Daniel Krashen
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View article: Conformal Blocks on Smoothings via Mode Transition Algebras
Conformal Blocks on Smoothings via Mode Transition Algebras Open
Here we introduce a series of associative algebras attached to a vertex operator algebra V of CFT type, called mode transition algebras, and show they reflect both algebraic properties of V and geometric constructions on moduli of curves. …
View article: Finiteness of formal pushforwards
Finiteness of formal pushforwards Open
Under mild hypotheses, given a scheme $U$ and an open subset $V$ whose complement has codimension at least two, the pushforward of a torsion-free coherent sheaf on $V$ is coherent on $U$. We prove an analog of this result in the context of…
View article: Classes in $\mathrm H_{p^m}^{n+1}(F)$ of lower exponent
Classes in $\mathrm H_{p^m}^{n+1}(F)$ of lower exponent Open
Let $F$ be a field of characteristic $p>0$. We prove that if a symbol $A=ω\otimes β_1 \otimes \dots \otimes β_n$ in $H_{p^m}^{n+1}(F)$ is of exponent dividing $p^{m-1}$, then its symbol length in $H_{p^{m-1}}^{n+1}(F)$ is at most $p^n$. In…
View article: Morita equivalences for Zhu's algebra
Morita equivalences for Zhu's algebra Open
Through the introduction of new ideals, and with the assistance of the $d$-th mode transition algebras $\mathfrak{A}_d$, for $d\in \mathbb{N}$, we show how Zhu's associative algebra $\mathsf{A}$, conventionally valued for tracking informat…
View article: Conformal blocks on smoothings via mode transition algebras
Conformal blocks on smoothings via mode transition algebras Open
Here we define a series of associative algebras attached to a vertex operator algebra $V$, called mode transition algebras, showing they reflect both algebraic properties of $V$ and geometric constructions on moduli of curves. One can defi…
View article: Brauertsch fields
Brauertsch fields Open
We prove a local-to-global principle for Brauer classes: for any finite collection of non-trivial Brauer classes on a variety over a field of transcendence degree at least 3, there are infinitely many specializations where each class stays…
View article: Local-Global Principles for Constant Reductive Groups over Semi-Global Fields
Local-Global Principles for Constant Reductive Groups over Semi-Global Fields Open
We study local-global principles for torsors under reductive linear algebraic groups over semi-global fields; that is, over one-variable function fields over complete discretely valued fields. We provide conditions on the group and the sem…
View article: Factorization presentations
Factorization presentations Open
Modules over a vertex operator algebra V give rise to sheaves of coinvariants on moduli of stable pointed curves. If V satisfies finiteness and semi-simplicity conditions, these sheaves are vector bundles. This relies on factorization, an …
View article: Transcendental splitting fields of division algebras
Transcendental splitting fields of division algebras Open
We examine when division algebras can share common splitting fields of certain types. In particular, we show that one can find fields for which one has infinitely many Brauer classes of the same index and period at least 3, all nonisomorph…
View article: Local-global principles for constant reductive groups over semi-global fields
Local-global principles for constant reductive groups over semi-global fields Open
We study local-global principles for torsors under reductive linear algebraic groups over semi-global fields; i.e., over one variable function fields over complete discretely valued fields. We provide conditions on the group and the semigl…
View article: Local–global principles for curves over semi‐global fields
Local–global principles for curves over semi‐global fields Open
We investigate local-global principles for Galois cohomology, in the context\nof function fields of curves over semi-global fields. This extends work of\nKato's on the case of function fields of curves over global fields.\n
View article: Local-global principles for tori over arithmetic curves
Local-global principles for tori over arithmetic curves Open
In this paper we study local-global principles for tori over semi-global\nfields, which are one variable function fields over complete discretely valued\nfields. In particular, we show that for principal homogeneous spaces for tori\nover t…
View article: Multiparty Non-Interactive Key Exchange and More From Isogenies on Elliptic Curves
Multiparty Non-Interactive Key Exchange and More From Isogenies on Elliptic Curves Open
We describe a framework for constructing an efficient non-interactive key exchange (NIKE) protocol for n parties for any n ≥ 2. Our approach is based on the problem of computing isogenies between isogenous elliptic curves, which is believe…
View article: A Tannakian approach to patching
A Tannakian approach to patching Open
We use Tannakian methods to show that patching for coherent sheaves implies patching for objects in any Noetherian algebraic stack with affine stabilizers. Among other things, this gives a straightforward way to prove patching for torsors …
View article: Multiparty Non-Interactive Key Exchange and More From Isogenies on Elliptic Curves
Multiparty Non-Interactive Key Exchange and More From Isogenies on Elliptic Curves Open
We describe a framework for constructing an efficient non-interactive key exchange (NIKE) protocol for n parties for any n ≥ 2. Our approach is based on the problem of computing isogenies between isogenous elliptic curves, which is believe…
View article: Local-global principles for zero-cycles on homogeneous spaces over arithmetic function fields
Local-global principles for zero-cycles on homogeneous spaces over arithmetic function fields Open
We study the existence of zero-cycles of degree one on varieties that are defined over a function field of a curve over a complete discretely valued field. We show that local-global principles hold for such zero-cycles provided that local-…
View article: Local-global Galois theory of arithmetic function fields
Local-global Galois theory of arithmetic function fields Open
We study the relationship between the local and global Galois theory of function fields over a complete discretely valued field. We give necessary and sufficient conditions for local separable extensions to descend to global extensions, an…
View article: Local-Global Principles for Zero-Cycles on Homogeneous Spaces over Arithmetic Function Fields
Local-Global Principles for Zero-Cycles on Homogeneous Spaces over Arithmetic Function Fields Open
We study the existence of zero-cycles of degree one on varieties that are defined over a function field of a curve over a complete discretely valued field. In particular, we show that local-global principles hold for such zero-cycles provi…
View article: Schubert cycles and subvarieties of generalized Severi-Brauer varieties
Schubert cycles and subvarieties of generalized Severi-Brauer varieties Open
We study twisted forms of Schubert cells in generalized Severi-Brauer varieties, and show that the codimension $2$ Chow groups of these varieties are torsion free in certain cases, using the topological filtration on their K-theory