The theory of matroids provides a unified abstract treatment of the concept of dependence in linear algebra and graph theory. In this talk we explain Bergman fans of matroids, and we investigate isomorphisms of Bergman fans for different fan structures. In particular, we introduce and study Cremona automorphisms.
A linear code is a vector subspace of 𝔽qn, where 𝔽q is a finite field with q elements. The family of linear error-correcting codes are specially important when one is attempting to transmit messages across a noisy communication channel. Data can be corrupted in transmission or storage by a variety of undesirable phenomenon, such as radio interference, electrical noise, scratch, etc.. It is useful to have a way to detect and correct such data corruption. An error-correcting code can correct more errors larger is its minimum distance. This course aims to introduce a family of error-correcting codes, the Algebraic Geometry Codes, and show how to use the theory of semigroups to improve the minimum distance of the code. This construction of codes make use of a function field in one variable over a finite field. We will show how the local information in one or two rational places, the knowledge of the semigroup in these places, can be used to improve the minimum distance of the code.
I will discuss the following conjecture: an irreducible ℚ̅ℓ-local system L on a smooth complex algebraic variety S arises in cohomology of a family of varieties over S if and only if L can be extended to an etale local system over some descent of S to a finitely generated subfield of complex numbers. I will describe the motivation for this conjecture coming from relative p-adic Hodge theory, known partial results, and possible approaches (not very successful so far) to formulating a purely p-adic (and thus hopefully more tractable) version of this conjecture. A large part of the talk will be expository, including material based on the ideas of Hélène Esnault, Raju Krishnamoorthy, and Josh Lam.
Multiplier ideals in characteristic zero and test ideals in positive characteristic are fundamental objects in the study of commutative algebra and birational geometry in equal characteristic. We introduced a mixed characteristic version of the multiplier/test ideal using the p-adic Riemann-Hilbert correspondence of Bhatt-Lurie. Under mild finiteness assumptions, we show that this version of test ideal commutes with localization and can be computed by a single alteration up to small perturbation.
We know thanks to the work of L. Giovenzana, A. Grossi, C. Onorati and D. Veniani that OG10-type hyperkähler manifolds do not admit any non-trivial symplectic automorphisms. What about non-regular symplectic birational transformations? Given a cubic fourfold V, one can construct a hyperkähler manifold XV of OG10-type following a construction of R. Laza, G. Saccà, C. Voisin. Such manifolds are known as LSV manifolds. It can be shown that any symplectic automorphism on V induces a symplectic birational transformation on XV. In a couple of works with L. Marquand, we study and classify all possible cohomological actions on the OG10-lattice which can be realised as symplectic birational transformations. By investigating further the induced action on cohomology, we exhibit a criterion to decide which of these actions can be realised as induced from a cubic fourfold on an associate LSV manifold.
In this talk, we relate the defect σ(X) := b4(X) − b2(X) of a singular cubic threefold X to various geometric properties of X. The question is motivated by the construction of the exceptional example of a Hyperkähler manifold of type O'Grady 10 from a cubic fourfold by Laza, Saccà and Voisin. By a result of Brosnan, the defect of hyperplane sections of the cubic fourfold is an obstruction for the LSV construction to work. The talk is based on a joint work in progress with Lisa Marquand.
In this lecture series I will explain how one can use deformation theory to study derived categories in positive characteristic.
I will start by giving an overview on what does it mean to 'lift' something 'to characteristic 0' and when is this possible. Then I will present a baby example: the study of the Fourier-Mukai partners of products of elliptic curves over algebraically closed fields of characteristic at least 5. After that, I will present Lieblich-Olsson deformation technique which allows us to deform derived equivalence. This is a very versatile tools with many applications (not just in positive characteristic!). I will conclude the series by going over some of these applications in greater details.
This will be a gentle introduction into Brauer groups and twisted sheaves. The emphasis will be on geometric aspects and eventually on moduli spaces of twisted sheaves on K3 surfaces. We will study the different ways to think about Brauer groups as groups of Azumaya algebras, Brauer-Severi varieties, twisted sheaves, 𝔾m-gerbes... How to translate from one to the other, how to define Chern classes, how to split Brauer classes, etc.
We develop a general approach to reduction along strong Dirac maps, which are a broad generalization of Poisson moment maps. We recover a number of familiar constructions and we give several new reduction procedures, including a multiplicative analogue of Whittaker reduction.
In a joint work with Alexei Skorobogatov we used a second-moment approach to prove asymptotics for the average of the von Mangoldt function over the values of a typical integer polynomial. As a consequence, we proved Schinzel's Hypothesis in 100% of the cases. In addition, we proved that a positive proportion of Châtelet equations have a rational point. I will explain subsequent joint work with Tim Browning and Joni Teräväinen that develops the method and establishes asymptotics for averages of an arithmetic function over the values of typical polynomials. Part of the new ideas come from the theory of averages of arithmetic functions in short intervals. One of the applications is that the Hasse principle holds for 100% of Châtelet equations. This agrees with the conjecture of Colliot-Thélène stating that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for rationally connected varieties.
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