The department has groups in algebraic geometry, and both
analytic and algebraic number theory.
These areas are among the oldest in mathematics, but are
still a source of vital ideas.
The department has experts in both differential equations and harmonic
Analysis is one of the oldest and broadest areas of mathematics.
It contains classical topics, familiar to most college students, such as the
theories of differentiation, integration and limits. Applications and
connections to other fields, both within and outside mathematics, are
numerous. The Rochester analysis group specializes mainly in harmonic
analysis and partial differential equations.
Harmonic analysis studies how
functions can be broken up into basic waves and applications thereof. The
Fourier transform and Fourier series are examples of topics included in
harmonic analysis. Applications are extensive in such fields as signal
processing, medical imaging and quantum mechanics. Partial differential
equations are equations that are frequently used to model real world
problems. Understanding the solutions, such as whether they exist, are
unique and how they behave in general, is the primary interest. Such
equations play a prominent role in physics, engineering, economics and
other disciplines. Partial differential equations are a rich source of
problems in harmonic analysis.
The geometry group works in a number of different areas.
Geometry has been one of the most popular fields of mathematics in recent decades.
It combines techniques from apparently unconnected areas, such as
the use of analytical tools for answering topological questions, and it has
strong relations with quantum field theory and string theory. Recent
results in geometry include the proof of the Poincare conjecture, by
G. Perelman in 2002-2003 following the analytic program of R. Hamilton,
and the differentiable sphere theorem, proved by S. Brendle and R. Schoen
In recent years probability has blossomed. Interactions with other fields such
as biology, computer science, finance, and especially statistical physics have led to
a broadening and deepening of the subject.
Although probability began in the early 1600's, the subject as we know it today dates
from the twentieth century, when powerful tools from abstract analysis brough new
rigor to the field. As scientists study nature in ever greater detail, they increasingly
find that random effects play an essential role in almost all descriptions of natural
phenomena. These new insights enriched probability, and spilled over into many other
parts of mathematics. At present the probability group focuses on the study of stochastic
partial differential equations. In addition, there are several associated faculty members
in mathematical physics.
The department has experts in algebraic topology, particularly in homotopy theory.
The 20th century has been called the century of topology.
now forms one of the foundations of modern analysis and geometry. The
development of algebraic topology led to the development of many of the key
concepts of algebra: homological algebra, category theory, Lie
groups/algebras and K-theory among them. The use of cobordism invariants
and intersection theory developed in differential topology form the
foundations for invariants such as the Seiberg-Witten invariants and
Gromov-Witten invariants so important in current day geometry. Today
topology continues to flourish as a pure endeavor. Symplectic topology and
Voevodsky's recent introduction of the techniques of homotopy theory into
algebraic geometry are among some examples. In addition, there are growing
connections with theoretical physics, dynamical systems, computer science
and DNA biology.