Research Description

General

Research in the department is conducted in a variety of fields in Mathematics and in Physics.
The research in Mathematics is mainly conducted in  Combinatoric, Functional analysis and Number
theory.  The research in Physics spans Astrophysics, Condensed matter physics, High energy physics,
Gravity and  String theory.  Research is also conducted in Mathematical education and in Physics
education.  We have described below the main research  fields the faculty are engaged  in .

Combinatorial Number Theory

Combinatorial number theory uses technics from combinatorics (like counting, inclusion-exclusion
arguments, probabilistic methods etc.) to attack number theoretic problems. Perhaps, the most
general settings of a typical problem in combinatorial number theory is this: what arithmetic
properties must a "dense" set of integers possess? The famous Szemeredy theorem, for instance,
asserts that if an infinite sequence of integers is "sufficiently dense" then it contains arithmetic
progressions of any prescribed length.
The field of combinatorial number theory was greatly influenced by Paul Erdos (1913--1996),
one the most remarkable mathematicians of all times. The so-called "Erdos problems" are an
imprescriptible part of combinatorial number theory.

Condensed Matter

Magnetism in non-magnetic metals.
Metals at high quantizing magnetic fields and low temperatures exhibit unusual magnetic properties under
conditions of the nonlinear de Haas-van Alphen effect of magnetic oscillations. The ground state of the
electron gas has a spatially nonuniform magnetic field with domains (called Condon domains), in which the
magnetic induction (and the magnetization) can take on one of two different values. The reason for this
collective effect is that an electron experiences the magnetic induction instead of the magnetic field.
When the magnetization shows oscillations, the domain state will reform periodically in each cycle of
oscillations. It turns out that, if the amplitude of magnetization oscillations becomes comparable with their
period, the current-current interaction between conduction electrons can lead to a phase transition of the
electron gas. The reasons underlying the formation of the domain structure are the same as in the case of
ferromagnets: for a region of values of the applied field, the uniform state of the specimen is either metastable
or absolutely unstable. The behavior of the system is analogous to the condensation of a gas, with the two
different states of magnetization playing the roles of the liquid and gas phases. The state with Condon domains
is physically similar to the intermediate state of type I superconductors, reconciling apparently conflicting
requirements of the electronic ground state and electrodynamical boundary conditions by forming a two-phase
system. There is a rich variety of magnetic metals that exhibit spontaneous magnetism of spin nature. We study
this phenomenon, which is a rare and extremely interesting example of magnetism of non-spin nature. High
magnetic fields can therefore magnetize such non-magnetic metals as silver, gold, copper and beryllium

Equilibrium and non-equilibrium phase transitions in solids and cell membranes. Proton superconductors.
The study of the physical properties of substances in a metastable state pertains to the general problem of phase
transitions and is one of the most important problems in physics. Metastable states arise in first-order phase
transformations in condensed matter, in nuclear matter, in a quark-gluon plasma, in inflational expansion of the
Universe at the early stages of its evolution, in the condensation and decomposition of an electron-hole liquid in
semiconductors, and in biological and chemically reacting systems far from thermodynamic equilibrium near the
threshold of self-organization. Grain boundaries, coherent interphase and antiphase boundaries and action potentials
are general nonequilibrium features of crystalline solids and strongly polarizable biological cell membranes. We
consider dynamics of interphase boundaries and metastable phases in ferroelectric solids and propagation of solitary
waves of excitation in strongly polarizable media, including mechanisms of action potential generation and traveling in
cell membranes. Proton conducting solids are of potential for fuel cells, steam electrolysis and sensors. In some
ferroelectric hydrogen-bonded superionic conductivity was discovered. We investigate the effect of structural and
ferroelectric phase transitions on conductivity of protonic superconductors.

Advanced materials and applied physics of solid state ("smart", multifunctional materials).
Smart materials possess a special functionality, such as sensing a change in its environmental conditions, actuating a
mechanical or electric response simultaneously, automatically modifying one or more of its property parameters, and
quickly recovering its original status once the environmental conditions are withdrawn. The materials have the ability
to modify their intrinsic structures, such as the band structure or the domain structure, in response to external change,
such as temperature, pressure, electric and magnetic field. Smart materials are multifunctional materials that serve as
sensors, actuators, control capabilities and memory cells. Several ferroelectric solid solutions are smart materials.
They possess nonlinear and hysteresis properties. In general, hysteresis means a memory effect. The ability of
ferroelectric materials to switch their polarization direction between two stable polarized states provides the basis for
the binary code-based nonvolatile ferroelectric random-access memories and new technology for storing data. Thus,
the development of practical ferroelectric memories is closely linked with progress in thin-film physics and engineering.
The memory properties depend, therefore, on the domain structure and the mobility of domain walls in thin films. It is of
importance to clear up the role of the domain walls in the polarization reversal in thin films. We develop a theory of the
domain wall motion in thin films as a reason for the polarization reversal. We study governing the fast rate kinetic of
creation and annihilation of metastable ferroelectric domains that may be used in the application to the binary code-based
nonvolatile ferroelectric random-access memories and new technology for storing data.

