The development of functional analysis, operator algebras, and related topics owes
much to the field of quantum mechanics. It has provided both relevant questions and
interesting concepts. Specically, energy and entropy estimates are routinely used in the
study of partial differential equations. Moreover, the use of analytic inequalities has been
perfected into an art which has generated many deep and insightful results. The goal
of this school is to introduce these topics to young mathematicians; namely those at the
PhD or postdoctoral level. The school will start with an introduction to quantum mechanics by Jan Wehr that will
review the basic concepts from a mathematical perspective. The main part of the school
will consist of three lectures given by Eric Carlen, Bruno Nachtergaele, and Robert Seiringer, all experts in their fields.

Eric Carlen

Trace Inequalities and Quantum Entropy
A state rho in quantum statistical mechanics is given by a density matrix, i.e. a positive definite symmetric operator with Tr rho = 1. The entropy S(rho) is then given by S(rho) = Tr rho ln rho,
and the relative entropy between two states rho and mu, is S(rho,mu) = Tr[rho ln rho  rho ln mu]. These quantities play an essential role in quantum statistical mechanics.
For example, they are used to express variational principles which characterize equilibrium
states, to describe correlations between different systems, and to illustrate properties of systems
interacting with heat baths.
In these types of applications, the model often consists of many interacting
subsystems. Inequalities such as the strong subadditivity of quantum entropy, originally due to Lieb and Ruskai, that governs how the entropy of the state of the whole system relates to the
entropy of the state of various subsystems, are of fundamental importance. Since then, other trace inequalities
allowing one to gain control over observables pertaining to the whole system in terms of simpler observables
have continued to be developed and applied. What has resulted from this study is a rich body
of inequalities and convexity results applicable to a wide range of functionals
relevant in quantum statistical mechanics. All the same, there remain a
number of open questions that are easy to state and understand, but whose proof
would likely lead to a better understanding of the field in general.
This course will give an introduction, at a level appropriate for beginning graduate students, to the theory of convex trace functions.
Classical results, such as Lieb's proof of the convexity of the WignerYanasi skew information rho mapsto Tr([rho^{1/2},H])^2,
will be discussed in detail as well as a variety of other results pertaining to more recent research.
With applications to quantum computing and information, this topic of current interest is surprisingly accessible.
The course will continue with some applications in quantum statistical mechanics, both equilibrium and nonequilibrium, and an introduction to open problems.

Bruno Nachtergaele

Quantum Entropy in Condensed Matter and Information Theory
Condensed matter physics is primarily concerned with lowtemperature properties of strongly interacting systems
of spins and electrons. This means that the focus is on
ground state properties and the nature of the lowlying excitations
of the model Hamiltonians describing the systems of interest.
Quantum information and compuation have added attention to
this subject with specific questions about models with properties
deemed to be suitable for the implemetation of quantum computation devices.
For fundamental reasons, quantum systems, unlike classical ones, generically fluctuate at zero temperature. It is therefore not a surprise that entropy is a very useful tool to study ground states
of quantum systems. In the context of quantum information theory,
quantum entropy has been used as an effective tool to quantify
entanglement, a notion describing the essential "quantumness"
of the state and that is directly related to its usefulness for quantum
computation.
This course will introduce the main mathematical techniques that are at the basis of several interesting developments in this area. Some are new, such as the dynamical equations introduced by Hastings (which he called quasiadiabatic evolution), and others have recently been refined and extended, such as LiebRobinson bounds and matrix product states, a.k.a. finitely correlated states. Several applications of these techniques will be discussed in detail,
such as theorems connecting ground state properties and aspects of the lowlying spectrum, and partial results about the area law conjecture for the local entropy of quantum ground states. Due attention will be given to the discussion of important open problems for future research.
Note added during the course: A $50 price has been promised for the first proof of the area law in dimension greater than 1!

Robert Seiringer

Inequalities for Schroedinger Operators and Applications
Schroedinger operators are of central importance in quantum mechanics. They constitute a
subfield of partial differential equations and have many applications besides quantum
mechanics, for instance in the study of turbulence and other nonlinear phenomena. A key to
understanding stability of quantum mechanical systems is the uncertainty principle, which in
mathematical terms is expressed in terms of Hardy and Sobolev inequalities for L^p norms of
gradients of functions on R^d. For large quantum systems, stability is much more subtle and
can be understood only via more refined types of uncertainty principles, which are known as
LiebThirring inequalities. These are bounds on sums of powers of the eigenvalues of
Schroedinger operators, which become relevant due to antisymmetry of manyparticle
wavefunctions, which is also known as the Pauli principle.
Recent research has focused on further understanding the relation between Hardy and Sobolev
inequalities on the one hand, and LiebThirring inequalities on the other hand. It turns out
that the usual LiebThirring inequalities can be improved by incorporating the Hardy term, and
also the Hardy inequality can be improved by a (subcritical) Sobolev term. In fact, these two
improvements are intimately related. The resulting inequalities have recently been applied
successfully to a longstanding problem in quantum mechanics, concerning the stability of
matter when both effects of special relativity and the interaction with electromagnetic fields
are taken into account. Further applications may be found in other areas, such as quantum
chemistry or even astronomy with stability of stars.
The purpose of this course is to explain the connections between Hardy, Sobolev and
LiebThirring inequalities. Moreover, their usefulness will be demonstrated in several
applications in analysis and mathematical physics.

Jan Wehr

Quantum Physics from Zero
Many young mathematicians have attended a basic undergraduate course on quantum mechanics as is usually taught by a physicist.
Such a class is typically insufficient to understand the necessary background and critical issues which will be at the heart of this school.
Since our goal is to reach a broad audience, we will provide this introductory minicourse to present and develop the main ideas in a clear and concise form.
Relevant topics include spaces of functions, the Schroedinger equation, positive linear functionals (the states in quantum mechanics), the unitary group of
evolution, and a few concrete models. A special emphasis will be given to those topics discussed within the other courses of the school and specific themes
which will arise in the lectures on current research projects.

