**Michael Freedman **

**Quantum Mathematics**

Opening talk on Monday afternoon

**The Double Semion Theory in Higher Dimensions**

Technical talk on Tuesday afternoon

**Bruno Nachtergaele**

*Gapped ground states and topological order in quantum spin systems: stability, classification, and invariants.*We will begin with a short introduction to the general mathematical framework for studying finite and infinite quantum spin systems as C*-dynamical systems. This setting will then be used to discuss several general theorems on ground states and equilibrium states of quantum spin systems, relating properties of their spectrum, the behavior of correlations, and broken and unbroken symmetries of the system. In the third lecture we will discuss the notion of gapped ground state phases, examples, and open problems.

Problem set 1, Lecture 2, Problem set 2, Lecture 3

**Robert Seiringer**

**Bose gases, Bose-Einstein condensation, and the Bogoliubov approximation**We present an overview of rigorous results on the low temperature properties of dilute quantum gases, which have been obtained in the past few years. The presentation includes a discussion of Bose-Einstein condensation, the excitation spectrum for trapped gases and its relation to superfluidity, as well as the appearance of quantized vortices in rotating systems. All these properties are intensely being studied in current experiments on cold atomic gases. We will give a description of the mathematics involved in understanding these phenomena, starting from the underlying many-body Schrödinger equation.

**Spiros Michalakis**

*Lieb-Robinson bounds and quasi-adiabatic evolution*The first lecture will be an introduction to Lieb-Robinson bounds for Hamiltonians with short-range interactions. The second lecture will build on the first to prove a powerful lemma on the transformation of short-range, gapped Hamiltonians into frustration-free, gapped Hamiltonians with exponentially decaying interactions. The final lecture will focus on the construction of Hasting's quasi-adiabatic evolution, and its properties.

**Sergey Bravyi**

**Topological quantum codes**

I will review some of the recent results on thermal stability of topological order and quantum self-correction. The discussion will be based on the formalism of stabilizer codes that provides a rich family of quantum spin models and enables a rigorous analysis of their properties. In the first lecture I will define topological order for pure and mixed states. I will use the 2D toric code to illustrate the definitions. In the second lecture I will discuss quantum error correction, dressed logical operators, and sketch a proof that the 4D toric code has topological order at a non-zero temperature. Finally, I will describe a partially self-correcting quantum memory based on the Haah's 3D Cubic Code, characterize its energy landscape, and derive lower bounds on the memory time.

**Daniel Stroock**

**Some applications of Gaussian measures to analysis**

**Determinants, Markov chains and Wilson's algorithm**

The goal of this lecture is to provide examples of the how determinants can simplify computations involving Markov chains on a finite state space. The first example will be the representation of stationary measures in terms of determinants of the transition probability matrix, and the second example will be a similar representation of the Green's function. Finally, these will be applied to give a proof of Wilson's algorithm for constructing the uniform distribution on the set of spanning trees in a connected graph, and, as a dividend, Kirchhoff's formula for the number of such trees.

# Héctor Bombín

**Topological subsystem codes and emergent gauge fields**

I will discuss some of the new avenues opened by topological subsystem quantum error-correcting codes, and in particular by gauge color codes. A closer look at the physics behind these codes reveals a surprising connection with lattice gauge theories, namely a 'hidden' kind of topological order.