Welcome and Opening Remarks

• Bianca Dittrich (Perimeter Institute)
• Katarzyna Rejzner (University of York)
• Theo Johnson-Freyd (Perimeter Institute)
• Sylvie Paycha (University of Potsdam)
• Reiko Toriumi (Okinawa Institute of Science and Technology)
• Anne Taormina (Durham University)

Division algebraic symmetry breaking

• Nichol Furey (Humboldt University of Berlin)

Can the 32C-dimensional algebra RCHO offer anything new for particle physics? Indeed it can. Here we identify a sequence of complex structures within RCHO which sets in motion a cascade of breaking symmetries: Spin(10) -> Pati-Salam -> Left-Right symmetric -> Standard model + B-L (both pre- and post-Higgs-mechanism). These complex structures derive from the octonions, then from the quaternions, then from the complex numbers. It should be noted that this pattern would not have been obvious within the standard formalism.

State sum models with defects

• Catherine Meusburger (University of Erlangen-Nürnberg)

"We explain how to construct a Turaev-Viro state sum model with defect planes, defect lines and defect points. This is work in progress with John Barrett."

Quantum information and black holes

• Johanna Erdmenger (Julius-Maximilians-Universität Würzburg)

"The concepts of quantum information theory play an important role in two seemingly distinct areas of physics: For studying the quantum properties of black holes as well as for devising quantum computing algorithms. Quantum entanglement and computational complexity may be mapped to  geometric quantities. This is intimately related to the holographic principle, according to which the information stored in a volume is encoded on its surface, as is the case for black holes.  In the talk I will describe the essential new concepts that relate quantum information to geometry and gravity. Technically, this involves generalising quantum information results to quantum field theories, i.e. from finite to infinite-dimensional Hilbert spaces. I will explain how the new  relations may be used to obtain both a further understanding of quantum black holes, as well as further advances for the theoretical foundations of quantum computing."

Mathematical Puzzles from Causal Set Quantum Gravity

• Sumati Surya (Raman Research Institute)

I will discuss some of the mathematical puzzles that arise from the causal set approach to quantum gravity. In this approach, any causal continuum spacetime is said to be emergent from an underlying ensemble of locally finite posets which represents a discretisation of the causal structure. If the discrete substructure is to capture continuum geometry to sufficient accuracy, then it must be "approximately" close to it. How can we quantify this closeness? This discreteness, while also preserving local Lorentz invariance, leads to a fundamental non-locality. This is not only an obstacle to a “traditional” initial value formulation, but also to the geometric interpretation of entanglement entropy. Is there an analytic way to quantify the remanent Planckian non-locality? These questions, as well as others arising from the quantum dynamics of causal, may be of potential interest to mathematicians, in particular Geometers and Combinatorists.

On generalized hyperpolygons

• Laura Schaposnik (University of Illinois at Chicago)

In this talk we will introduce generalized hyperpolygons, which arise as Nakajima-type representations of a comet-shaped quiver, following recent work joint with Steven Rayan. After showing how to identify these representations with pairs of polygons, we shall associate to the data an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet. We shall see that, under certain assumptions on flag types, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system.

Exploring spacetime beyond classicality

• Renate Loll (Radboud University Nijmegen)

The physics of General Relativity is deeply intertwined with the mathematics of Lorentzian differentiable manifolds. The latter provide excellent models of spacetime across a vast range of physical scales, encoding gravitational interactions into the curvature properties of smooth metric spaces. However, describing geometry in terms of the infinitesimal line element "ds" does not seem appropriate in the quantum regime near the Planck scale. -- Coming from different perspectives, geometers, classical relativists and quantum gravity researchers are actively investigating more general geometric settings, abandoning smoothness or even continuity of the metric space. The crucial question is what aspects of standard (pseudo-)Riemannian geometry survive, and what this tells us about the "essence" of curvature and geometry and about the ultimate nature of physical spacetime. Can we learn from each others' insights?

