PROF IVAN AVRAMIDI'S
RESEARCH
For a more detailed description of my research see Professional
Research
Summary
What follows is an
explanation that I owe to many people
who asked me the same
question:
``If you are a
physicist (as you claim), how did
you end up
working at a Math Department (why are you allowed to teach Math)?''
My research interests are non-uniformly distributed among the following areas of Theoretical Physics and Mathematics:
- Geometric Analysis on Manifolds
- Mathematical Physics
- Quantum Field Theory, Quantum Gravity and Gauge Theories
- Differential Geometry
Cosmology. I
got
my
M.S. degree in 1979 in
Theoretical Physics with the Astrophysics option.
In my Master Thesis I
studied the influence of Hawking
radiation of the
primordial black holes on the Standard Model of the Universe, in
particular,
on the recombination of hydrogen.
In 1974 Hawking
discovered that the black
holes are not so black after all. The black holes were called black
since it
was believed that nothing, not even light, can escape its extreme
gravitational
attraction once it gets sufficiently close to the black hole. By taking
into
account the quantum nature of matter Hawking was able to show that this
is not
exactly true and there must be a radiation coming from the black hole.
This
radiation is negligible for large black holes formed by collapse of a
gravitating bodies like stars, but must be very strong even dominant
for smaller
so called primordial black holes that were formed from the density
fluctuations in the early Universe.
The open problem was
how many such black
holes could be in the Universe, and if they are really there, how would
they
affect the observable astronomical features of the standard model of
the
Universe, like the cosmic background radiation. The number of
primordial
black holes is determined by the amplitude of the density fluctuations
in the
early Universe. If this amplitude is too large, there would be too
many primordial black holes that would distort the standard evolution
of the
Universe too much. I studied this question and
obtained certain restrictions on the amplitude of the density
fluctuations
in the early Universe.
Quantum Field
Theory and Quantum Gravity.
I got my
PhD in Theoretical and Mathematical Physics in 1987 .
As a PhD student I
started working in Quantum Field
Theory
and Quantum Gravity. This was a natural transformation since I
wanted to understand the Hawking radiation or the quantum theory
of the black holes. As the black holes get smaller the radiation
intensifies
tremendously until the black hole explodes in a fireball. What
exactly happens with the black hole when it gets too small
nobody knows for sure. The Hawking calculation
applies only to a semi-classical regime when one can neglect the
quantum effects of the gravity itself. However,
at the very extreme, so called Planckian, scales this approximation
is not valid any more.
That is why to really
understand the explosion of the
black holes one needs a theory that describes the quantum properties of
the gravitation as well. In general relativity the gravity is described
by the metric properties of the space-time. Therefore, quantization of
gravity means quantization of the space-time, a scary idea if you hear
it for the first time. Such a theory does not exist yet. The marriage
of Quantum Theory and General Relativity is probably the greatest
challenge of theoretical physics of all times. Among the modern
possible candidates for Quantum Gravity are: Superstring Theory,
M-Theory, Non-commutative Geometry, Loop Gravity,
Euclidean Path Integrals, Lattice Triangulation, etc..
(a complete list would be too long).
My work during this
time
was closely related to the so called Background
Field Method and Effective Action Approach .
I also studied the high-energy
(or short-distance) behavior of an alternative theory of gravitation,
called
Higher Derivative Quantum Gravity . This theory exhibits
much
richer structure than the general relativity. In some aspects it is
better
than general relativity (it is renormalizable), in others it is worse
(it violates unitarity, one of the basic principles of quantum theory).
Another problem I
studied was the problem of
Interaction of Higher-Spin Fields with Gravity . It turns
out that to be consistent such interaction
must be non-local.
Non-Abelian
Gauge Field Theories.
There is also another very interesting phenomenon in Quantum Field
Theory,
called the ``Confinement of Color''. This is a phenomenon in hadron
physics.
Hadrons are elementary particles, like protons, neutrons, pi-mesons
etc,
that interact ``strongly'' with each other. These interactions are
described
(very successfully) by the QCD (Quantum Chromodynamics). According to
this
theory all hadrons are made of Quarks that interact with each other via
Gluons. There are three different types of Quarks; to distinguish
between them they are all assigned a certain number called Color (that
takes
only three different values).
In the same way as QED (Quantum Electrodynamics) is the dynamics of the
electric
charge, QCD is the dynamics of color.
There is huge
difference between the QED
and QCD though. If you take two electrons and push them together then
the
interaction increases at small distances (in fact it becomes infinite).
