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.
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.
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.
This talk is concerned with generators for the bounded derived category of coherent sheaves over a noetherian scheme X of prime characteristic p when the Frobenius morphism is finite. It is shown that for any compact generator G of D(X), the e-th Frobenius pushforward of G classically generates the bounded derived category whenever pe is larger than the codepth of X, an invariant that is a measure of the singularity of X. From this, we can establish a canonical choice of strong generator when X is separated. The work is joint with Matthew R. Ballard, Srikanth B. Iyengar, Alapan Mukhopadhyay, and Josh Pollitz.
How do we describe the topology of the space of all nonconstant holomorphic (respectively, algebraic) maps F: X → Y from one complex manifold (respectively, variety) to another? What is, for example, its cohomology? Such problems are old but difficult, and are nontrivial even when the domain and range are Riemann spheres. In this talk I will explain how these problems relate to other parts of mathematics such as spaces of polynomials, arithmetic (e.g., the geometric Batyerv-Manin type conjectures), algebraic geometry (e.g., moduli spaces of elliptic fibrations, of smooth sections of a line bundle, etc) and if time permits, homotopy theory (e.g., derived indecomposables of modules over monoids). I will show how one can fruitfully attack such problems by incorporating techniques from topology to the holomorphic/algebraic world (e.g., by constructing a new spectral sequence).
In the 1960s, Grothendieck dreamt that algebraic varieties can be linearized in a universal way, leading to his philosophy of motives. Subsequent ideas of many mathematicians (especially Beilinson and Deligne) led to a beautiful conjectural framework surrounding the notion of a motive. In the last decade, thanks to the discovery of perfectoid geometry and subsequent developments, some aspects of this framework have also been realized unconditionally in the context of p-adic motives on p-adic varieties. In these lectures, I will survey some of this landscape, with an emphasis on the concrete applications that have guided the theoretical developments.
