People
Robert Lemke Oliver
Assistant Professor
Contact Info:
Tufts University
Department of Mathematics
503 Boston Avenue
BromfieldPearson
Room 116
Medford, MA 02155
Email @tufts.edu:
robert.lemke_oliver
Phone: 6176270436
Personal site
Expertise:
Number theory
Research:
My research is in analytic number theory, but I am often considering
problems of an arithmetic flavor. Analytic number theory essentially
began with Riemann's seminal 1860 memoir, in which he showed that
the apparent randomness and noise in the distribution of prime
numbers is controlled by the zeros of a single complex function, the
socalled zeta function. Riemann's memoir outlined a program that
led to a proof of the "prime number theorem," and it is also the
source of what we now call the Riemann hypothesis, one of the seven
Millennium prize problems and arguably the most important unsolved
problem in mathematics. Riemann's zeta function is the prototype of
a class of functions called Lfunctions, which together can capture
finer, or sometimes just different, information about the primes.
For example, studying Lfunctions lets us prove statements like 25
percent of primes end in the digit 1; more generally, whenever
primes are grouped into classes (e.g., by their last digit),
Lfunctions are often the "right" tool to prove that each class gets
its fair share. I am fascinated by these sorts of equidistribution
problems, as well as their natural outgrowth.
Another central focus of my research is problems related to
"elliptic curves." An elliptic curve is simply a curve in the xyplane
cut out by the equation \(y^2=x^3+ax+b\), for some coefficients a and
b. While this is a very simple equation (e.g., it can be studied in
firstyear calculus), all of the standard analytic tools go out the
window when we ask only for the rational solutions to the
equation. In general, whenever we ask about only the rational
solutions to a polynomial equation, this is called a Diophantine
problem. It turns out that elliptic curves are an interesting
boundary case within the class of Diophantine problems: they
sometimes admit infinitely many rational solutions, and sometimes
only finitely many. However, it turns out that even if there are
infinitely many solutions, there is a finite set of solutions from
which all others can be generated; in particular, the set of
rational points has the structure of a finitelygenerated abelian
group. In principle (though not always in practice), for any given
elliptic curve, we can at least determine the number of generators
needed, and often the actual generating set as well. But how does
the number of generators vary as we vary the coefficients defining
the elliptic curve in some way? How many generators are needed for a
"typical" elliptic curve? Are there elliptic curves that need more
than 100 generators, or more than your favorite arbitrarily large
number? These are questions of active study.
The bridge between elliptic curves and prime numbers is through
Lfunctions and the theory of modular forms. A modular form is a
complex function satisfying certain dazzling symmetry properties
that almost no function should be able to satisfy. There is a
natural recipe, reliant on these symmetries, that produces from a
modular form an Lfunction; that is, modular forms are a natural
source of Lfunctions, which in turn naturally encode information
about the primes. Remarkably, there is also a (not at all obvious!)
way of starting from an elliptic curve and producing a modular form
which "knows" everything about the elliptic curve. This is the
socalled modularity theorem, one of the crowning achievements of
20th century number theory, and whose proof resolved Fermat's Last
Theorem. Thus, modular forms are attached to elliptic curves, but
they also arise naturally in combinatorics (e.g., the partition
generating function is a modular form), in studying quadratic forms,
and in many other arenas I know less about. My current research in
the area is focused on the application of modular forms to elliptic
curves, but I have previously thought about and remain interested in
these other applications as well.
