New proofs of theorems of Kathryn Mann

New Proofs of (New) Direct Product Theorems

Direct product theorems are formal statements of the intuition, “It is harder to solve multiple instances of a problem than to solve a single instance.” For this talk, the amplification of hardness will be in reducing the probability of success for a feasible algorithm. So a direct product result for circuits might say, “If no small circuit can compute Boolean function f(x) with probability at ...

New Proofs of Euclid's and Euler's Theorems

In this note we give a new proof of the existence of infinitely many prime numbers. There are several different proofs with many variants, and some of them can be found in [1, 3, 4, 5, 6]. This proof is based on a simple counting argument using the inclusionexclusion principle combined with an explicit formula. A different proof based on counting arguments is due to Thue (1897) and can be found...

New Proofs of Basic Theorems in Calculus

In this note we present new proofs of three basic theorems in calculus. Although these theorems are well-known, in each proof we obtain something which seems to be unknown. We start with the Heine-Cantor theorem about uniform continuity and obtain explicitly the optimal delta for the given epsilon. We then proceed with the Weierstrass extreme value theorem and present two proofs of it: the “env...

A Pushing-Pulling Method: New Proofs of Intersection Theorems

New Proofs of the Torelli Theorems for Riemann Surfaces

In this paper, by using the Kuranishi coordinates on the Teichmüller space and the explicit deformation formula of holomorphic one-forms on Riemann surface, we give an explicit expression of the period map and derive new differential geometric proofs of the Torelli theorems, both local and global, for Riemann surfaces.