{"id":11,"date":"2026-01-19T11:05:45","date_gmt":"2026-01-19T11:05:45","guid":{"rendered":"https:\/\/introductoryschool.univie.ac.at\/?page_id=11"},"modified":"2026-10-13T06:24:03","modified_gmt":"2026-10-13T06:24:03","slug":"program","status":"publish","type":"page","link":"https:\/\/introductoryschool.univie.ac.at\/?page_id=11","title":{"rendered":"Program"},"content":{"rendered":"\n

The Introductory School will include minicourses and lectures. The program is available on the tab “Schedule”<\/strong>. Please go to https:\/\/www.esi.ac.at\/events\/e589\/<\/a> and scroll down to days 14.-18. September.<\/p>\n\n\n\n

Mini-course by Ilse Fischer<\/h2>\n\n\n\n

Alternating Sign Matrices and Plane Partitions<\/h3>\n\n\n\n

Abstract:<\/strong> Alternating Sign Matrices (ASMs) are famously difficult to enumerate. Even more elusive\u2014perhaps impossible\u2014are satisfying bijective proofs of the numerous equinumerosity results involving them. In the first half of this course, we will review several foundational combinatorial constructions for ASMs and related objects that are key to bijective proofs of related results.
These constructions include the spider move (used to enumerate domino tilings of the Aztec diamond), Wieland gyration (central to the ingenious proof of the Razumov\u2013Stroganov conjecture by Cantini and Sportiello), graphical condensation (a key tool behind many beautiful results on the enumeration of lozenge tilings), and the Robinson\u2013Schensted\u2013Knuth (RSK) correspondence.
In the second half of the course, we will introduce Robbins polynomials, generalizations of Schur polynomials that serve as multivariate generating functions for ASMs. These polynomials satisfy Cauchy- and Littlewood-type identities that extend the classical ones. This is particularly exciting because Littlewood-type identities play a central role in several non-bijective proofs of the equinumerosity results mentioned above, and the classical Cauchy and Littlewood identities admit elegant bijective proofs via RSK. We will also see that Robbins polynomials arise as special cases of the fully inhomogeneous spin Hall\u2013Littlewood symmetric rational functions introduced by Borodin and Petrov, and we will explain how Littlewood-type identities for Robbins polynomials have been lifted to this much more general framework.
The second part of the course is based on joint work with Moritz Gangl, Hans H\u00f6ngesberg, and Florian Schreier-Aigner.<\/p>\n\n\n\n

Mini-course by Rick Kenyon<\/h2>\n\n\n\n

Dimers and webs<\/strong> <\/h3>\n\n\n\n

Abstract:<\/strong> An n-web is an n-valent bipartite ribbon graph. (Planar) webs arise in representation theory: they are combinatorial devices used to understand invariants in tensor products of irreducible representations. Webs on surfaces can likewise be used to parameterize the character- or representation-variety of the surface.
Intriguingly, webs on surfaces also arise in the n-dimer model of statistical mechanics. In these lectures we discuss webs and their connection with the probability and statistical mechanics of the dimer model, giving a vast generalization of the classical Kasteleyn theorem which counts dimer covers (perfect matchings) of planar graphs.
These lectures are based on joint work with Dan Douglas, Nicholas Ovenhouse, Haolin Shi, Haihan Wu. <\/p>\n\n\n\n

Mini-course by Ioan Manolescu<\/h2>\n\n\n\n

GFF convergence for the height function of the six-vertex model<\/strong><\/h3>\n\n\n\n

Abstract: <\/strong>The goal of this mini-course is to prove the convergence of the height-function of the six-vertex model to an explicit multiple of the Gaussian Free Field (GFF) in the range of parameters $-1 \\leq \\Delta \\leq -1\/2$.
In this range, the BKW correspondence relates the the six-vertex model to critical planar FK-percolation with cluster-weight $q \\in [1,4]$. The latter was recently proved to become rotationally invariant at large scales, a property which extends to the multi-point correlations of the six-vertex model. Harnessing the expression of the multi-point correlations via the transfer matrix, we deduce constraints on the spectrum of the transfer matrix. Combining these with qualitative features of the six-vertex model, we conclude that the multi-point correlations of the six-vertex height function are asymptotically harmonic, and thus that they converge to their GFF correspondents. 
It should be mentioned that the GFF convergence only holds for the full-plane model. Moreover, the multiple of the GFF to which the six-vertex converges (which depends on $\\Delta$) is obtained via a Bethe-Ansatz estimate of certain eigenvalues of the transfer matrix. This computation acts as an indication of scale invariance for the limit of the height function, and is the only place where explicit eigenvalue computations are used. 
The course will start with an introduction to planar FK-percolation, the six-vertex model and their correspondence. We will then aim to give a brief overview of the convergence result above, illustrating how percolation techniques (such as those of FK-percolation) combine with transfer matrix computations. If time permits, we will discuss consequences of the convergence for FK-percolation.  
Based on joint work with H. Duminil-Copin, K.K. Kozlowski and P. Lammers https:\/\/arxiv.org\/abs\/2603.06268<\/a><\/p>\n\n\n\n

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Part of the thematic program:<\/p>\n\n\n\n