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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
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 an+bn=cn 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 Poincar (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.
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, Birkhuser Verlag, Basel, 2009 (with A. HYPERLINK "http://www.ams.org/mathscinet/search/publications.html?pg1=IID&s1=264146" 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, Birkhuser, 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 Feireisls 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. Muchas 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 Kristalys 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 Morosanus 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 Morosanus 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 Szamuelys book [TS1] is still under review, but its title reveals the topics discussed in the book.
Tamas Szamuelys 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
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