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:

   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.

Ivan Avramidi