Recent and Emerging_New Developments in Mathematics and Contributions b

Gheorghe Morosanu, Professor and Dept HeadDepartment of Mathematics and its Applications Central European University Budapest, June 29, 2010 Recent and Emerging/New Developments in Mathe

Gheorghe Morosanu, Professor and Dept Head

Department of Mathematics and its Applications

Central European University

Budapest, June 29, 2010

Recent and Emerging/New Developments in Mathematics

and Contributions by Department Members

1 The most interesting developments of the last twenty years and

emerging/new developments in Mathematics

Mathematics is a special science with a long nice history In particular, Carl

Friedrich Gauss (1777-1855) reffered to it as the queen of sciences While

Mathematics has its own internal development (this part being usually

known as Pure Mathematics), it is also used as an essential tool in many

fields, such as: biology, chemistry, ecology, economics, engineering, medicine, physics, and many others This part of Mathematics that is concerned with applications of mathematical knowledge to other fields is

usually known as Applied Mathematics In fact, there is no clear border

separating the two parts, and they influence each other

The systematic study of mathematics began with the ancient Greeks between

600 and 300 BC Mathematics has since been greatly extended, culminating with an explosion of mathematical knowledge in the last half the 20th

century and the last decade

On the one hand, some mathematicians are interested in solving theoretical problems and conjectures For example, Andrew Wiles (born 1953), building

on the work of others, proved in 1995, with the assistance of his former

student Richard Taylor (born 1962), the well-known Fermat's Last Theorem (conjectured in 1637 by Pierre de Fermat (1601- 1665)): no three positive integers a, b, and c can satisfy the equation a n + b n = c n for any integer value of n greater than two Despite of its simple statement, it took 358 years

to solve it Another example: Thomas Callister Hales (born 1958) is known for his 1998 computer-aided proof of the Kepler conjecture, named after Johannes Kepler (1571-1630), a centuries-old problem in discrete geometry

which states that the most space-efficient way to pack spheres is in a pyramid shape It is worth mentioning that the Clay Mathematics Institute (CMI) of Cambridge, Massachusetts, established in 2000 seven Millenium Prize Problems, some of the most important open problems that have

resisted solution for many years The Board of Directors of CMI designated a

$7 million prize fund for the solution to these problems, with $1 million allocated to the solution of each problem Note that one of the seven problems, posed in 1904 by Henri Poincaré (1854-1912), was solved in 2003

by Russian mathematician Grigori Perelman (born 1966), who surprisingly declined to accept the award The other six problems are still open One of them is also a long standing one: it had been formulated in 1859 by Bernhard Riemann (1826-1866), and then also included into a list of 23 open problems proposed in 1900 by David Hilbert (1862-1943) Other open questions are related to more recent developments, such as the one

formulated by Stephen A Cook (born 1939), connected to Theoretical Computer Science, and another one on Navier-Stokes existence and smoothness, formulated by Charles Fefferman (born 1949), connected to Fluid Dynamics

On the other hand, more mathematicians are working toward the extension

of the existing mathematical methods, thus contributing to the development

of both pure and applied mathematics In the last decades we have witnessed

an explosion of knowledge in all sciences leading to a tremendously increasing need of mathematical tools That explains the great development

of different areas in Applied Mathematics (such as, applied differential equations, applied functional analysis, applied statistics, financial mathematics, mathematical biology, numerical analysis, variational methods) and even the creation of new areas (e.g., computational fluid dynamics, cryptology, mathematical psychology, mathematical sociology)

Many theoretical advances in mathemtics have been reported in the last twenty years, in both classical and modern fields of mathematics It is enough to look at the work of the recipients of different awards, such as the

Wolfe Prize, the Fields Medal, and the Abel Prize (the prize amount = 740,000 Euro) – equivalent to the Nobel Prize – established on 1 January

2002 by the Niels Henrik Abel Memorial Fund, named after Norwegian mathematician N.H Abel (1802-1829) The Abel Prize laureates are:

 2003: Jean-Pierre Serrre (France), for playing a key role in shaping

the modern form of many parts of mathematics, including topology, algebraic geometry and number theory

 2004: Michael F Atiyah (UK/Lebanon) and Isadore M Singer

(USA), for for their discovery and proof of the index theorem,

bringing together topology, geometry and analysis, and their

outstanding role in building new bridges between mathematics and

theoretical physics.

