Teaching and learning mathematics
Asymptotical methods in quantum mechanics and wave diffraction
Optimization problems on graphs
A good mathematical problem is the heart of mathematical instruction. It attracts student's interests, activates and empowers students' mathematical knowledge, connects different topics and ideas, and makes the lesson dynamic and memorable. How can one recognize or create such problems?
This question has many facets and involves consideration from different perspectives.
My interest in this topic originates from many years of teaching practice and outreach activities in mathematics.
My joint position at the Faculty of Education and Department of Mathematics gives me a unique opportunity to explore the nature of good mathematical problems and their role in teaching mathematics in a greater extent.
My research projects and articles aim to address this question through the following lenses:
1. Cognitive and historical developments of mathematical knowledge.
2. Challenge and mathematical giftedness.
3. Mathematical paradoxes, intuitive and rigorous mathematical logic.
4. The role of technology in assisting mathematical thinking.
5. Interconnectedness and unity of mathematical knowledge.
6. Teachers' use and interpretation of mathematical problems.
My research in this area was greatly informed and advanced by my participation in
two ICMI study conferences: “Challenging mathematics in and beyond the classroom” in
2006 and “Proof and proving in Mathematics Education” in 2009 as well as other meetings
including Canadian Mathematics Education Forum in 2009, Congress of European Society
for the Research in Mathematics Education in 2011 and annual meetings of the
Canadian Math Education study group since 2007. In 2011 Canadian Math Education Study Group
met at Memorial University, so I had a pleasure to help with local organization of this event.
Three of my current projects received funding: (1) High-school mathematics teachers’
perceived impact on their practices of students’ problem-solving enrichment activities
(Research and Development, 2009; $ 5,745); (2) The interconnecting-problem approach in
teaching mathematics: From theory to practice (VP SSHRC, 2010; $ 6,694); (3) Teaching
Euclidean geometry with dynamic geometry software (Instructional Development, 2010 and 2011; $
10,000).
Mechanics of the world as we see it is very well described
by classical mechanics formulated in the mathematical form by
Newton, enriched by Lagrange and extensively developed in the
XIX century. At the beginning of the XX-th century, however,
the solid foundation of the classical mechanics was shaken and
eventually it was replaced by two mathematically sophisticated theories
more adequately describing phenomena in a very large and in an extremely
small scale, respectively - Relativity Theory and Quantum Mechanics.
In spite of the fact that the basic equation of quantum mechanics,
the Schroedinger equation, looks completely unlike any equations
of classical mechanics, there is a profound relationship between
the two theories called semiclassical approximation.
It reflects the fact that quantum dynamics reduces to the classical one
for anything larger than micro-particles. Semiclassical approximation
also provides a way to approximately describe dynamics in the
micro-universe in familiar terms of classical (macro-)physics.
My research activity within the semiclassical science relates to the
following:
(A) Equations for quantum averages.
Approximate solutions of the Schroedinger-like equations
are constructed in a special form (WKB asymptotics with complex
phase), called trajectory-coherent states (TCS).
Observable quantities (position, momentum, higher momenta)
corresponding to initial states that possess certain localization
properties in the phase space, satisfy an infinite system of ordinary
differential equations (with time as the independent variable).
With my thesis supervisors V.V.Belov, V.G.Bagrov, A.Yu.Trifonov
in Moscow and Tomsk,
I studied properties of such systems, and possibility of their
truncations. One interesting example of the truncated ODE w.r.t.
quantum averages is the Wong equation in the theory of isospin. This approach was then extended
in my papers to the non-linear case of Hartree-type
equation, which plays a fundamental role in the modeling of physical phenomena such
as the theory of Bose-Einstein condensate.
(B) Spectral line shape theory
In my postdoctoral research with Profs. T.Osborn and G.Tabisz in
the University of Manitoba, I worked on application on
the mathematical modeling of light scattering by gases, which goes
well beyond the simple model of atomic state transitions. It occurs
that the semiclassical approach requires both a large amount of computations
of trajectories of ODE systems and analysis of the global behaviour
of such trajectories (preservation of wave packets). It is an exciting
research field, which makes use of the theory of dynamical systems,
symplectic geometry, differential geometry, and numerical analysis.
(C) Shortwave diffraction
The problem of electromagnetic wave diffraction from an obstacle has a different physical
nature, but allows application of similar mathematical techniques when asymptotical
behavior is concerned. In shortwave diffraction, once again, we deal with rapidly oscillating
functions and thus a WKB-like approach can be applied in this case. The challenge of numerical
computations requires development and study of new techniques such as acceleration of
series convergence. This research is being done at MUN in collaboration with Dr. S. Sadov.
My research in mathematical physiscs is currently supported by NSERC grant ”New asymptotics
in non-linear quantum mechanics and shortwave diffraction, (2006-2011; $ 9,880 per year).
Optimization problems on graphs
Optimization problems on graphs are associated with
human economical activities rather than with
fundamental laws of nature. The classical travelling
salesman problem, various tasks of industrial
planning, and Internet routing all have to do with
optimization on graphs.
It is not uncommon for a graduate of the Dept. of
Applied Mathematics of the Moscow Institute of
Electronics and Mathematics (MIEM)
to combine research in partial differential equations
and discrete mathematics.
Unified treatment of evolution equations of
mathematical physics and optimization problems on
networks and graphs, proposed by R.Bellman in the
1950s, attained a popularity and was further developed
in both conceptual and formal levels by the school of
Professor V.P. Maslov, who was the founder of the
department.
Bellman proposed a general procedure for solving
optimization problems of such kind. Maslov developed
an algebraic formalism that makes the Bellman equation
look like a linear system (over a certain semiring).
This framework provides a convinient general approach
to various optimization problems and an interesting
new insight on classical algorithms e.g. for
the shortest path problem.
During my work in MIEM, I took part in an applied
project and gave a formulation in algebraic terms of
the Ford-Falkerson algorithm for the maximal flow
problem.
Edge searching algorithms aim to construct a search strategy to catch an intruder hidden in a graph.
This topic was investigated by me at MUN in collaboration
with Drs. D. Pike, D. Dyer and our PhD student O.Yasar.