The situation in QCD is exactly the opposite. If you take two Quarks
and push
them together, then the interaction disappears at the smaller distances
(and
increases at large distances). The Confinement of Color is a hypothesis
that states that the interaction becomes infinite at a certain distance
comparable with
the size of the proton. This means that color particles, both
Quarks and Gluons, are confined (or jailed) inside hadrons. Because of
the
infinite potential barrier they do not have any chance to escape. They
can
only escape in ``white'' companies of two or three particles (pairs, or
triples) that form other hadrons. Color particles can never be observed
alone. So, hadrons act like a sort of a ``black hole'', but only for
color.
There is a challenging
problem of deriving the
Confinement Hypothesis from the
Quantum Chromodynamics. This is the problem of investigating the true
non-perturbative structure of the vacuum state of the theory, the state
with the least energy. It turns out that there are some non-trivial
field configurations with the energy less than that of the naive
``empty''
vacuum. Such a vacuum can lead to the confinement property. However,
this
problem is still far from being completely solved. I studied the
Vacuum Structure of the non-Abelian Yang-Mills Theory by
analyzing the effective
potential and found some indications of the non-trivial vacuum with
some
chromo-magnetic fields (Savvidy type vacuum).
Geometric Analysis and Mathematical
Physics.
Another major transformation happened when I started thinking seriously
about the effective action in quantum field theory. The effective
action is a
certain functional that, in principle, describes all quantum phenomena.
It can only be computed in the semi-classical approximation (called
loop expansion). Even the first quantum corrections to classical theory
turn out to be too complicated to be computed exactly. They are
described by the Green functions, the spectra and the determinants of
self-adjoint second-order partial differential operators.
A very powerful tool to
study such problems is the Heat
Kernel Method.
In particular, the
Asymptotic Expansion of the Heat Kernel provides
very important information about the effective action. It turns out
that
there are many different areas in physics (Quantum Field Theory,
Statistical Physics, Solitons) and mathematics (Spectral Geometry,
Conformal Geometry, Differential Geometry, Spectral Asymptotics, Index
Theorems,
Completely Integrable Hamiltonian Systems ) that lead
to the Partial Differential Operators of the same type (most
importantly
Laplace type and Dirac type) and can be dealt with by the same method,
the Heat Kernel Method. That is why, a large portion of my research
was concerned with the calculation of the
Asymptotic Expansion of the Heat Kernel . In some cases I
was able to obtain some Non-Perturbative
Results
that go beyond the standard asymptotic expansion. I
studied
a variety of problems, including Laplace type operators, Higher-Order
Operators
, non-Laplace
Type Operators ,
boundary value problems with all kinds of boundary conditions
(recently Oblique
Boundary Conditions ,
Discontinuous Boundary Conditions ).
Financial
Mathematics.
A few years ago I realized that what I have been doing for the most of
my life, the Heat Kernel Method, is exactly what is needed in Financial
Mathematics. In fact, in 2007 I was invited to give a series of
lectures on this subject at an investment bank NATIXIS in Paris.
Roughly speaking, financial securities, like stocks, for example,
exhibit random (stochastic) behavior. This is very similar to the
random
behavior of molecules of a heated gas (or the fluctuations of atoms in
a heated solid). However, the macroscopic properties of gases and
solids, like diffusion and heat conductivity, are described by
deterministic
partial differential equation called the diffusion equation and the
heat equation. Similarly, even though the prices of the underlying financial
assets are random, the financial instruments called derivatives (such as options and futures) are described
by deterministic partial differential equations which are essentially some
versions of the heat (or diffusion) equation.
These are exactly the same type of equations I have been
studied
my whole life. This inspired my natural interest in the financial
mathematics and the finance, in general.
Permanent thinking about the Quantum Gravity lead me finally to a class of problems that could be collectively described as an Interplay between Geometric Analysis, Mathematical Physics and Differential Geometry :
- the properties of the spaces of Riemannian metrics on a manifold and their relation with topology,
- continuous
deformations of the differentiable structures,
the Riemannian metric and the corresponding deformation of the
differential operators and their spectra,
- discrete transformations of the metrics (dualities) and the corresponding transformations of the spectra of differential operators,
- spectral geometry of submanifolds (an interplay of the isometries and heat trace asymptotics),
- isospectral deformations and their relation with integrable systems,
- non-commutative Riemannian Geometry, and even more weird stuff of that sort.
