## as a part of a research project in:

- combinatorics,
- algebraic combinatorics, enumerative combinatorics, asymptotic combinatorics,
- representation theory,
- asymptotic representation theory,
- random matrix theory,
- Voiculescu's free probability theory,
- mathematical phycics,

## there is a number of open research jobs:

- scholarships for BSc / MSc students,
- scholarship for a PhD student,

*Dynamic
asymptotic combinatorics*: project summary

*[for a version of the summary with beautiful
pictures see the PDF below with the full project
documentation] *

Many combinatorial structures can be viewed as
discrete versions of continuous geometric objects such
as *‘surfaces’* or *‘shapes’*. Good
examples to keep in mind are provided by *Young
diagrams*, *Young tableaux* as well as
by *alternating sign matrices*. It is then
natural to ask about the asymptotic behavior of these
discrete shapes and surfaces as the size tends to
infinity. In many cases, the shape of the typical
random element approaches a continuous limit. In such
a case we say the model has a *“limit shape”*.
It should be stressed that such questions concerning
the limit shape are usually formulated in the
probabilistic setup in which the considered class of
combinatorial objects is equipped with some natural
probability measure.

Many combinatorial structures can be naturally
equipped with some sort of dynamics which gives rise
to the questions about existence of some ** “dynamic
limit shape” **which is in our focus.
A good concrete example to keep in mind is the
celebrated

*Schützenberger’s jeu de taquin*: from the initial standard Young tableau one removes the south-west corner box (which contains the number 1, see Figure 2a); in this way an empty cell (hole) is created. We either slide down the box directly above the hole or we slide left the box directly right to the hole, choosing the box which carries a smaller number. We continue the sliding; in this way the empty cell moves successively towards the boundary of the tableau, see Figure 2b. Sample questions about the dynamical limit shape might be:

*is there some typical trajectory along which the sliding occurs?*Suppose we iterate the above jeu de taquin transformation;

*does this discrete model of draining a two-dimensional sandpile converge to some hydrodynamic limit?*

The goal of this research proposal is to ** investigate
the asymptotics of large random combinatorial
structures and their limit shapes and to
investigate their connection to (asymptotic)
representation theory, random matrix theory,
Voiculescu’s free probability theory, ergodic
theory, statistical mechanics and mathematical
physics**. Special attention will be
given to dynamical aspects of such asymptotic
behavior, in particular to trajectories of particles
and trajectories of second class particles from the
interacting particle systems, dynamical
transformations of large combinatorial structures and
their hydrodynamic limits. Specific examples which we
have in mind are concentrated in (but not limited to)
the class of combinatorial objects related to
algebraic combinatorics and representation theory such
as random Young diagrams and tableaux as well as
combinatorial algorithms such as
Robinson–Schensted–Knuth algorithm and
multidimensional Pitman transform as well as
Schützenberger’s jeu de taquin, promotion and
evacuation.

## details

the jobs are open in the beautiful medieval town of Toruń

the positions are funded by a *"Dynamic
asymptotic combinatorics"* grant MAESTRO-ST
of National Center of Science

the details of the grant proposal are available here (hint: have a look on page 24)

if you are interested, please contact me directly by email