIR-style epidemic modeling framework with the ability to distinguish different regions and their corresponding behaviors. This modeling framework is then combined with data assimilation and machine learning techniques to reconstruct the historical growth trajectories of COVID-19 confirmed cases in two counties in Wisconsin. The associations between the spread of COVID-19 and business foot traffic, race and ethnicity, and age structure are then investigated. The results reveal that, in a college town (Dane County), the most important heterogeneity is age structure, while, in a large city area (Milwaukee County), racial and ethnic heterogeneity becomes more apparent. Scenario studies further indicate a strong response of the spread rate to various reopening policies, which suggests that policy makers may need to take these heterogeneities into account very carefully when designing policies for mitigating the ongoing spread of COVID-19 and reopening.

 

The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve with a marked point, which is trivial when the curve is hyperelliptic with a marked Weierstrass point. The image of the Ceresa cycle under a certain cycle class map provides a class in étale cohomology called the Ceresa class. Describing the Ceresa class explicitly for non-hyperelliptic curves is in general not easy. We present a "combinatorialization" of this problem, explaining how to define a Ceresa class for a tropical algebraic curve, and also for a topological surface endowed with a multiset of commuting Dehn twists (where it is related to the Morita cocycle on the mapping class group). We explain how these are related to the Ceresa class of a smooth algebraic curve over ℂ((t)), and show that the Ceresa class in each of these settings is torsion.