HTML and/or PDF files in the folder talklist-mt
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1. Five talks given in London, Ontario, October 2005, The pdf file contains the abstracts of the talks. Click on titles for the actual talks: MTabstracts09-04-05LO.pdf

2. Dihedral Groups: MT view of Modular curve cusps London, Ont., Talk 1, Oct. 17, 2005, revised extensively for Istanbul, June 17, 2008. While cusps of modular curves are special among those for general Modular Towers, understanding general cusps has benefited from using the M(odular)T(ower) viewpoint to see modular curve cusps. Modular curve cusps have just two types, g-p', and p-cusps: the 3rd type, o-p' is missing. In good cases, as with alternating groups replacing dihedral groups, and the prime p=2, where the o-p' disappear at level 1, we aren't surprised that a modular curve-like uniformity of cusps sets in at higher levels. In both cases H(arbater)-M(umford) cusps and their shifts capture much information. App. E1 illustrates how to use the sh-incidence cusp pairing on reduced Hurwitz spaces by computing it for the curves Xj(p2), j=0, 1. London1-ModCurves.html %-%-% London1-ModCurves.pdf

3. Regular realizations of p-projective quotients and modular curve-like towers, Talk at the pro-p profinite Group Theory Conference 05/25/06 in Oberwolfach Germany. An exposition on Modular Towers and its relation to the Strong Torsion Conjecture of Abelian Varieties: mfried-ow05-25-06.html %-%-% mfried-ow05-25-06.pdf

4. How Pure-cycle Nielsen classes test the Main Modular Tower Conjecture, Talk on 10/26/06 RIMS Conference on Profinite Arithmetic Geometry. An exposition of how the "Fried-Serre" lifting invariant finds 2 cusps whose existence gives the Main MT Conjecture: rims-fried10-26-06.html %-%-% rims-fried10-26-06.pdf

5. Connectedness of moduli spaces of Riemann Surface covers, Talk given in the Eisenbud-Osserman seminar at Berkeley, 10/17/07: Berk-connMod10-17-06.html %-%-% Berk-connMod10-17-06.pdf

6. Maximal Frattini quotients of p-Poinare Duality Groups, at Davidson AMS meeting on 03/03/07, Davidson N. Carolina. Has a completely group theoretic formulation of the Main Conjecture of Modular Towers, and examples that are serious tests for the Strong Torsion Conjecture: amsdav03-03-07FratpPoin.pdf

7. Finite group theory and Connectedness of moduli spaces of Riemann Surface covers, Colloquium Talk at Univ. of Michigan, 03/27/07. The talk failed at UM because no group theorists showed, and it was primed for them. We will try this one again elsewhere: michcoll03-27-07.html %-%-% michcoll03-27-07.pdf

8. Atomic Orbital-type cusps on Alternating Group Modular Towers, talk in the session Applicatons of Algebraic Stacks, Western Ontario Canada, 8-10 December, 2007. The MT goal: Show the sh-incidence (cusp pairing) matrix in action applied to infinitely many MTs (starting from a connectedness result of Liu and Osserman) where the Main Conjectures are proved: lonOnt12-08-07.html %-%-% lonOnt12-08-07.pdf

9. Updating an Abel-Gauss-Riemann Program, UC Irvine colloquium, May 22, 2008, expanded for Istanbul, June 18, 2008, based on it following London1-ModCurves.pdf: Riemann produced many tools that allow generalizing Abel's production of modular curves. A missing ingredient was how to generalize Galois' introduction and analysis of the higher level curves we call X0(pk+1), k≥ 0. While Shimura's projects are related, they don't capture the most useful part of Riemann's program. Using moduli spaces of covers requires deducing properties of the spaces from their cusps. New techniques for identifying cusps of Hurwitz spaces combine with connectedness results to identify advantageous cusps. Our applications here show modular curve-like properties for Modular Towers over alternating group Hurwitz spaces. ucicoll05-22-08.html %-%-% ucicoll05-22-08.pdf

10. Conway-Fried-Parker-Voelklein connectedness results London, Ont., Talk 2, Oct. 18, 2005, revised extensively for Istanbul, June 19, 2008 as the 3rd talk after London1-ModCurves.pdf and ucicoll05-22-08.pdf. Revised further for U. of Wisconsin, Oct. 8, 2009. By combining Clebsch's result from 1872 on 2-cycle Hurwitz spaces with the Spin lifting invariant describing connected components of 3-cycle Hurwitz spaces, we see the rational for the CFPV Theorem. The talk improves CFPV, so it extends all the classical results. It also includes the full definition of g-p' representative, a generalization of H(arbater)-M(umford) representatives. When conjugacy classes are repeated suitably often, there will be but one Hurwitz space component containing all g-p' reps. Then, if the conjugacy classes form a rational union, that component will be defined over Q. Applying the g-p' theorem avoids special knowledge of Schur multipliers. London2-AltGps.html %-%-% London2-AltGps.pdf

11. Istanbul Summer School 9–20 June 2008, Geometry and Arithmetic of Moduli Spaces of Coverings.
  1. Preliminary tools and working knowledge
  2. Construction of moduli spaces and stacks of coverings (Hurwitz spaces)
  3. Geometry and arithmetic of moduli spaces of coverings
  4. Connected Hurwitz space components and inverse Galois theory
  5. Grothendieck-Teichmueller tower and Galois actions
  6. Modular Towers
The colorful flyer indicates that even Turkish mathematicians are savvy to use their country's exotica. The html file gives the program subdivisions and the relation of the M(odular) T(ower) program and my three talks to the rest of that program. Istanbul06-09-08.html

12. Frattini towers and the shift-incidence cusp pairing: Modular curves are the most famous example of the title. As moduli space towers they exhibit a "Frattini property," based on their monodromy groups as covers of the j-line. Using the goals of Serre's "l-adic representations" book I will treat, in parallel, two cases of general ideas.
  1. Modular curves here derive from the semi-direct product of Z/2 acting through multiplication by -1 on Z; and
  2. the equally rich case from Z/3 acting irreducibly on Z2.
This view has modular curves as families of sphere covers attached to dihedral groups. In this case we see something familier – their cusps and the monodromy on homology in a fiber – in a new way. Then, with analogous methods, we outline the 2nd case to show how the tools extend. To take Serre's Open Image Theorem beyond modular curves, to general moduli of abelian varieties, has failed to master the limiting effect of correspondences – read motives – of arithmetic monodromy on special tower fibers. Our Z/3 case shows how Frattini data in our Hurwitz space approach helps tame that structure. UCIsh-incModCurves04-15-10.pdf

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