Mordell-Lang in positive characteristic

We give a new proof of the Mordell-Lang conjecture in positive characteristic for finitely generated subgroups. We also make some progress towards the full Mordell-Lang conjecture in positive characteristic

Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration

We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type $\operatorname{SL}_n$ and $\operatorname{PGL}_n$. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore we prove for $d$ coprime to $n$, that the number of rank $n$ Higgs bundles of degree $d$ over a fixed curve defined over a finite field, is independent of $d$. This proves a conjecture by Mozgovoy--Schiffman in the coprime case.Comment: Various part of the article have been revise

Algebraic zip data

An algebraic zip datum is a tuple \CZ := (G,P,Q,\phi) consisting of a reductive group $G$ together with parabolic subgroups $P$ and $Q$ and an isogeny $\phi\colon P/R_uP\to Q/R_uQ$. We study the action of the group $E := \{(p,q)\in P{\times}Q | \phi(\pi_{P}(p)) =\pi_Q(q)\}$ on $G$ given by $((p,q),g)\mapsto pgq^{-1}$. We define certain smooth $E$-invariant subvarieties of $G$, show that they define a stratification of $G$. We determine their dimensions and their closures and give a description of the stabilizers of the $E$-action on $G$. We also generalize all results to non-connected groups. We show that for special choices of \CZ the algebraic quotient stack $[E \backslash G]$ is isomorphic to $[G \backslash Z]$ or to $[G \backslash Z']$, where $Z$ is a $G$-variety studied by Lusztig and He in the theory of character sheaves on spherical compactifications of $G$ and where $Z'$ has been defined by Moonen and the second author in their classification of $F$-zips. In these cases the $E$-invariant subvarieties correspond to the so-called "$G$-stable pieces" of $Z$ defined by Lusztig (resp. the $G$-orbits of $Z'$).Comment: 42 pages, added some references, to appear in Doc. Mat

Graded and filtered fiber functors on Tannakian categories

We study fiber functors on Tannakian categories which are equipped with a grading or a filtration. Our goal is to give a comprehensive set of foundational results about such functors. A main result is that each filtration on a fiber functor can be split by a grading fpqc-locally on the base schem

Causes, consequences, and management of tree spatial patterns in fire-frequent forests

2022 Summer.Includes bibliographical references.Increasingly, restoration treatments are being implemented to dually meet wildland fire hazard reduction alongside ecological objectives. Restoration treatments however deviate from conventional fuels treatments by emphasizing the re-creation of forest structure present prior to EuroAmerican settlement, notably the retention of single and grouped trees interspersed between canopy openings. As these historical forests persisted over cycles of fire returns, it is assumed that restoring these historical complex tree spatial patterns will, in turn, restore historical ecological processes. This includes more benign fire behavior that results in only partial tree mortality, allowing persistent and partial retention of forest cover over cycles of fire return. The qualitative description of historical forest structure, lacks, however, a clear process-based explanation detailing the interactions of heterogeneous forest structures and fire. While fires were historically frequent, it is unclear what role fire played in the genesis and maintenance of tree spatial patterns. If models of tree spatial dynamics can be improved and the interactions between tree spatial patterns and fire can be elucidated, forest managers will have an improved understanding of the implications of restoration-based fuels hazard reduction treatments both during fire-free periods and during fire events. The aims of this dissertation were to: 1) explore the causes of tree spatial patterns in dry fire-frequent forests; 2) investigate the consequences of tree spatial patterns on potential fire behavior and effects; 3) determine how alternate silvicultural strategies targeted at manipulation of tree spatial patterns can influence fire behavior and effects. In Chapter 2, I explored spatial patterns of tree regeneration over 44 years in absence of fire. In cooler periods, regeneration preferred clustering in openings, including openings following overstory mortality and away from overstory trees. Mortality risk of regeneration was heightened nearer overstory trees. In warmer periods, these trends reversed, likely because of a 'nurse effect' from the overstory. In anticipation of climate change, these results suggest silviculturists may benefit by capturing regeneration mortality in within openings while keeping regeneration near the overstory. In Chapter 3, I found that regenerating trees also form heterogeneous patterns following stand-replacing fires. In these sparse, early seral forests, all species were spatially aggregated, partly attributable to the influence of topography and beneficial interspecific attractions between ponderosa pine and other species. Results from this study suggest that scale-dependent, and often facilitatory, rather than competitive, processes act on regenerating trees. In Chapter 4, I studied the interaction between fire and tree spatial patterns, both historically and in modern forests. Tree mortality in the historical period was clustered and density-dependent because tree mortality was greater among small trees, which tended to be assembled in tightly spaced clusters. Tree mortality in the contemporary period was widespread, except for dispersed large trees, because most trees were a part of large, interconnected tree groups. Postfire tree patterns in the historical period, unlike the contemporary period, were within the historical range of variability found for the western United States. This divergence suggests that decades of forest dynamics without significant disturbances have altered the historical means of pyric pattern maintenance. In Chapter 5, I examined how fuels treatment designs with different manipulations of tree spatial patterns may influence treatment effectiveness. I simulated fires on hypothetical cuttings which manipulated the arrangement of crown fuels horizontally and vertically, either increasing the distance between tree crowns or not, and either removing small trees or not. All cutting methods reduced fire behavior and severity, but the results confirm possible tradeoffs between ecological restoration and hazard reduction; treatments that separated tree crowns reduced severity the most because these treatments reduced crown fire spread. But these can easily be overcome where restoration treatments incorporate small tree removal, because this action limits crown fire initiation. Managers could also incorporate managed fires to reduce surface fuel loads and use more aggressive cuttings to further gains in hazard reduction, regardless of cutting method used

Graded and Filtered Fiber Functors on Tannakian Categories

We study fiber functors on Tannakian categories which are equipped with a grading or a filtration. Our goal is to give a comprehensive set of foundational results about such functors. A main result is that each filtration on a fiber functor can be split by a grading fpqc-locally on the base scheme

Taxonomies of Model-theoretically Defined Topological Properties

A topological classification scheme consists of two ingredients: (1) an abstract class K of topological spaces; and (2) a taxonomy , i.e. a list of first order sentences, together with a way of assigning an abstract class of spaces to each sentence of the list so that logically equivalent sentences are assigned the same class.K, is then endowed with an equivalence relation, two spaces belonging to the same equivalence class if and only if they lie in the same classes prescribed by the taxonomy. A space X in K is characterized within the classification scheme if whenever Y E K, and Y is equivalent to X, then Y is homeomorphic to X. As prime example, the closed set taxonomy assigns to each sentence in the first order language of bounded lattices the class of topological spaces whose lattices of closed sets satisfy that sentence. It turns out that every compact two-complex is characterized via this taxonomy in the class of metrizable spaces, but that no infinite discrete space is so characterized. We investigate various natural classification schemes, compare them, and look into the question of which spaces can and cannot be characterized within them