#### The most interesting developments of the last 20 years and emerging/new developments in Mathematics

Mathematics is a special science with an impressively long history. Carl Friedrich Gauss (1777-1855), often said to be the greatest mathematician since antiquity, referred to it as *the* *queen of sciences*. While mathematics has its own internal development, usually known as *pure mathematics*, it is also an essential tool in many fields, including biology, chemistry, ecology, economics, engineering, medicine, physics and many others. The area of mathematics that is concerned with applications of mathematical knowledge to other fields is usually referred to as *applied mathematics*. Despite these distinctions, 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 in 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, some of which date back centuries. For example, Andrew Wiles, building on the work of others, proved in 1995, with the assistance of his former student Richard Taylor, the well-known *Fermat's Last Theorem* (conjectured in 1637 by Pierre de Fermat). Despite the apparent simplicity of this theorem, it wasn’t solved for 358 years.

In 2000, the Clay Mathematics Institute (CMI) of Cambridge, Massachusetts, established seven Millenium Prize Problems, representing some of the most important open problems that have resisted solution for many years. A prize of $1 million has been offered for the solution of each problem. One of the seven problems, posed in 1904, was solved in 2003 by Russian mathematician Grigori Perelman; however, the other six problems are still open. One of them is also a longstanding one: it was formulated in 1859 by Bernhard Riemann and then also included in a list of 23 open problems proposed in 1900 by David Hilbert. 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 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 and variational methods,* *and even the creation of new areas, such as computational fluid dynamics, cryptology, mathematical psychology and mathematical sociology.

Many theoretical advances in mathemtics have been reported in the last 20 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* (740,000 Euros)—equivalent to the *Nobel Prize*—established in 2002 by the *Niels Henrik Abel Memorial Fund. *The *Abel Prize* laureates are honored for their noteworthy work across a broad spectrum of areas. Examples of these awards include: (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; (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;

- 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;
- 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. As 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, including 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 and stochastic processes. As one can see, these areas are in accordance with the general current trend in mathematics and its applications, as described above.

Most of our faculty members’ papers and books have been published by top-ranking journals and leading publishers. Some of the highlights are as follows:

Eduard Feireisl (with A. Novotny), *Singular Limits in Thermodynamics of Viscous Fluids* (*Birkhauser Verlag: Basel,* 2009); Eduard Feireisl, *Dynamics of Viscous Compressible Fluids* (*Oxford** University Press: Oxford,* 2004). Both of these are concerned with the system of Navier-Stokes equations, which is the topic of one of the seven Millennium Prize Problems. P.B. Mucha’s review of book states: “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 in *Mathematical Topics in Fluid Mechanics **(Vol. 2*, Oxford University Press: New York, 1998).”

Alexandru Kristaly (with V. Radulescu and C. Varga), *Variational Principles in Mathematical Physics, Geometry, and Economics* (Cambridge University Press: Cambridge, 2010); in *Calculus of Variations*. Kristaly's work combines theoretical results and applications to mathematical physics, geometry and economics.

Gheorghe Morosanu (with L. Barbu), *Singularly Perturbed Boundary Value Problems* (Birkhauser: Basel-Boston-Berlin, 2007). This monograph 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 is 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. Gheorghe Morosanu (with V. M. Hokkanen), *Functional Methods in Differential Equations* (Chapman & Hall/CRC: Boca Raton-London-New York-Washington, DC, 2002). This monograph emphasizes 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.

Denes Petz, *Quantum Information Theory and Quantum Statistics* (Springer-Verlag: Berlin, 2008). B.C. Sanders says of this publication, “This book rigorously covers topics in quantum information theory and quantum statistics. Overall, the mathematical explanations are clear, concise and self-contained … a useful compendium and reference for quantum information theory topics rather than as a textbook that is read from cover to cover.” Denes Petz *(with F. Hiai),* *The Semicircle Law, Free Random Variables* (*American Mathematical Society: Providence, RI, 2000); R*eviewer 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* (American Mathematical Society: 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. Denes Petz (with M. Ohya), *Quantum Entropy and its Use *(*Springer-Verlag: Berlin,*1993). Reviewer G.A. Raggio says of this publication: “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, *Galois Groups and Fundamental Groups* (Cambridge University Press: Cambridge, 2009). As the title suggests, this book discusses* *Galois groups and fundamental groups. Tamas Szamuely (and P. Gille), *Central Simple Algebras and Galois Cohomology* (Cambridge University Press: Cambridge, 2006). This book provides a detailed proof of the celebrated Merkurev-Suslin Theorem.