 2005: Peter D Lax (Hungary/USA), for his groundbreaking

contributions to the theory and application of partial differential equations and to the computation of their solutions.

 2006: Lennart Carleson (Sweden), for his profound and seminal

contributions to harmonic analysis and the theory of smooth dynamical systems.

 2007: S.R Srinivasa Varadhan (India/USA), for his fundamental

contributions to probability theory and in particular for creating a unified theory of large deviation.

 2008: John G Thompson (USA) and Jacques Tits

(Belgium/France), for their profound achievements in algebra and in particular for shaping modern group theory.

 2009: Mikhail Gromov (Russia/France), for his revolutionary

contributions to geometry.

 2010: John Tate (USA), for his vast and lasting impact on the theory

of numbers.

As far as the future of mathematics is concerned, the most notable trend is

the great expansion of mathematics and its applications While computers

become more and more important and powerful, mathematics remains the main theoretical support for all sciences

2 Contributions by our Department Members

Our faculty members have been involved in major research areas of pure and

applied mathematics, such as: algebra, algebraic geometry, asymptotic analysis, bioinformatics, calculus of variations, combinatorics, computational biology, cryptography, difference equations, discrete mathematics, evolutions equations, fluid mechanics, geometry, number theory, numerical analysis, optimization, ordinary and partial differential equations, probability theory, quantum mechanics, statistics, stochastic processes As you can see, these areas are in accordance with the general

current trend in mathematics and its applications, as described above

Here is a list of our faculty members, including adjunct professors, and the research areas they have covered during the last two decades:

 Marianna BOLLA, adjunct professor (BME, Budapest):

combinatorics, probability and statistics

 Carsten CARSTENSEN, distinguished visiting professor (Humboldt University, Berlin): numerical analysis of partial differential

differential and integral equations, optimization, variational

inequalities

 Laszlo CSIRMAZ, CEU professor (part time): cryptography, graph

theory

 Matyas DOMOKOS, adjunct professor (Renyi Institute): algebra

(group theory)

 Gabor ELEK, adjunct professor (Renyi Institute): combinatorics and

functional analysis

 Eduard FEIREISL, distinguished visiting professor (Czech Academy

of Sciences, Prague): fluid mechanics

 Ervin GYORI, adjunct professor (Renyi Institute): graph theory

 Gergely HARCOS, adjunct professor (Renyi Institute): number

theory

 Pal HEGEDUS, CEU assistant professor: algebra (group theory)

 Gyula KATONA, adjunct professor (Renyi Institute): combinatorics,

cryptology

 Alexandru KRISTALY, adjunct professor (Babes-Bolyai University,

Cluj-Napoca, Romania): calculus of variations, optimization

 Laszlo MARKI, adjunct professor (Renyi Institute): algebra

(categories, rings, semigroups)

 Istvan MIKLOS, adjunct professor (Renyi Institute): bioinformatics

 Gheorghe MOROSANU, CEU professor: applied functional

analysis, calculus of variations, differential equations, fluid

mechanics, singular perturbation theory

 Andras NEMETHI, adjunct professor (Renyi Institute): singularity

theory, algebraic geometry and (low-dimesional) topology

 Peter P PALFY, adjunct professor (Renyi Institute): algebra (group

theory)

 Denes PETZ, adjunct professor (Renyi Institute): operator algebras,

quantum information theory, quantum statistics

 Tamas SZAMUELY, adjunct professor (Renyi Institute): arithmetic

and algebraic geometry

 Balazs SZEKELY, adjunct professor (BME, Budapest): stochastic

processes

 Robert SZOKE, adjunct professor (ELTE, Budapest): differential

geometry (Riemann manifolds)

It is worth pointing out that most of our faculty members’ papers and books have been published by top-rank journals and leading publishers In my opinion, the most notable contributions have been reported in book form by

 [EF1] Eduard FEIREISL, SingularLimits in Thermodynamics of

Viscous Fluids, Birkhäuser Verlag, Basel, 2009 (with A Novotny).

