at the

American Institute of Mathematics, San Jose, California

organized by

Hajo Leschke, Alexander V. Sobolev, and Wolfgang Spitzer

In the recent ten or so years, the theory of these operators (TM and WHO) attracted a great deal of renewed attention from mathematical and theoretical physics communities due to the mathematical beauty of problems involved, and due to new applications in physics. For example, novel developments in quantum information theory and statistical physics concern the asymptotics of the (quantum) entanglement entropy of thermal equilibrium states, the overlap of ground states in the Anderson orthogonality catastrophe and the emptiness formation probability in quantum spin models. These quantities can be deduced from suitable Szego limit theorems, that is, from asymptotic results for traces of certain functions of TM and WHO.

The workshop will focus on the following problems.

- Double scaling limits for TM and WHO with symbols which, apart from the scaling parameter $L$, depend on another parameter, say on temperature $T$, as $L \rightarrow \infty$ and $T \downarrow 0$ in a coordinated way.
- Asymptotics for block TM and WHO with Fisher-Hartwig singularities.
- The study of multi-term asymptotics for TM and WHO with the aim to detect topological entropy, that is, a term depending, for instance, on the Euler characteristics of the spatial domain.
- Extension to TM and WHO which are translation invariant with respect to a group different from (all) Euclidean translations. Examples include periodic potentials, magnetic fields and random operators.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

Double-scaling limits of Toeplitz determinants and Fisherâ€“Hartwig singularities

by Jani A. Virtanen

Fredholm Toeplitz operators with VMO symbols and the duality of generalized Fock spaces with small exponents

by Zhangjian Hu and Jani A. Virtanen , *Proceedings of the Royal Society of Edinburgh Section A: Mathematics*

Full Szego-type trace asymptotics for ergodic operators on large boxes

by Adrian Dietlein

Loops in SL(2,C) and Factorization

by Estelle Basor, and Doug Pickrell

Complex groups and root subgroup factorization

by Doug Pickrell

Fredkin Spin Chain

by Olof Salberger, and Vladimir Korepin