Henri Moscovici (The Ohio State University) Noncommutative geometry by (one) example. Sponsored by the Meyer Fund
Apr. 21, 2015 12:10pm (MATH …
Kempner
Michael Fried (UC Irvine)
X
By 1872 we knew the space of compact surfaces of genus was connected. I discuss applications extending this. Consider pairs with , nonconstant and analytic, mapping to the sphere, , both with r branch points. A locally path connected topology on such pairs uses moving the branch points of the s. We ask:
When is connected to ?
We find the s must have the same geometric monodromy group with the same conjugacy classes attached to the branch points. Sometimes these invariants suffice; we get one connected component. Sometimes we need more sophisticated invariants to describe components.
Two problems on rational functions with coefficients, in a number field , allow me to show the beginning tools that make this work.
Find all with the Schur cover property: For infinitely many residue fields of , , with running over the residue field (including ), is one-one.
Find all , with indecomposable over , but is a composition of degree rational functions over the complexes.
We find is one of two ``easy" (centerless) groups: The dihedral group of order ( odd) or another slightly larger group, and all conjugacy classes are the same. Then we see that modular curves include all rational functions giving solutions. Finally, the solutions come from the two fiber types over the -line in Serre's Open Image Theorem.
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This talk is based on \S 6.1 and \S 6.2 of ``The place of exceptional covers among all diophantine relations," J. Finite Fields 11 (2005) 367--433, arXiv:0910.3331v1 [math.NT]
The goal is to review and contextualize the evolution of my thoughts and interactions with tropical geometry. This talk is based on collaborative work always with Hannah Markwig, and at different times with each one of Aaron Bertram, Paul Johnson and Dhruv Ranganathan.
Back in 2007, Hannah Markwig approached me after being told by Paul Johnson that her tropical covers smelled like cut and join. Deciphering Paul’s oracle was the beginning of a fruitful and ongoing collaboration, that is pulling me closer and closer to the tropical world.
Over the course of the years, we have been studying Hurwitz theory and Gromov-Witten theory, first using tropical geometry as a powerful combinatorial tool, and then trying to understand what is the conceptual reason for the remarkably tight connection between the boundary geometry of moduli spaces of curves and maps and the piecewise linear objects in tropical geometry. The introduction of the analityc point of view, brought to the moduli space of curves by Abramovich, Caporaso and Payne, offered not only a much sought for conceptual perspective, but also opened up the way for further investigation.
TROPICALIZING A HURWITZ THEORIST
Apr. 21, 2015 3pm (Math 350)
Algebraic Geometry
Michael Fried (UC Irvine)
X
Talk 1: Alternating groups and Lift Invariants
Riemann developed theta functions to generalize Abel's Theorem – constructing analytic functions on an elliptic curve – to an arbitrary compact Riemann surface X. Attempts to make the thetas canonical, lead to two types: even and odd. Further, we get several of each type if X has genus > 1. We show how Riemann's thetas arise in considering components of sphere cover families. Our example is of spaces of r-branch point, 3-cycle, degree n, covers.
Main Theorem: These spaces have one (resp. two) component(s) if r=n-1 (resp. r ? n). By improving a Fried-Serre result, we can explicitly recognize in which component a 3-cycle cover belongs.
This generalizes: With covers of a fixed branch type, we can have several components. Still, we can often use a Lift Invariant to separate them.
Application 1: Fibers of the absolute spaces of 3-cycle covers with + (resp. -) lift invariant carry canonical even (resp. odd) thetas when r is even (resp. odd). Sometimes we can assure the even thetas produce non-zero theta-nulls (odd ones are always zero). These Hurwitz-Torelli functions are like automorphic functions on the family of covers.
Application 2: The Alternating group, An, n > 3, embeds in, On+, the nxn determinant 1 orthogonal matrices. We produce a mysterious covering group, Spinn, by pulling it back to the universal cover of On+.
