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We give a proper fractional extension of the classical calculus of variations by considering variational functionals with a Lagrangian depending on a combined Caputo fractional derivative and the classical derivative. Euler-Lagrange…
The aim of this paper is to present a very simple original, purely formal, proof of Quillen's adjunction theorem for derived functors, and of some more recent variations and generalizations of this theorem. This is obtained by proving an…
We develop the $L$-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of quadratic twist of modular $L$-functions using multiple Dirichlet series under the generalized Riemann…
We prove the precise inversion of adjunction formula for quotient singularities and klt Cartier divisors. As an application, we prove the semi-continuity of minimal log discrepancies for klt hyperquotient singularities.
This is a first instalment of much larger work about relations between birational geometry and moduli of triples. The extraction of work is mainly related to Theorem 6. It is a weak version of Kawamata's Conjecture 1 and an important…
We study positivity properties of the moduli (b-)divisor associated to a relative log pair $(X,B)/Y$ with relatively trivial log canonical class.
We prove effective forms of the Sato-Tate conjecture for holomorphic cuspidal newforms which improve on the author's previous work (solo and joint with Lemke Oliver). We also prove an effective form of the joint Sato-Tate distribution for…
We develop a topological approach to prove the generalized Lax conjecture using the fact that determinants of sufficiently big symmetric linear pencils are able to express the rigidly convex sets of RZ polynomials of any degree $d$.…
We introduce the notion of basic slc-trivial fibrations. It is a generalization of that of Ambro's lc-trivial fibrations. Then we study fundamental properties of basic slc-trivial fibrations by using the theory of variations of mixed Hodge…
In this paper we adopt a geometric point of view regarding a famous conjecture due to Littlewood in diophantine approximation of real numbers. Following the spirit of the geometric theory of continued fractions, we give a sufficient…
Gradient-based techniques are becoming increasingly critical in quantitative fields, notably in statistics and computer science. The utility of these techniques, however, ultimately depends on how efficiently we can evaluate the derivatives…
We develop an intersection theory for a singular hemitian line bundle with positive curvature current on a smooth projective variety and irreducible curves on the variety. And we prove the existence of a natural rational fibration structure…
As an application of the residue functions corresponding to the lc-measures developed by the authors, the proof of the injectivity theorem on compact K\"ahler manifolds for plt pairs by Matsumura is improved in this article to allow…
Gauge theories with fermions in adjoint and fundamental representations are relevant for many different applications including composite Higgs models and general aspects of the confinement problem. We present first results from simulations…
A particular case of the Jacobian conjecture is considered and for small dimensional cases a computational approach is offered
Variational inference is computationally challenging in models that contain both conjugate and non-conjugate terms. Methods specifically designed for conjugate models, even though computationally efficient, find it difficult to deal with…
We prove a few cases of a conjecture on the invariance of cohomological support loci under derived equivalence by establishing a concrete connection with the related problem of the invariance of Hodge numbers. We use the main case in order…
In this article, we establish a strategy to the abundance conjecture for K\"ahler varieties via induction on algebraic dimension. Our strategy is to reduce the abundance conjecture for K\"ahler varieties to the abundance conjecture for…
This paper demonstrates the existence of $\mathbb{Q}$-complements for algebraically integrable log-Fano foliations on klt ambient varieties. Additionally, we investigate properties of algebraically integrable Fano foliations such as a…
The paper [GLZ] "L-functions of Carlitz modules, resultantal varieties and rooted binary trees" is devoted to a description of some resultantal varieties related to L-functions of Carlitz modules. It contains a conjecture that some of these…