# Recent progress in Mathematics and Statistics

The colloquium series invites high calibre researchers to introduce recent progress in emerging topics and new developments in established topics in mathematics, statistics and other related fields at a technical level accessible to a broad spectrum of audience in the School. It intends to cultivate new ideas and promote collaborations across multiple disciplines. The colloquium is held at 4-5pm, first Wednesday of every month, starting from June, 2022.

Please contact Rachel Wang , Ben Goldys , and Laurentiu Paunescu for further information.

Wednesday 1 June 2022 (4-5pm, Carslaw Lecture Theatre 275)

**Speaker:** Professor Stephen Bartlett (School of Physics, University of Sydney)

**Title:** Quantum Memories and Schrödinger’s Cat

**Abstract:** (click to expand)

Quantum information is very fragile, but clever quantum engineers aspire to use error correction to keep information intact. Topologically ordered phases—wherein the most exotic properties of quantum physics such as entanglement are protected within a strongly-interacting material—are currently being commandeered as quantum error-correcting codes for today’s quantum architectures. I’ll introduce these as well as a new generation of theoretical materials that promise to self-correct themselves. Much like a real-world example of Schrödinger’s Cat, a self-correcting quantum memory can protect quantum information in a thermal environment for an arbitrarily long time, without the need for active error correction. I’ll demonstrate that symmetry can assist in giving self-correction in 3D spin lattice models. In particular, I will present quantum codes corresponding to a 2D symmetry-enriched topological (SET) phase that appears naturally on the boundary of an exotic 3D symmetry-protected topological (SPT) phase.
**About the speaker:**
Stephen Bartlett
is a theoretical quantum physicist and Professor in the School of Physics, the University of Sydney. He leads a team pursuing both fundamental and applied research in quantum information theory, including the theory of quantum computing. He is a Chief Investigator in the Australian Research Council Centre of Excellence in Engineered Quantum Systems (EQUS), where he leads a research program on Designer Quantum Materials. He is the inaugural Lead Editor of the APS journal *PRX Quantum*.

Tuesday 2 August 2022 (3-4pm, Carslaw Lecture Theatre 157-257) **temporary change of day and time**

**Speaker:** Professor Todd Oliynyk (School of Mathematics, Monash University)

**Title:** Stable big bang singularity formation in general relativity

**Abstract:** (click to expand)

Since the 1920's, it has been known that the spatially homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes generically develop curvature singularities in the contracting time direction along spacelike hypersurfaces, known as *big bang singularities*, both in vacuum and for a wide range of matter models. For many years, it remained unclear if the big bang singularities were physically relevant. It was thought by some that big bang singularities were due to the unphysical assumption of spatial homogeneity and that they would disappear in non-homogenous spacetimes, or in other words, big bang singularities were

*unstable*under nonlinear perturbations as solutions to the Einstein field equations. A partial resolution to this situation came in 1967 when Hawking established his singularity theorem that guarantees a cosmological spacetime will be geodesically incomplete for a large class of matter models and initial data sets, including highly anisotropic ones.

While Hawking's singularity theorem guarantees that cosmological spacetimes are geodesically incomplete (i.e. at least one observer will experience something pathological at a finite time in the past) for a large class of initial data sets, it is silent on the cause of the geodesic incompleteness. It has been widely anticipated that the geodesics incompleteness is due to the formation of curvature singularities, and it is an outstanding problem in mathematical cosmology to rigorously establish the conditions under which this expectation is true and to understand the dynamical behaviour of cosmological solutions near singularities.

In this talk, I will begin by introducing the FLRW and Kasner solutions of the Einstein-scalar field equations, which are exact, spatially homogeneous solutions that play a distinguished role in the analysis of big bang singularities. After briefly providing context for the FLRW and Kasner solutions in the historical development of the field of cosmology, I will define what it means for a FLRW/Kasner big bang singularity to be

*stable*. With this notion in hand, I will then discuss the recent influential FLRW and Kasner big bang stability proofs of Rodnianski-Speck and Fournodavlos-Rodnianski-Speck. One aspect of these stability results that I will pay particular attention to is their global nature. To conclude the talk, I will discuss some recent work done in collaboration with Florian Beyer where we improve the Rodnianski-Speck FLRW big bang stability result by establishing that the FRLW big bang is locally stable, which is a significantly stronger notion of stability with important physical consequences that I will briefly discuss. Time permitting, I will also briefly discuss open questions and future directions for research.

**About the speaker:**
Todd Oliynyk
is a mathematical physicist and Professor in the School of Mathematics at Monash University. His main research interests are in mathematical relativity, partial differential equations and geometric analysis. He was awarded the Australian Mathematical Society Medal in 2011, an Australian Research Council Future Fellowship in 2012 and a Fulbright Senior Scholarship in 2017, and he is a Fellow of the Australian Mathematical Society.

Tuesday 4 October 2022 (4-5pm, Carslaw Lecture Theatre 373)

**Speaker:** Professor Georg Gottwald
(School of Mathematics and Statistics, The University of Sydney)

**Title:** Levy flights as an emergent phenomenon in a spatially extended system

**Abstract:** (click to expand)

Anomalous diffusion and Levy flights, which are characterized by the occurrence of random discrete jumps of all scales, have been observed
in a plethora of natural and engineered systems, ranging from the motion of molecules to climate signals.
Mathematicians have recently unveiled mechanisms to generate anomalous diffusion, both stochastically and deterministically. However, there exists to the best of our knowledge no explicit example of a spatially extended system which exhibits anomalous diffusion without being explicitly driven by Levy noise.

We provide the first explicit example of a stochastic partial differential equation which albeit only driven by normal Gaussian noise supports anomalously diffusive propagating front solutions. This is an entirely emergent phenomenon without explicitly built-in mechanisms for anomalous diffusion. This is joint work with Chunxi Jiao.

**About the speaker:**
Georg Gottwald
is a Professor in the School of Mathematics at the University of Sydney.

Wednesday 2 November 2022 (4-5pm, Carslaw Lecture Theatre 157-257)

**Speaker:** Professor Geordie Williamson
(School of Mathematics and Statistics, The University of Sydney)

**Title:** What can the working mathematician expect from deep learning?

**Abstract:** (click to expand)

Deep learning (the training of deep neural nets) is a very
simple idea. Yet it has led to many striking applications throughout
science and industry over the last decade. It has also become a major tool
for applied mathematicians. In pure mathematics the impact
has so-far been modest. I will discuss a few instances where it has
proved useful, and led to interesting results in pure mathematics. I will also
reflect on my experience as a pure mathematician interacting with deep
learning. Finally, I will discuss what can be learned from the
successful examples that I understand, and try to guess an answer to
the question in the title.
(Deep learning also raises interesting mathematical questions, but
this talk won't be about this.)
**About the speaker:**
Geordie Williamson
is the Director of the Sydney Mathematical Research Institute and Professor of Mathematics at the University of Sydney.