Serre's originally sought regular realizations of Spinn as a Galois group. We expand this problem to see famous results on modular curves as extremely special cases of the Inverse Galois Problem.
This talk is based on Alternating groups and moduli space lifting Invariants, Arxiv #0611591v4. Israel J. Math. 179 (2010) 57–125 (DOI 10.1007/s11856-010-0073-2).
Talk 2: An Open Image Theorem based on the Small Heisenberg Group
My Talk 1 was about alternating groups, Hurwitz spaces from 3-cycles, and how lift invariant values gave a complete description of the Hurwitz space components. That probably didn't strike anyone as a very modular curve-like topic. For one, even if you studied modular curves, you didn't hear about lift invariants.
When r = 4 conjugacy classes, C, define a reduced Hurwitz space, then, we find each component is an upper half-plane quotient. Further, by changing variables we find it ramifies only over the points 0, 1 and ? of the classical j-line, P1j.
Part A: Main Modular Tower Conjecture: For simpler notation I often use the case r = 4. We use K to denote a number field.
I will show each of the 1st Talk spaces are level 0 of a sequence that resembles
For p a prime, we say a set of group elements is p' if each has order prime to p. The construction produces a sequence like (*) for any p perfect group G, and any p' classes C whose elements generate G. For r = 4, with G = G0,
(**) M(G, p, C): … ? H(Gk+2, p, C) ? H(Gk+1, p, C) ? … ? H(G, p, C) ? P1j.
Our Main M(G, p, C) Conjecture: There are no K points on H(Gk+1, p, C) for k large.
A Modular Tower on M(G, p, C) is any (nonempty) projective sequence of components on (**). The lift-invariant of the 1st Talk gives a method to identify all Modular Towers on M(G, p, C).
The M(G, p, C) conjecture shows us why the inverse Galois problem is so hard. It generalizes that Modular curves, outside their cusps, have no K points at high levels.
When r = 4, the main conjecture holds. We can see that case by locating a level on each Modular Tower whose genus exceeds 1. Then, that case follows from Weil's theorem. We use p-Frattini properties of Modular Tower cusps.
Part B: Using Part A to Generalize Serre's Open Image Theorem:
I constructed all modular curves from two ''easy'' groups: D = ZxsZ/2 (for dihedral), semidirect product of Z and Z/2 (order 2 group), and 2D = (Z)2xsZ/2. What happens when we replace D and 2D with G = (Z)2xsZ/3 and 2G = (Z)4xsZ/3, p? 3?
We find here a lift invariant value appears in a Heisenberg group, instead of in a Spin group. Each value gives one component. But, we have a new phenomenon: There are several Harbater-Mumford components (definition from Talk 1). Those always have trivial lift invariant. So, separating covers in these components requires a new idea.
An OIT starts by generalizing to Modular Towers a 1967 Serre result for modular curves. I explained Serre's result in the colloquium talk. I expand here to the conjectured property.
Our M(G, p, C) monodromy statement: Any Modular Tower has geometric monodromy (over P1j when r = 4) that is eventually p-Frattini, and for almost all p it is actually p-Frattini.
We base this talk on The Small Heisenberg Group, and l-adic representations from Hurwitz Spaces.
When studying Gabor systems, we often assume that the collection of time-frequency shifts forms a lattice. In this talk, we will look at Gabor systems where the time-frequency shifts are supported on a quasicrystal. Since quasicrystals do not generally form groups, we cannot study them using the same techniques as lattice Gabor frames. Instead we will study these Gabor frames by using a groupoid which is naturally associated to a quasicrystal. By studying certain twisted groupoid C* -algebras, we are able to show the connections between the physics of quasicrystals and Gabor theory. In particular, we are able to prove a non-existence result, showing that it is impossible to construct a tight multiwindow Gabor frame for certain quasicrystals if the generators are assumed to be in the modulation space M^1 .
Gabor frames for quasicrystals Sponsored by the Meyer Fund
