1,721,567 research outputs found
Gauss: The Last Entry
Full text linkWe present one of the shortest examples of a statement with a visionary impact: we discuss an expectation by Gauss. His idea preludes developments only started more than a century later. Several proofs were given for the prediction by Gauss. We show where this statements fits into modern mathematics. We give a short proof, using methods, developed by Hasse, Weil and many others. Of course this is history upside down: instead of seeing the Last Entry as a prelude to modern developments, we give a 20-th century proof of this 19-th century statemen
The Riemann-Hurwitz formula
Full text linkLet ϕ : S → T be a surjective holomorphic map between compact Riemann surfaces. There is a formula relating the various invariants involved: the genus of S, the genus of T, the degree of ϕ and the amount of ramification. Riemann used this formula in case T has genus zero. Contemporaries referred to this general formula as ”Riemann’s theorem”. Proofs were given by Zeuthen and Hurwitz. We discuss this formula in its historical context, and in modern generalizations
On what has been called Leibniz’s rigorous foundation of infinitesimal geometry by means of Riemannian sums
No full textA number of scholars have recently maintained that a theorem in an unpublished treatise by Leibniz written in 1675 establishes a rigorous foundation for the infinitesimal calculus. I argue that this is a misinterpretation
Strong Approximation and a Conjecture of Harpaz and Wittenberg
Full text linkWe study strong approximation for some algebraic varieties over Q which are defined using norm forms. This allows us to confirm a special case of a conjecture due to Harpaz and Wittenberg
Mısır’ın Tavuk Fabrikaları
No full textMısır’da, 13. asırdan beri enteresan bir gelenek devam etmekte. On binlerce yumurta, gübre fırınlarında kuluçkaya yatırılmakta… Seyahatnamelerle dünyaya yayılan bu hadise Avrupa’da onlarca makaleye konu olmuş. Hatta, günümüzdeki bebek kuvözlerine de ilham olduğu söylenmekte
In Defence of Geometrical Algebra
No full textThe geometrical algebra hypothesis was once the received interpretation of Greek mathematics. In recent decades, however, it has become anathema to many. I give a critical review of all arguments against it and offer a consistent rebuttal case against the modern consensus. Consequently, I find that the geometrical algebra interpretation should be reinstated as a viable historical hypothesis
Holomorphic modular forms and cocycles
No full textThis is a slightly expanded version of my lecture at the conference Modular forms are everywhere at Bonn, May 2017, taking into account remarks by Don Zagier and Shaul Zemel
Monodromy of A-hypergeometric functions
No full textUsing Mellin-Barnes integrals we give a method to compute elements of the monodromy group of an A-hypergeometric system of differential equations. The method works under the assumption that the A-hypergeometric system has a basis of solutions consisting of Mellin-Barnes integrals. Hopefully these elements generate the full monodromy group, but this has only been verified in some special cases
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