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Cryptography and Schur's Conjecture Oct. 2005, Version of a talk given Houston University, U. of Montana at Bozeman, and London Ontario to large (50+) audiences of graduate students. Introduces Riemann's existence theorem as a discovery tool – as I discovered it myself in 1968 – by proving the original Schur (1919) Conjecture for polynomials. This describes the exact polynomials that can be used for an RSA type cryptography scheme. The talk conclusion points to the deeper equivalence of versions of Schur's Conjecture for rational functions with pieces of Serre's Open Image Theorem. The two hour lecture Poincare series coming from Cryptology questions about Exceptional Towers takes an elementary approach to bringing that topic to life. Crypt-Schur09-22-05.pdf

Exceptional covers and Davenport Pairs: At the Oberwolfach conference late January of 2001, could have been subtitled "Determining Chow motives from Weil vectors." The role of Galois Stratification and the ring of Chow Motives owes much to the vision of the paper of Denef-Loeser, "Definable sets, Motives and p-adic integrals," referred to in the paper. It also owes much to specific understanding from such practical problems as exceptional covers and Davenport pairs. obwffin01-22-2001.pdf

The Exceptional Tower of a variety over a finite field, Berkeley Number Theory Seminar, 10/18/06. Three aspects of exceptional covers stand out. 1. Their collection over a given space (and given finite field) form a category with fiber products having a canonical (infinite) profinite group with permutation representation as monodromy group. 2. Their generalization to p(ossibly)r(educible)-exceptional covers not only includes Davenport pairs – the arithmetic version of spaces with equal Laplacian eigenvalues – but also relations on Poincare series. 3. Basic questions on Davenport pairs jump the dihedral group monodromy groups of the most known exceptional covers to groups whose classification guided early applications of finite simple groups: S I.a → Articles: R(egular)I(nverse)G(alois)P(roblem) and arithmetic of covers (outside Modular Towers) → Variables Separated Polynomials and Moduli Spaces. exccov10-18-06.html %-%-% exccov10-18-06.pdf

Poincare series coming from Cryptology questions about Exceptional Towers, Mich. Number Theory Seminar, 03/26/07. Even a sophisticated number theory audience will know little group theory. Riemann's Existence Theorem turns nonabelian Riemann surface covers (algebraic relations between variables) over to the efficiency of group theory. This talk is a model for showing serious finite group theory using easy finite groups. Connecting the topics of motives, cryptography and Serre's OIT sounds like a stretch, until you realize one big issue: How do accessible covering spaces connect to the Grothendieck category of Chow motives. michnt03-26-07.html %-%-% michnt03-26-07.pdf

Iteration dynamics from Cryptology on Exceptional Covers, UCI No. Theory seminar, May 20th, 2008. This starts with a survey of the exceptional tower over a variety over a finite field, and a brief history describing all the exceptional rational functions. The rest of the talk ties these two topics with a serious challenge: Describing the periods of the exceptional covers mod p that come from the G(eneral) L(inear) part of Serre's O(pen) I(mage) T(heorem). uciNTh05-20-08.html %-%-% uciNTh05-20-08.pdf

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