## Type of Work

Article (61) Book (0) Theses (9) Multimedia (0)

## Peer Review

Peer-reviewed only (70)

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## Campus

UC Berkeley (10) UC Davis (0) UC Irvine (14) UCLA (4) UC Merced (13) UC Riverside (0) UC San Diego (1) UCSF (24) UC Santa Barbara (1) UC Santa Cruz (0) UC Office of the President (19) Lawrence Berkeley National Laboratory (2) UC Agriculture & Natural Resources (0)

## Department

Research Grants Program Office (RGPO) (19) California Breast Cancer Research Program (5) University of California Research Initiatives (UCRI) (1) Multicampus Research Programs and Initiatives (MRPI); a funding opportunity through UC Research Initiatives (UCRI) (1)

Department of Emergency Medicine (UCI) (1) Institute for Clinical and Translational Science (1)

## Journal

Proceedings of the Annual Meeting of the Cognitive Science Society (13) Western Journal of Emergency Medicine: Integrating Emergency Care with Population Health (1)

## Discipline

Social and Behavioral Sciences (19) Life Sciences (1) Medicine and Health Sciences (1)

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BY - Attribution required (10) BY-NC - Attribution; NonCommercial use only (1)

## Scholarly Works (70 results)

This thesis is devoted to constructing the triangulated categories of motives over fs log schemes with rational coefficients and their six operation formalism. Throughout the introduction, let $\Lambda$ be a fixed ring. For simplicity, assume also that every log scheme we deal with in the introduction is a noetherian fs log schemes over the spectrum of a fixed prime field or Dedekind domain.

Given a singular scheme X over a field k, we consider the problem of resolving the singularities of X by an algebraic stack. When X is a toroidal embedding or is etale locally the quotient of a smooth scheme by a linearly reductive group scheme, we show that such &ldquo stacky resolutions &rdquo exist. Moreover, these resolutions are canonical and easily understandable in terms of the singularities of X.

We give three applications of our stacky resolution theorems: various generalizations of the Chevalley-Shephard-Todd Theorem, a Hodge decomposition in characteristic p, and a theory of toric Artin stacks extending the work of Borisov-Chen-Smith. While these applications are seemingly different, they are all related by the common theme of using stacky resolutions to study singular schemes.

In developing homotopy theory in algebraic geometry, Michael Artin and Barry Mazur studied the

In order to shed light on Orlov’s conjecture that derived equivalent smooth, projective varieties have isomorphic Chow motives, we examine the zeta functions of derived equivalent varieties over finite fields; in this setting Orlov’s conjecture predicts equality of zeta functions. It is demonstrated that derived equivalent smooth, projective varieties over finite fields that are abelian or satisfy a certain condition on their cohomology. This condition is satisfied, for example, by a surface or Calabi–Yau 3–fold.

One of our approaches to comparing the zeta functions of derived equivalent varieties over finite fields comes from using the Lefschetz Fixed Point Theorem to turn the question into one of comparing the -adic ́etale cohomology of varieties. Cohomology groups are not in general preserved under the action of Fourier–Mukai equivalences on cohomology, but cohomological structures we call even and odd Mukai–Hodge structures, which are a realization of the Mukai motive, are preserved. Investigation into when isomorphism of these cohomological structures implies equality of zeta functions gives us our cohomological condition for equality of zeta functions.

We also develop a relative version of the map Fourier–Mukai transforms induce on cohomology and define a relative notion of even and odd Mukai–Hodge structures, and show these structures are preserved in a situation arising from the derived equivalence of smooth, projective varieties with semiample (anti-)canonical bundles. Using this result, it is demonstrated that when derived equivalent smooth, projective varieties have semiample (anti-)canonical bundles, the fibers over any fixed geometric point in their shared (anti-)canonical variety must also have isomorphic even and odd Mukai–Hodge structures. Hence, for any such varieties over finite fields, if their geometric fibers satisfy any of the conditions identified for isomorphism of Mukai–Hodge structures to imply equality of zeta functions, then the varieties themselves also have equal zeta functions.

Abramovich, Corti and Vistoli have studied modular compactifications of stacks of curves equipped with abelian level structures arising as substacks of the stack of twisted stable maps into the classifying stack of a finite group, provided the order of the group is invertible on the base scheme. Recently Abramovich, Olsson and Vistoli extended the notion of twisted stable maps to allow arbitrary base schemes, where the target is a tame stack, not necessarily Deligne-Mumford.

In this dissertation, we use this extended notion of twisted stable maps to extend the results of Abramovich, Corti and Vistoli to the case of elliptic curves with level structures over arbitrary base schemes. We prove that we recover the compactified Katz-Mazur regular models, with a natural moduli interpretation in terms of level structures on Picard schemes of twisted curves. Additionally, we study the interactions of the different such moduli stacks contained in a stack of twisted stable maps in characteristics dividing the level.

We develop the notion of stratifiability in the context of derived categories and the six operations for stacks in the work of Laszlo and Olsson. Then we reprove Behrend's Lefschetz trace formula for stacks, and give the meromorphic continuation of the L-series of stacks defined over a finite field. We give an upper bound for the weights of the cohomology groups of stacks, and as an application, prove the decomposition theorem for perverse sheaves on stacks with affine diagonal, both over finite fields and over the complex numbers. Along the way, we generalize the structure theorem of mixed sheaves and the generic base change theorem for stacks. We also give a short exposition on the lisse-analytic topoi of complex analytic stacks, and give a comparison between the lisse-etale topos of a complex algebraic stack and the lisse-analytic topos of its analytification.