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Kobi Kremnizer

Publications and source records attributed to Kobi Kremnizer.

At least 19 recordsLinked to original sources
arXiv2025

A Cousin Complex for the Quantum Projective Space

Grothendieck constructed a Cousin complex for abelian sheaves on an arbitrary topological space. In a special setting, its dual called the BGG resolution is applicable in representation theory. Arkhipov proposed a complex whose dual is only suitable for representation theory of quantum groups at roots of unity of prime order. It is desirable to get one which works for quantum groups at all roots of unity. For a quantum projective space, we provide such a complex.

math.QA↗
arXiv2025

A Localization Theorem for Finite W-algebras

Following the work of Beilinson-Bernstein and Kashiwara-Rouquier, we give a geometric interpretation of certain categories of modules over the finite W-algebra. As an application we reprove the Skryabin equivalence.

math.RT↗
arXiv2024

Zero-free half-planes of the ζ-function via spaces of analytic functions

In this article, we introduce a general approach for deriving zero-free half-planes for the Riemann zeta function $ζ$ by identifying topological vector spaces of analytic functions with specific properties. This approach is applied to weighted $\ell^2$ spaces and classical Hardy spaces $ H^p $ ($ 0<p\leq2 $). As a consequence precise conditions are obtained for the existence of zero-free half planes for the $ζ$-function.

math.NT↗
arXiv2022

Analytic geometry over F_1 and the Fargues-Fontaine curve

This paper develops a theory of analytic geometry over the field with one element. The approach used is the analytic counter-part of the Toen-Vaquie theory of schemes over F_1, i.e. the base category relative to which we work out our theory is the category of sets endowed with norms (or families of norms). Base change functors to analytic spaces over Banach rings are studied and the basic spaces of analytic geometry (like polydisks) are recovered as a base change of analytic spaces over F_1. We end by discussing some applications of our theory to the theory of the Fargues-Fontaine curve and to the ring Witt vectors.

math.AG↗
arXiv2022

Stein Domains in Banach Algebraic Geometry

In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a contribution towards the foundations of derived analytic geometry.

math.FA↗
arXiv2021

Analytic Hochschild-Kostant-Rosenberg Theorem

Let $R$ be a Banach ring. We prove that the category of chain complexes of complete bornological $R$-modules (and several related categories) is a derived algebraic context in the sense of Raksit. We then use the framework of derived algebra to prove a version of the Hochschild-Kostant-Rosenberg Theorem, which relates the circle action on the Hochschild algebra to the de Rham-differential-enriched-de Rham algebra of a simplicial, commutative, complete bornological algebra. This has a geometric interpretation in the language of derived analytic geometry, namely, the derived loop stack of a derived analytic stack is equivalent to the shifted tangent stack. Using this geometric interpretation we extend our results to derived schemes.

math.AG↗
arXiv2020

On the Sheafyness Property of Spectra of Banach Rings

Let R be a non-Archimedean Banach ring, satisfying some mild technical hypothesis that we will specify later on. We prove that to R one can associate a homotopical Huber spectrum Spa^h(R) via the introduction of the notion of derived rational localizations. The spectrum so obtained is endowed with a derived structural sheaf O_{Spa^h(R)} of simplicial Banach algebras for which the derived Tate-Cech complex is strictly exact. Under some hypothesis we can prove that there is a canonical morphism of sites Spa(R) -> |Spa^h(R)| that is an equivalence in some well-known examples of non-sheafy Banach rings. This permits to use the tools of derived geometry to understand the geometry of Spa(R) also when H^0(O_{Spa(R)}) is not a sheaf.

math.AG↗
arXiv2020

Generic probabilistic modelling and non-homogeneity issues for the UK epidemic of COVID-19

Coronavirus COVID-19 spreads through the population mostly based on social contact. To gauge the potential for widespread contagion, to cope with associated uncertainty and to inform its mitigation, more accurate and robust modelling is centrally important for policy making. We provide a flexible modelling approach that increases the accuracy with which insights can be made. We use this to analyse different scenarios relevant to the COVID-19 situation in the UK. We present a stochastic model that captures the inherently probabilistic nature of contagion between population members. The computational nature of our model means that spatial constraints (e.g., communities and regions), the susceptibility of different age groups and other factors such as medical pre-histories can be incorporated with ease. We analyse different possible scenarios of the COVID-19 situation in the UK. Our model is robust to small changes in the parameters and is flexible in being able to deal with different scenarios. This approach goes beyond the convention of representing the spread of an epidemic through a fixed cycle of susceptibility, infection and recovery (SIR). It is important to emphasise that standard SIR-type models, unlike our model, are not flexible enough and are also not stochastic and hence should be used with extreme caution. Our model allows both heterogeneity and inherent uncertainty to be incorporated. Due to the scarcity of verified data, we draw insights by calibrating our model using parameters from other relevant sources, including agreement on average (mean field) with parameters in SIR-based models.

stat.AP↗
arXiv2020

Comparison of different exit scenarios from the lock-down for COVID-19 epidemic in the UK and assessing uncertainty of the predictions