Extremal Graph Theory

In Extremal Graph Theory one is interested in finding the extremal values of parameters that
guarantee that a graph has a certain given property. An example of a famous problem in this
area is the "Four Color Theorem". Given a planar graph (a set of points and lines connecting
some of the points such that no two lines are crossing), what is the minimum number of colors
required in order to color the points such that any two points connected by a line have distinct
colors? The answer is that four colors always suffice, even if the graph is infinite. There are
trivial examples showing that this is best possible.

Functional Analysis

under construction

Packing  and covering Problems

In packing and covering problems one wishes to pack/cover the elements of a given
combinatorial structure by some other fixed combinatorial structure. For example, given
a set of 7 points in the plane with all 21 straight lines connecting all the pairs of points,
can you always pack the 21 lines with 7 triangles (i.e. each line appears in exactly one
triangle) The answer is yes. This is known as a Steiner Triple System.

Physics Education

The research in physics education in the department is based on cognitive constructivism,
which has strong empirical support and indicates important directions for changing instruction.
It implies that teachers need to be cognizant of representational, motivational and epistemic
dimensions, which can restrict or promote student learning.  A critical part of this conception
of physics education is that students must be provided  with opportunities and materials to
develop skills to participate in epistemic interchanges, and the classroom community must
have the appropriate features of an objective epistemic community. The present research
concentrates on :
(1) Dealing  with junior high school  students alternative frameworks of some basic astronomy
concepts. One of the main objectives of this research will be to develop new teaching methods
and  accessories that will help them to change their intuitive preconceptions and get the
accepted scientific concepts.
(2) Checking how microcomputer-based laboratories and computer simulations can
influence the physical conceptual understanding and the attitudes towards physics of
high-school students.

Probabilistic Methods in  Combinatorics

Most problems in combinatorics are of the form: "Given a combinatorial object having some
property X, prove (or disprove) that it always has property Y." A (positive) probabilistic
proof of such problems is a proof whose outline is the following: We take the combinatorial
object having property X, then we do some random sampling associated with the subobjects
of the combinatorial object, and show that with positive probability property Y holds.
 

Quantum Condensed  Matter

Many particle systems in the condensed states of matter (solids and liquids) exhibit a diverse
array of phenomena  originating from quantum mechanics. Quite astonishingly, some of these
effects such as magnetism in Iron and Nickel  and conduction properties of metals, are
manifested at room temperature and can be attributed to the quantum mechanical nature of the
electrons - which are the lightest constituents of the matter atoms. However, quantum effects
become  particularly pronounced in conditions of low temperatures and in reduced dimensions,
and lead to peculiar and fascinating  phenomena. Prominent examples include superconductivity
(electrical conduction without resistance), quantization of the conductance through thin wires
in multiples of a fundamental constant, and a similar quantization in a two-dimensional  electron
gas subject to a strong magnetic field (the quantum Hall effect). These effects are accessible
to measurements thanks to the advanced technology of the last few decades. The main goal
of our theoretical research in this field is to establish an understanding of the underlying
mechanisms responsible for this class of phenomena. The frontier in this field includes two major
research topics, described in detail below: coherent electron transport, and strongly correlated
systems.

The first topic is concerned with effects originating from the ``wavy" nature of the electrons,
primarily indicated by the phenomenon of interference. As a consequence of quantum
interference, electrical transport through small devices is not governed by Ohm's law, and the
conductance can exhibit an oscillating dependence on a magnetic field which can be thought
of as a kind of `interference pattern'. In large (macroscopic) systems, interference strongly
affects the electrical transport in a disordered medium (such as a heavily doped semiconductor).
In particular, sufficiently strong disorder can lead to `localization' of the electron wave-functions,
and consequently to a phase transition from conductor to insulator. In the presence of magnetic
fields, electronic transport in disordered systems becomes particularly complex
and is one of the main directions of active studies in this field.

The topic of ``strongly correlated systems" is concerned with the interplay between quantum
mechanical effects and another aspect of condensed matter systems, which becomes most
important in low-dimensions: the interaction between the many particles (such as
electrostatic repulsion between electrons) are very significant. In certain cases, the
interactions are so strong that, in attempt to reduce its energy to a minimum, the system
organizes itself in a `correlated state', which can not be simply described as a collection of
the original particles. Instead, one should define collective objects (`quasi-particles') which
act as the `natural' building blocks of the system. For example, interacting electrons restricted
to a one-dimensional channel act as if their charge and spin separate into two independent
carriers, of charge only (`holons') and spin only (`spinons'). Another example is the fractional
quantum Hall effect, which in essence can be understood in terms of composite objects
(hybrids of charge and magnetic flux) and elementary quasi-particles which have a fraction
of the electron charge. Currently, the most challenging problems regarding electron transport
phenomena in the frontier of this field include the understanding of superconductivity  observed
in a certain class of materials in relatively high temperatures; the superconductor-metal-insulator
transitions characterizing thin dirty-metal films at low temperatures; and the anomalous metallic
behavior of certain two-dimensional electron systems in semiconductor devices.
 

