Measuring Complexity Growth in Dynamical Systems
Joseph H. Silverman (Brown University)
Dynamics is the study of iteration of a function f : X –> X. A coarse measure of the complexity of f is its degree. The average degree of the iterates of f is called the *dynamical degree* of f. Formally,
DynDeg(f) = lim (deg f^n)^{1/n}.
Similarly, a coarse measure of the complexity of an arithmetic object, such as an integer, is its height, which is the number of bits required to store the object on a computer. The *arithmetic degree* of a point x in X is the average height of the points in its orbit. Formally,
ArithDeg(f,x) = lim (height f^n(x))^{1/n}.
In this talk I will discuss dynamical and arithmetic degrees and a fundamental conjecture that relates them, and, as time permits, describe recent work on the variation of dynamical degrees in families.
8:00 – 3:45 Keller Hall 313
(instead of as previously announced 8:00-12:30 Keller Hall 302, 12:30-3:45 Keller Hall 301)
Potential times for sessions
Local organizers Khan, Kjos-Hanssen, Beros, Ross will be teaching at some times.
Wednesday
8:30 – 9:20
9:30 – 10:20 (Khan teaching)
10:30 – 11:20
11:30 – 12:20 (Kjos-Hanssen teaching; Ross teaching logic)
12:30 – 1:20 (Khan teaching)
1:30 – 2:20 (Kjos-Hanssen teaching; Ross teaching logic)
2:30 – 3:20
Title: Taxicabs and Sums of Two Cubes: An Excursion in Mathematics
Speaker: Joseph H. Silverman (Brown University)
Some numbers, such as
9 = 1^3 + 2^3 and 370 =3^3 + 7^3,
can be written as a sum of two cubes. Are there any numbers that can be written like this in two (or more) different ways? This elementary question will lead us into a beautiful area of mathematics where number theory, geometry, algebra, and calculus interact in surprising ways. The talk will be accessible to undergraduates at all levels.
8:00-3:45 Keller Hall 313
Potential times for sessions
Local organizers Khan, Kjos-Hanssen, Beros, Ross will be teaching at some times.
Friday
8:30 – 9:20
9:30 – 10:20 (Khan teaching)
10:30 – 11:20
11:30 – 12:20 (Kjos-Hanssen teaching; Ross teaching logic)
12:30 – 1:20 (Khan teaching)
1:30 – 2:20 (Kjos-Hanssen teaching; Ross teaching logic)
2:30 – 3:20
Speaker:Joseph H. Silverman (Brown University)
Title: The Ubiquity of Elliptic Curves
Abstract: Elliptic curves are amazing mathematical objects that have rich geometric, algebraic, and analytic structures. They appear frequently in mathematics and physics with applications ranging from cryptography to string theory to Fermat’s Last Theorem. In this survey talk I will explain what elliptic curves are and briefly describe some of their many uses in mathematics and physics.
Speaker: David R. Stoutemyer (UHM, ICS Department)
Title: A mathematical magic show: Demo and secrets of a prototype AskConstants website that can turn your floating-point results into the exact formulas that they want to be.”
Abstract: There are at least three existing websites that already do what the title promises:
https://isc.carma.newcastle.edu.au/
http://mrob.com/pub/ries/index.html
http://www.wolframalpha.com/
I will describe their techniques and some additional ones used in the AskConstants program. As a contest, if you email dstout at this university a floating-point constant or a float-free constant expression such as sqrt (3) / 2 + 3 * pi / 5, that I can convert to a float, then I will summarize how AskConstants and other such programs perform on that example. If you send a float, try to compute it using at least n-digit arithmetic where n is several more than the total number of digits, operators and functions in its float-free source. Good sources of ideas are calculus text definite integrals, algebraic numbers, values of special functions at rational arguments or at rational multiples of pi, constants from research literature and collections such as Steven Finch’s Encylopedia of Constants:
http://www.people.fas.harvard.edu/~sfinch/