## 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,

*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