 [EF2] Eduard FEIREISL, Dynamics of Viscous Compressible

Fluids, Oxford University Press, Oxford, 2004

 [AK] Alexandru KRISTALY, Variational Principles in

Mathematical Physics, Geometry, and Economics, Cambridge

University Press, 2010 (with C Varga and V Radulescu)

 [GM1] Gheorghe MOROSANU, Singularly Perturbed Boundary

Value Problems, Birkhäuser, Basel-Boston-Berlin, 2007 (with L

Barbu)

 [GM2] Gheorghe MOROSANU, Functional Methods in Differential

Equations, Chapman & Hall/CRC, Boca Raton-London-New

York-Washington, D.C., 2002 (with V.-M Hokkanen)

 [DP1] Denes PETZ, Quantum Information Theory and Quantum

Statistics, Springer-Verlag, Berlin, 2008

 [DP2] Denes PETZ, The Semicircle Law, Free Random Variables,

American Mathematical Society, Providence, RI, 2000 (with F Hiai)

 [DP3] Denes PETZ, Quantum Entropy and its Use, Springer-Verlag,

Berlin, 1993 (with M Ohya)

 [TS1] Tamas SZAMUELY, Galois Groups and Fundamental

Groups, Cambridge University Press, 2009

 [TS2] Tamas SZAMUELY, Central Simple Algebras and Galois

Cohomology, Cambridge University Press, 2006 (with Ph Gille)

Comments on the above books

Both Eduard Feireisl’s books [EF1] and [EF2] are concerned with the

system of Navier-Stokes equations that is the topic of one of the seven millennium prize problems (see above) Book [EF1] is still under review Let

me reproduce partly P.B Mucha’s review of book [EF2]: “The main goal of this book is to prove the existence of weak solutions to the full system of evolutionary Navier-Stokes equations for compressible viscous heat-conductive fluids for arbitrary data in N-dimensional domains From the mathematical point of view, achieving this aim is a serious challenge This book is the first monograph dealing with these types of issues for the full Navier-Stokes system, and it can be viewed as an extension of the results of

P.-L Lions [Mathematical topics in fluid mechanics Vol 2, Oxford Univ.

Press, New York, 1998.”

Alexandru Kristaly’s book [AK] (with C Varga and V Radulescu) belongs

to Calculus of Variations It combines theoretical results and applications to

mathematical physics, geometry, economics

Gheorghe Morosanu’s monograph [GM1] (with L Barbu) gathers results,

mainly obtained by the authors, on the asymptotic analysis of some boundary value problems describing important applications (waves, fluid flows, diffusion) The novelty of the book consists in extending the singular perturbation theory to nonlinear problems by using appropriate tools from

functional analysis, partial differential equations, and the theory of evolution equations

The purpose of Gheorghe Morosanu’s monograph [GM2] (with V.-M.

Hokkanen) is to emphasize the importance of functional methods in the study of a broad range of applications, including various hyperbolic and parabolic boundary value problems The use of functional methods leads to better results as compared to the ones obtained by classical techniques, and sometimes more appropriate mathematical models may be derived as a byproduct of our approach, thus reaching a concordance between the physical sense and the mathematical definition for the solutions of concrete problems

On Denes Petz’ book [DP1], B.C Sanders says: “This book rigorously

covers topics in quantum information theory and quantum statistics Overall, the mathematical explanations are clear, concise and self-contained … I recommend this book as a useful compendium and reference for quantum information theory topics rather than as a textbook that is read from cover to cover.”

On Denes Petz’ book [DP2] (with F Hiai), reviewer D.Y Shlyakhtenko

says: “This book is about free entropy, a new and rapidly developing subject with connections to diverse areas of mathematics: noncommutative probability theory, operator algebras and random matrices The first book on

free probability theory [D V Voiculescu, K J Dykema and A Nica, Free random variables, Amer Math Soc., Providence, RI, 1992 has become the

standard introductory text for the subject However, much has happened in free probability theory since the publication of that book; as a result, there are general areas of free probability not covered by that book”, but covered

by this one

On Denes Petz’ book (with M Ohya), reviewer G.A Raggio says: “This is

the first book on quantum entropy; it presents an up-to-date, comprehensive and very broad mathematical treatment of entropy for quantum systems in its many guises and forms.”

Tamas Szamuely’s book [TS1] is still under review, but its title reveals the

topics discussed in the book

Tamas Szamuely’s book [TS2] (with Ph Gille) provides a detailed proof of

the celebrated Merkurev-Suslin Theorem The authors assume familiarity with standard algebraic concepts and basic knowledge in algebraic geometry, which are nevertheless recalled in an Appendix

It is difficult to rank the above books with respect to their quality I can say that those signed by E Feireisl and myself belong to Applied Mathematics, even if they use advanced methods from functional analysis

Sincerely,

Gheorghe Morosanu