A gentle introduction to (modular tensor) categories

• Ana Ros Camacho (Cardiff University)

In this talk we will introduce categories, a notion that packages mathematical objects of any kind and provides an abstract language to study them. We will build up our way towards so-called modular tensor categories, which roughly speaking are categories with a tensor product, duals, and quite a bit of extra categorical structure. They arise in (rational) conformal field theory and its study poses many interesting questions on their classification, internal structure and generalizations. I will give an overview of these questions and some current lines of research in this topic.

Conformal correlators and AdS2/CFT1

• Valentina Forini (City University of London)

From gauge fields to direct connections on gauge groupoids

• Alessandra Frabetti (Université Claude Bernard Lyon 1)

"Geometrically, a gauge theory consists of a spinor bundle describing the matter fields, associated to some principal bundle whose gauge group rules the internal symmetries of the system. The gauge fields are the local expressions of a principal connection inducing a covariant derivative which settles the dynamics of the matter fields. Principal connections can be seen as parallel displacements on the fibres of the principal bundle along curves on the base manifold. In this talk I shortly present a generalisation of gauge fields given by direct connections on gauge groupoids, based on a work in progress with S. Azzali, Y. Boutaib, A. Garmendia and S. Paycha."

Why women leave: Model women and models of discrimination

• Susama Agarwala (Johns Hopkins University )

"In this talk, I posit two concepts from the economics literature as hypotheses for the observed data on women in academia. This talk includes time for discussion about how these concepts can inform our approach to mentoring junior women. "

Change the coefficients of conditional entropies in extensivity

• Asuka Takatsu (Tokyo Metropolitan University)

The Boltzmann--Gibbs entropy is a functional on the space of probability measures. One characterization of the Boltzmann--Gibbs entropy is given by the Shannon--Khinchin axioms, which consist of continuity, maximality, expandability and extensivity. The extensivity is expressed in terms of the linear combinations of conditional probabilities. Replacing the coefficients in the linear combinations with a power function provides a characterization of the Tsallis entropy. I talk about the impossibility to replace the coefficients with a non-power function.

On hidden quantum group symmetries in CFT

• Eveliina Peltola (Bonn University)

"I discuss applications of a hidden $U_q(\mathfrak{sl}_2)$-symmetry in CFT with central charge $c \leq 1$ (focusing on the generic, semisimple case, with $c$ irrational). This symmetry provides a systematic method for solving Belavin-Polyakov-Zamolodchikov PDE systems, and in particular for explicit calculation of the asymptotics and monodromy properties of the solutions. Using a quantum Schur-Weyl duality, one can understand solution spaces of such PDE systems in a detailed way. The solutions, in turn, are useful both for CFT questions and for rigorous understanding of the connections of 2D CFT with critical models of statistical physics."

Entropy and energy fluctuations in non-equilibrium quantum statistical mechanics

• Annalisa Panati (University of Toulon)

"Non-equilibrium statistical mechanics has seen some impressive developments in the last three decades, thank to the pioneering works of Evans, Cohen, Morris and Searles on the violation of the second law, soon followed by the ground-breaking formulation of the Fluctuation Theorem by Gallavotti and Cohen for entropy fluctuation in the early nineties. Their work was by vast literature, both theoretical and experimental The extension of these results to the quantum setting has turned out to be surprisingly challenging and it is still an undergoing effort. Kurchan’s seminal work (2000) showed the measurement role has to be taken in account, leading to the introduction of the so called two-time measurement statistics (also known as full counting statistics). However in this context, the lack of a trajectory notion leads to both conceptual and technical problems, or phenomena with no classical counterpart, as underlined by some of our recent results. In this talk I will review some of the key concept involved in the Fluctuation Theorem and its extensions to the quantum setting; I will present some recent results exploring the role played by ultraviolet regularity conditions (joint work with R.Raquépas and T.Benoist )."