Conference

Emmy Noether was a giant of mathematics whose work tied together two fundamental concepts: conservation laws and symmetries in nature. But who was she, and why does her work still have such impact? Mathematician Peter Olver explores Noether’s life and career, and delves into the curious history of her famous theorems. Physicist Ruth Gregory looks at the lasting impact of Noether’s theorem, and how it connects with the Standard Model and Einstein’s general relativity.

Some of the most violent events in the Universe are accompanied by spectacular warpages of space-time that travel to us in the form of gravitational waves. Gravitational waves were predicted by Einstein's theory of general relativity over a century ago, but scientists have not yet detected them directly. Learn about how we search for these tiny space-time ripples and decode the unique information they carry about mysterious events in space as far back in time as the first moments after the Big Bang.

A new quantum age is dawning in which humans will be able to build and control scalable quantum systems with amazing properties and applications. Aside from enabling revolutionary future technologies, quantum information science is providing powerful new tools for attacking deep problems in fundamental physical science. In particular, the recent convergence of quantum information and quantum gravity is sparking exciting progress on some old and very hard questions.

Determining causal relationships is central to scientific understanding. Knowledge of such relations permit us not only to predict how a system will behave naturally, but also how it would behave under different hypothetical circumstances, including those where we exert control over some component. In the context of quantum theory, the problem of figuring out what causes what is particularly vexing. One of the central results in the foundations of quantum theory, Bell's theorem, can be understood as demonstrating that it is impossible to provide a causal explanation of the correlations that arise for entangled quantum systems without resorting to fine-tuning. Impossible, that is, using the standard framework of causal models. A new quantum notion of causal model, however, holds promise for achieving such an explanation. It also has practical applications, allowing one to infer causal relationships from observed correlations in scenarios where classically one could not. Correlation does not imply causation. Except in a quantum world, where sometimes it does.

I will discuss the state of particle physics today and emphasize the implications of the discovered H-boson in high energy physics and cosmology and the connections with the future discovery and precision machines and observatories world-wide.

The theory of resurgence connects perturbative and non-perturbative physics. Focusing on certain one-dimensional quantum mechanical systems with degenerate harmonic minima, I will explain how the resurgent trans-series expansions for the low lying energy eigenvalues follow from the exact quantization condition via the uniform WKB approach. In the opposite spectral region (with high lying eigenvalues), in contrast to the divergent asymptotic expansions expressed as trans-series, the relevant expansions are convergent. However, due to the poles in the expansion coefficients, they contain non-perturbative contributions which can be identified with complex instantons. I will demonstrate that in each spectral region there are striking relation between perturbative and non-perturbative expansions even though the nature of these expansions are very different. Notably, the quantum mechanical examples that I will discuss encode the vacua of certain supersymmetric gauge theories in their spectra.

The first lecture will be devoted to the review of the classical theory of the Witten Laplacian, the second -- to the concepts of resurgent analysis. The third -- to applications of the resurgent analysis to the Witten Laplacian. Time permitting, we will touch upon some foundational questions of resurgent analysis.

These lectures will focus on the geometry of ambitwistor string theories. These are infinite tension analogues of conventional strings and provide the theory that leads to the remarkable formulae for tree amplitudes that have been developed by Cachazo, He and Yuan based on the scattering equations. Although the bosonic ambitwistor string action is expressed in space-time, it will be seen that its target is classically `ambitwistor space', the space of complexified null geodesics in the complexification of a space-time. The lectures will review Ambitwistor constructions from the 70's and 80's that extend the Penrose-Ward twistor constructions for self-dual Yang-Mills and gravitational fields in four dimensions to arbiitrary fields in general dimension. LeBrun showed that the conformal geometry of a space-time is encoded into the complex structure of ambitwistor space. The linearized version encodes linear fields on space-time into sheaf cohomology classes on ambitwistor space. In the case of momentum eigenstates, these give the `scattering equations' that underly the CHY formulae and the ambitwistor string can be used to compute amplitudes via these formulae. If there is time, the lectures will discuss how different matter theories can be obtained, different geometric realizations of ambitwistor space lead to different formulae, the relationship between the asymptotic symmetries of space-time and Weinberg's soft theorems concerning the behaviour of amplitudes when momenta become small, and/or extensions of the ideas to loop amplitudes.

I will give an overview of the algebro-geometric approach to Feynman integral in perturbative quantum field theory and the occurrence of motives and periods in parametric Feynman integrals in momentum space, focusing on joint work with Paolo Aluffi.