Hermitian generalization of the Rarita-Schwinger operators

We introduce two new linear differential operators which are invariant with respect to the unitary group SU（n）. They constitute analogues of the twistor and the Rarita-Schwinger operator in the orthogonal case. The natural setting for doing this is Hermitian Clifford Analysis. Such operators are constructed by twisting the two versions of the Hermitian Dirac operator 6z_ and 6z_ and then projecting on irreducible modules for the unitary group. We then study some properties of their spaces of nullsolutions and we find a formulation of the Hermitian Rarita-Schwinger operators in terms of Hermitian monogenic polynomials.     

Counting rational points on cubic curves

We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals. The bounds are uniform in the curve and involve the rank of the corresponding Jacobian. The method used in the proof is a combination of the “determinant method” with an m-descent on the curve.     

Recovering plane curves of low degree from their inflection lines and inflection points

In this paper we consider the following problem: is it possible to recover a smooth plane curve of degree d ≥ 3 from its inflection lines? We answer the posed question positively for a general smooth plane quartic curve, making the additional assumption that also one inflection point is given, and for any smooth plane cubic curve.     

Tautological Classes of the Stack of Rational Nodal Curves

This is the second in a series of three papers in which we investigate the rational Chow ring of the stack 𝔐₀ consisting of nodal curves of genus 0. Here we define the basic classes: the classes of strata and the Mumford classes.     

Sustainable Access to Acridin-9-(10H)ones with an Embedded m-Terphenyl Moiety Based on a Three-Component Reaction

A Ce(IV)-catalyzed three-component reaction between chalcones, anilines and β-ketoesters followed by a microwave-assisted thermal cyclization afforded 1,3-diaryl-1,2-dihydroacridin-9(10H)-ones. Their microwave irradiation in nitrobenzene, acting both as solvent and oxidant, afforded fully unsaturated 1,3-diarylacridin-9(10H)-ones, which combine acridin-9-(10H)one and m-terphenyl moieties. Overall, the route generates three C-C and one C-N bond and has the advantage of requiring a single chromatographic separation.     

Merging lithium carbenoid homologation and enzymatic reduction: A combinative approach to the HIV-protease inhibitor Nelfinavir

An effective stereocontrolled synthesis of the HIV protease inhibitor Nelfinavir is reported. Two transformations were identified crucial for achieving success: the formation of a densely functionalized α-chloroketone via the homologation of a Weinreb amide with chloromethyllithium (LiCH₂Cl), followed by its erythro selective reduction into the corresponding chiral chlorohydrin. A commercially available enzyme P2-C02 was particularly well suited for this purpose, affording the key alcohol (in an excellent 99% de), which was then smoothly converted into the active biologically active agent.     

Fano varieties in index one Fano complete intersections

Let be a smooth complex complete intersection such that . Let f : S → X be a generically finite morphism from a smooth projective variety to X. Under some positivity assumption on the anticanonical divisor of S, if 2 ≤ dim S ≤ dim X − 2 we prove that the deformations of f are contained in a subvariety of codimension at least 2.     

Syzygies of Multi-Variable Higher Spin Dirac Operators on $$\mathbb{R}^{3}$$

This paper is a short report on the generalization of some results of our previous paper [12] to the case of spin j/2 Dirac operators in real dimension three for arbitrary odd integer j. We use an explicit formula for the local expression of such operators to study their algebraic properties, construct the compatibility conditions of the overdetermined system associated to the operator in several spatial variables, and we prove that its associated algebraic complex, dual do the BGG sequence coming from representation theory, has substantially the same pattern as the Cauchy-Fueter complex. The author is a member of the Eduard Čech Center and his research is supported by the relative grants.