Գրիգոր Բարսեղյան
Հետաքրքրությունների ոլորտ:
- Complex Analysis (particularly Nevanlinna value distribution theory and Geometric functions theory
- Differential geometry particularly Minimal surfaces
- Algebraic Geometry
- Bio and econ.mathematics
- real and complex differential equations.
For achievements, see the "Other" section below.
Three PhD theses
Visited nearly 40 institutions, among them was visiting professor in HKUST, Hong Kong (1997, 2001); in UNED Madrid, Spain (2005); Research Fellow in ICTP, Italy (2002-2009); Research in Pairs in Oberwolfach, Germany (2009); Marie Curie Fellow in University Colledge London (2013-2015); Leading visiting professor in Guangzhow universuty, China (2017).
Grants, appointments, invitations:
Content of this item.
Part 1. Tree groups of results that may have a significant impact on mathematics.
Part 2. Shortly about five trends.
Part 3. BOOKS, EDITED VOLUMES, LIST OF RESEARCH PAPERS.
PART 1. Tree groups of results that may have a significant impact on mathematics.
PART 1, Item 1. On a big gap in mathematics and studies of Gamma-lines and level sets.
Studies of zeros of real and complex functions f(z) of one variable constitute an essential part of real and complex analysis. In the complex analysis, these notions are studied in classical Nevanlinna-Ahlfors theories <cite>Nev</cite> as well as for a huge number of particular classes of functions. It is quite explainable: zeros, more generally a-points of f(x), play an important role in mathematics and admit some important physical interpretations.
Taking this into account, we have a completely inexplicable situation for functions u(x,y) of two real variables.
The zeros of u(x,y), that is, the solutions u(x,y)=0, are levels of functions u(x,y) that denote important concepts in many applied sciences, so one might expect that they should have been studied in detail. Meanwhile, the real situation is almost the opposite. Level sets in mathematics have been studied very little. Since level sets are mainly curves, their investigation must primarily involve consideration of length, curvature, and the number of connecting components.
The length of level sets was first studied in the frame of Gamma-lines <cite>Bars2002Book</cite> for imaginary parts of meromorphic functions. At present this topic is counted usually as a theory of Gamma-lines, see, for instance, «MathSciNet» , review MR2002433 (2005f:30001) related to my book [BarsBook2002] published in 2002.
Solutions of equations u(x,y)=0 (more generally u(x,y)=const) arise very often in pure and applied mathematics. Remember that the level sets of u-A admit a lot of interpretations (streaming line, potential line, isobar, isoterm) in different applied fields of engineering, physics, environmental problems etc. Notice that in "non degenerating cases" these solutions are some curves. In the case u(x,y) is a polynomial in two variables P(x,y) the solutions P(x,y)=const were studied very largely in the frame of Hilbert's problem 16, part 1; namely problem was to give bounds for the number of connected components of the solutions As to other geometric aspects of the solutions P(x,y)=const they were almost non touched. Also, till end of 1970s, we have no any study related to level sets of any large class of real functions u(x,y), even harmonic functions or more generally functions Rew or |w|. This was a big gap since solutions of Rew=A∈(-∞,∞), or of |w|=R∈(0,∞), have important interpretations in different applied sciences.
At present, we obtain similar results for arbitrary smooth enough real functions of two variables, see in <cite>BarsSuk2004</cite> and <cite>Bars2008</cite>.
Thus, studies of Gamma-lines enlarge the applicability of complex analysis (where, just a-points of functions w(z), i.e. solutions of w(z)=a, were studied before). Respectively studies of level sets of real functions open perspectives for considerations of numerous physical and other processes in many applied sciences.
PART 1, Item 2. The idea of complex conjugation allows us to transfer many studies of complex analysis to differential geometry.
In short, the idea is to establish relationships between the surfaces M in R³ and some complex functions w(z) associated with M; accordingly, the idea is to apply complex functions w(z) in studying surfaces M. In particular, real functions u(x,y) also constitute a surface; thus the idea also applies to real functions.
In some ways, this idea is similar to the idea of Riemann surfaces; in fact, this is an analog of the idea of Riemann surfaces which works in differential geometry and real analysis.
Application of this idea leads to some results in differential geometry. In particular, analogs of the main theorems of classical Nevanlinna theory (in complex analysis) are obtained that are valid for generalized minimal surfaces (in geometry).
We also pose some problems that, in my opinion, can lead to a new crossroads between complex analysis, differential geometry, and real analysis.
Part 1, item 3. Some (more than 20) new principles related to the basic concepts in mathematics.
The results of a general nature related to basic concepts in mathematics (usually referred as principles) were established mostly before 20th century and quite few in 20th century. We mean here primarily such basic concepts as arbitrary enough smooth real functions of one and two variables, plane curves and surfaces in R³, and arbitrary meromorphic (particularly analytic) functions in a given domain.
Some (more than 20) new principles concerning the mentioned concepts were established in my works. All these results are novel, i.e. have no predecessors. The results have led to some new trends (for which I got an "International Marie Curie Award" in 2012).
Among them, we mention:
1. two generalities related to arbitrary smooth real functions of one variable, see [BarsBook2002], chapter.4, [Bars2009], item 2;
2. a generality related to arbitrary smooth real functions of two variables, see [Bars2010b];
3. a generality related to arbitrary smooth curve in the plane - principle of angles, see [Bars2009] (plenary lecture at the 6th ISAAC Congress;
4. a generality related to arbitrary smooth surface in R³, see [Bars2017];
5. more than 15 generalities related to arbitrary analytic or meromorphic functions in a given domain and even more generalities related to functions in the complex plane, see [Bars1978], [Bars1981], [Bars1981a], [Bars1983], [Bars1986], [Bars1999], [Bars2010], [Bars2013, [Bars2017], [Bars2018], especially see my book [BarsBook2002] and survey [Bars2019]. Among them principle of logarithmic derivatives and principle of simple a-points, see the first section in [Bars2019].
We can add here also some results related to arbitrary complex polynomials, see, for instant, [Bars2018b].
BOOKS, EDITED VOLUMES.