Quantum Field Theory

The two major pillars of modern physics are quantum mechanics and the theory of relativity.
An unavoidable consequence of quantum mechanics and of the special theory of relativity is
the theory of relativistic quantum fields. Quantum field theory is capable of explaining the
creation and annihilation of particles and their interactions in between (which cannot be done
solely by either of its ancestors). Indeed, The Standard Model of elementary particle physics,
namely, the theory which describes all known ``elementary" particles and their interactions
under the electro-weak and the strong forces (i.e., all known forces but gravity), is a quantum
field theory. This quantum field theory was vindicated by experimental checks to very high
precission, down to distances of the order of 10^{-17} cm (one ten-thousandth the size of
a proton). Thus, quantum field theory is a major framework of high energy physics (and to
this day, and with the current capabilities of experimental particle physics, it is the
only theoretical framework of high energy physics confirmed by experiment).

Aspects of quantum field theory pursued here include non-perturbative aspects of quantum
field theory such as vacuum (i.e., ground state) structure and the related issues of dynamical
symmetry breaking, the effects of external conditions (such as boundary conditions or the
influence of external background fields) on the physical contents of quantum field theories,
color confinement, extended objects (solitons and instantons, QCD strings, random  surfaces
and their random geometries) and fluctuations around them.

An important arena in which many of these issues may be studied are the so-called ``large N"
vector and matrix models. Large N models are quantum field theories whose fields are vectors
and/or matrices with a very large number N of components. These components are arranged
such that there is large amount of symmetry in these models, which enables exact solutions of
these quantum field theories as N tends to infinity. Examples include  random non-hermitean
matrix models (which are kind of lobotomized field theories living in zero space and zero time
dimensions - i.e.,at a point), which became popular in recent years. Another example is the
formation of self- consistent solitons (a.k.a. ``fermion bags") in certain interacting field
theories, which can serve as a concrete model to  formation of particles offinite size and internal
structure (such as the proton, for example), out of point-like elementary excitations of the
quantum fields involved (the so called quarks and gluons, in the case of the proton).

Quantum field theory is quite robust: its methods are applicable in other physical disciplines,
and most notably - in condensed matter physics.

String Theory, M-Theory and Black Holes

String theory is at the moment the most promising candidate for a unified description of the
fundamantal particles and forces in nature including gravity. In string theory the myriad of
particles types is replaced by a single fundamental building block, a string. The string can
vibrate and different vibration modes of the string represent the different paricle types. One
such vibration represents the graviton the particle carrying the force of gravity. One can
study the interaction of gravitons in string theory, which can not be done in any other way.
So string theory (or more exactly superstring theory) resolves one of the most enigmatic
problems of the twentieth century theoretical physics: how to reconcile between quantum
mechanics and general relativity. Amazingly string theory predicts that we live in more
than four space-time dimensions . The dimensions we do not see are curled up so small we
can not see them in todays experiments. Studying string theory has taught us a lot about the
quantum nature of space-time, but more understanding of its prediction are needed.

In the last couple of years our understanding of string theory has grown. Previously there
were five different consisitent superstring theories all living in ten-dimensions. They are
now understood to be different manifestation of a theory called M-theory, which lives in
eleven space-time dimensions. Further it was realised that string theory contains not only
strings but also extended objects of variouse dimensions called D-branes (a membrane is
a two dimensional object). Previously it was possible to study graviton scattering in string
theory, but many other interesting physical phenomena, like black holes, were out of the
range  of the techniques that were available.  It was realised that D-branes are the building
blocks of a class of black holes. This was then used to analyse properties of these black
holes, and even count the microscopic states reponsible for black hole entropy. Even more
recently studies in string theory have led to a realisation that  string theory (quantum gravity)
may have an alternative description in terms of quantum field theories that do not involve
gravity at all. A strange notion connected to these ideas is called holography. In short the
idea is that one of the dimensions we think we live in is actually an illusion, other degrees
of freedom conspire to make nature look that way at low energy. This is like a hologram which
looks three-dimensional but is actually only a two-dimensional picture. This connection
between a quantum field theory and string theory can be used in principle to learn much
more about the underlying degrees of freedom of space-time and about quantum effects
associated with them.

To learn more you can try the following links:    Official string theory web    String theory
Superstring and fundamental theory    String theory in a nutshell

Zero sum problems in Combinatorics

In Zero-sum problems, one assigns integer weights to the elements of a combinatorial
structure and queries whetherthere exists a substructure, having total weight zero
(modulo p). For example, suppose we have a square matrix of order n whose elements
are either zero or one (these are the weights). For which values of n is it true that you
can always find, say, a square submatrix of order 4, having row sum zero (mod 2) and
column sum zero (mod 2). The answer is n >= 10 and the proof is highly nontrivial.
For n=9 there are examples showing that the assertion is false.