# Potential Theory and Arithmetic Dynamics

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**A conference in honor of Robert Rumely**

Saturday March 25 and Sunday March 26, 2017

Saturday March 25 and Sunday March 26, 2017

All talks to be in Boyd Graduate Studies Building, Room 328

**Saturday, March 25, 2017**

**9:30-10:30 Ted Chinburg, University of Pennsylvania**

**Title: **Some applications of Rumely's work on capacity theory**Abstract: ** In this talk I will sketch some applications of Rumely's work on capacity theory to cryptography and to Arakelov theory. The cryptographic applications have to do with the computational difficulty of finding auxiliary polynomials which can be used to factor large integers into a product of primes. This is joint work with N. Heninger, B. Hemenway and Z. Scherr. The applications to Arakelov theory have to do with the successive minimal of the lattices of global sections of powers of ample metrized line bundles. This is joint work with Q. Guignard and C. Soule.

**10:45-11:45 Paul Fili, Assistant Professor, Department of Mathematics, Oklahoma State University**

**Title: ** Equidistribution and unlikely intersections**Abstract:** Building on the ideas in the proof of Baker and Rumely's equidistribution theorem, Baker and DeMarco established an adelic equidistribution theorem for preperiodic parameters of a family of dynamical systems. Using this they were able to prove a conjecture of Zannier on unlikely intersections in dynamical systems. In this talk, we'll discuss the techniques involved in making such a result effective. We'll introduce a metric of mutual energy for adelic measures associated to the Arakelov-Zhang pairing. Using this metric and potential theoretic techniques involving discrete approximations to energy integrals, we'll prove an effective bound for the Zannier problem.

Break for lunch

**1:30-2:30 Matt Baker, Professor and Director of Undergraduate Studies, Georgia Tech School of Mathematics**

**Title**: The Secret Life of Graphs**Abstract**: My collaboration with Bob Rumely had a profound influence on the direction of my mathematical career. I will try to chronicle the development of my interests in non-archimedean dynamics, Berkovich spaces, divisors on graphs, and tropical geometry, all of which owe a profound debt to Bob's originality.

**2:45-3:45 David Krumm, Visiting Assistant Professor of Mathematics and Statistics, Colby College**

**Title**: Galois groups in a family of dynatomic polynomials**Abstract**: We will discuss a method for determining the Galois groups and factorization types of all the one-variable specializations of any irreducible polynomial in two variables over the rationals. As an application we prove new results concerning the dynamics of quadratic polynomials with rational coefficients.

Break for coffee

**4:15-5:15 John Doyle, Visiting Assistant Professor, Department of Mathematics, University of Rochester**

**Title**: Dynamical modular curves for quadratic polynomial maps**Abstract**: Given a positive integer n, there is an algebraic curve Y_1(n) that parametrizes quadratic polynomials (up to equivalence) together with a marked point of period n -- this is somewhat analogous to the classical modular curves for torsion points on elliptic curves. More generally, given a finite directed graph G (satisfying certain conditions), one can define a curve Y_1(G) which parametrizes quadratic polynomials together with a collection of preperiodic points that form a graph isomorphic to G. I will describe what is known about such curves, and I will discuss a few applications in the literature and in ongoing work.

## 6:00 Banquet at the Holiday Inn. Please register Here

**Sunday, March 26, 2017**

**8:30-9:30 Carl Pomerance, John G. Kemeny Parents Professor Emeritus, Dartmouth College**

**Title**: The first dynamical system**Abstract**: For a natural number n, let s(n) denote the sum of the positive divisors of n that are smaller than n. Introduced by Pythagoras 2500 years ago, it is perhaps the first function ever studied in mathematics. Additionally, Pythagoras suggested iterating s, finding some 1-cycles and 2-cycles. It has been conjectured (over a century ago) that there are no unbounded orbits, and there is a "counter" conjecture (from over 40 years ago), that most orbits starting at even numbers are unbounded. Amazingly, the first number in doubt is 276. I will report on some recent developments concerning the distribution of numbers in a cycle, numbers missing from the range of s, and some numerical and statistical results on the two conjectures.

**9:45-10:45 Ken Jacobs, Ralph Boas Assistant Professor, Northwestern University**

**Title**: Archimedean Perspectives from non-Archimedean Dynamics**Abstract**: Much research in arithmetic dynamics has focused on translating classical results in complex dynamics to the non-Archimedean setting. One notable exception to this pattern is Rumely's recent work in the development of several new arithmetic-dynamic equivariants, including the crucial measures. These equivariants have been useful in studying both the dynamics of individual maps as well as the parameter space of maps of a fixed degree. In this talk, we will present Archimedean analogues of Rumely's equivariants and show that they share many of the same features as the associated objects in the non-Archimedean setting.

**11:00-12:00 Xander Faber, CCS**

**Title: **Generalizing the Ax--Tate Theorem and Applications to Semi-Stable Reduction**Abstract:** The Ax--Tate theorem (generalized by Sen) describes the fixed points of the action of G_p = Gal(\bar Q_p / Q_p) on C_p, the completion of an algebraic closure of Q_p. The G_p-action naturally extends to an action on the Berkovich projective line over C_p, and Bob Rumely and I have been working toward a complete description of the G_p-invariant locus in this more general setting. I will report on our progress, as well as on applications of our work for the theory of semi-stable reduction of elliptic curves and dynamical systems.

How to get to the University of Georgia