We model further development of the COVID-19 epidemic in the UK given the current data and assuming different scenarios of handling the epidemic. In this research, we further extend the stochastic model suggested in \cite{us} and incorporate in it all available to us knowledge about parameters characterising the behaviour of the virus and the illness induced by it. The models we use are flexible, comprehensive, fast to run and allow us to incorporate the following: -time-dependent strategies of handling the epidemic; -spatial heterogeneity of the population and heterogeneity of development of epidemic in different areas; -special characteristics of particular groups of people, especially people with specific medical pre-histories and elderly. Standard epidemiological models such as SIR and many of its modifications are not flexible enough and hence are not precise enough in the studies that requires the use of the features above. Decision-makers get serious benefits from using better and more flexible models as they can avoid of nuanced lock-downs, better plan the exit strategy based on local population data, different stages of the epidemic in different areas, making specific recommendations to specific groups of people; all this resulting in a lesser impact on economy, improved forecasts of regional demand upon NHS allowing for intelligent resource allocation.

q-bio.PE↗
arXiv2020

Fr\'echet Modules and Descent

We study several aspects of the study of Ind-Banach modules over Banach rings thereby synthesizing some aspects of homological algebra and functional analysis. This includes a study of nuclear modules and of modules which are flat with respect to the projective tensor product. We also study metrizable and Fr\'{e}chet Ind-Banach modules. We give explicit descriptions of projective limits of Banach rings as ind-objects. We study exactness properties of projective tensor product with respect to kernels and countable products. As applications, we describe a theory of quasi-coherent modules in Banach algebraic geometry. We prove descent theorems for quasi-coherent modules in various analytic and arithmetic contexts.

math.FA↗
arXiv2019

Beyond perturbation 2: asymptotics and Beilinson-Drinfeld Grassmannians in differential geometry

We prove that for any k greater or equal to 2, given a smooth compact k-dimensional manifold and a multiplicative k-1-gerbe on a Lie group, together with an integrable connection, there is a line bundle on the corresponding Beilinson-Drinfeld Grassmannian having the factorization property. We show that taking global sections of this line bundle we obtain a factorization algebra.

math.DG↗
arXiv2018

Hall categories and KLR categorification

This paper is the first step in the project of categorifying the bialgebra structure on the half of quantum group $U_{q}(\mathfrak{g})$ by using geometry and Hall algebras. We equip the category of D-modules on the moduli stack of objects of the category $Rep_{\mathbb{C}}(Q)$ of representations of a quiver with the structure of an algebra object in the category of stable $\infty$-categories. The data for this construction is provided by an extension of the Waldhausen construction for the category $Rep_{\mathbb{C}}(Q)$. We discuss the connection to the Khovanov-Lauda-Rouquier categorification of half of the quantum group $U_{q}(\mathfrak{g})$ associated to the quiver $Q$ and outline our approach to the categorification of the bialgebra structure.

math.RT↗
arXiv2018

Beyond perturbation 1: de Rham spaces

It is shown that if one uses the notion of infinity nilpotent elements due to Moerdijk and Reyes, instead of the usual definition of nilpotents to define reduced $C^\infty$-schemes, the resulting de Rham spaces are given as quotients by actions of germs of diagonals, instead of the formal neighbourhoods of the diagonals.

math.DG↗
arXiv2017

A Tannakian Reconstruction Theorem for IndBanach Spaces

Classically, Tannaka-Krein duality allows us to reconstruct a (co)algebra from its category of representation. In this paper we present an approach that allows us to generalise this theory to the setting of Banach spaces. This leads to several interesting applications in the directions of analytic quantum groups, bounded cohomology and galois cohomology. A large portion of this paper is dedicated to such examples.

math.FA↗
arXiv2017

Quasi-coherent sheaves in differential geometry

It is proved that the category of simplicial complete bornological spaces over $\mathbb R$ carries a combinatorial monoidal model structure satisfying the monoid axiom. For any commutative monoid in this category the category of modules is also a monoidal model category with all cofibrant objects being flat. In particular, weak equivalences between these monoids induce Quillen equivalences between the corresponding categories of modules. On the other hand, it is also proved that the functor of pre-compact bornology applied to simplicial $C^\infty$-rings preserves and reflects weak equivalences, thus assigning stable model categories of modules to simplicial $C^\infty$-rings.

math.DG↗
arXiv2015

Non-Archimedean analytic geometry as relative algebraic geometry

We show that Berkovich analytic geometry can be viewed as relative algebraic geometry in the sense of Toën--Vaquié--Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category we can define a topology on certain subcategories of the of the category of affine schemes with respect to this category. By examining this topology for the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in analytic geometry. This gives a functor of points approach to non-Archimedean analytic geometry and in this way we also get definitions of (higher) non-Archimedean analytic stacks. We demonstrate that the category of Berkovich analytic spaces embeds fully faithfully into the category of varieties in our version of relative algebraic geometry. We also include a treatment of quasi-coherent sheaf theory in analytic geometry. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.

math.AG↗
arXiv2015

Integrated Information-induced quantum collapse

We present a novel spontaneous collapse model where size is no longer the property of a physical system which determines its rate of collapse. Instead, we argue that the rate of spontaneous localization should depend on a system's quantum Integrated Information (QII), a novel physical property which describes a system's capacity to act like a quantum observer. We introduce quantum Integrated Information, present our QII collapse model and briefly explain how it may be experimentally tested against quantum theory.

quant-ph↗