

I study ergodic theory. To get
an idea of what this is, consider the following
example. Suppose you were interested in
determining the "average" temperature of water
in a lake. One thing you could do is put a
bunch of thermometers at various spots in the
lake, take readings from each of these
thermometers (at the same instant), and average
these readings. Another thing you could do
is to take one thermometer, attach it to a fish
and take readings from this one thermometer
every minute, averaging the readings you get.
A key principle of ergodic theory is that these
two methods of averaging will coincide, so long
as the movement of the fish is reasonable (by
"reasonable", I mean that the fish should
eventually swim in all parts of the lake).
Ergodic theory concerns itself (among other
things) with describing the "average" behavior
of systems that change over time by examining
probabilistic and statistical properties of
dynamical systems (a dynamical system is any
mathematical model for a quantity that changes
as time passes; think of the location of the
fish as the dynamical system in the above
example). Historically speaking, the
motivating problems of ergodic theory come from
physics, but ideas in ergodic theory have been
applied in other areas of mathematics as well as
in biology, chemistry, economics, and other
fields. For more on dynamical systems and
ergodic theory, click here
and here.
My past work includes:
 complex dynamics of
quadratic rational maps
 descriptive dynamics and
ergodic theory of continuous semigroup actions
 applications of ergodic
theory to number theory, particularly the
complexity theory of multiple ergodic averages
My current work explores the "speedup
equivalence" of actions of commuting
transformations in both the measurepreserving
and topological categories.
Outside of ergodic theory, I am also interested
in connections between math and games, and in
applications of mathematics and statistics to
sports analytics, and have worked with students
on projects in these areas in the past.
Click here for my
papers and click here
to see slides from my talks.


I am always interested in working
with students in independent study courses
and on research projects.
In the past I have directed independent
studies in:
 dynamical systems
and chaos
 measure theory
and Lebesgue integration
 vector calculus and
differential geometry
 mathematical biology
 topology
 statistical
methods in sports analytics
Several of my students have presented
posters at local, regional and national
events and four have had papers published
(see below).
If you are interested in an independent
study or undergraduate research project (I
have lots of ideas, and I am also willing to
learn new math to meet your interests), email me
or stop by my office. 

Papers by my
undergraduate students


Michael Nadrowski, "An
analysis of methods used to measure major college
football recruiting classes and assign star ratings to
recruits"
Mathematics and Sports 3 (2022), 120.
Jonathon Wilson, "Entropy of LEGO(R)
jumper plates" in J. Beineke and J.
Rosenhouse (Eds) The Mathematics of Various
Entertaining Subjects Volume 3,
Princeton
University Press, Princeton, NJ (2019), 287311.
Anzhane' Lance, "Dynamics
of the family l(z + 1/z + 1)", preprint (2018)
Allie Wicklund, "A sexage,
densitydependent matrix model for whitetailed deer
populations incorporating annual harvest",
preprint (2016)
Tyler George, "Eergodicity and speedups of ergodic systems" RoseHulman Undergraduate Math. J. 16
(2015) 7287.
Keith Goldner, "A Markov model of football" J. Quant. Anal. Sports
8 (2012) (online, 16 pp.)

