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 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.

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.

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.

under construction

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.

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.

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.

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.