# 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, usually the first Wednesday of every month, starting from June, 2022.

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

Wednesday 9 August 2023 (4-5pm, Carslaw Lecture Theatre 173)

**Speaker:** Prof. Askold Khovanskii
The University of Toronto

**Title:** Algebraic Geometry and Convex
Geometry

**Abstract:** (click to expand)

Newton polyhedra relate algebraic geometry and singularity theory with the geometry of convex polyhedra within the framework of toric geometry. This connection is useful in both directions. On the one hand, it provides explicit answers to problems in algebra and singularity theory in terms of convex polyhedra. For instance, according to the Bernstein-Khovanskii-Koushnirenko (BKK) theorem, the number of solutions of a generic system of n equations in (C^*)^n with fixed Newton polyhedra is equal to the mixed volume of the Newton polyhedra multiplied by n!. This suggests that there should be an analog of the famous Alexandrov- Fenchel inequalities from the theory of mixed volumes in algebraic geometry. (These inequalities can be considered as a broad generalization of the classical isoperimetric inequality.) On the other hand, algebraic theorems of a general nature (such as the Hirzebruch-Riemann-Roch theo- rem) suggest unexpected results in the geometry of convex polyhedra. The theory of Newton-Okounkov bodies connects algebra and geometry in the broad framework of general algebraic varieties. This relationship is useful in many directions. It suggests the existence of birationally invariant theory of intersection of divisors and provides elementary proofs of Alexandrov-Fenchel inequalities in the theory of intersections and their local versions for the multiplicities of intersections of ideals in local rings. Alexandrov-Fenchel geometric inequalities easily follow from their algebraic analogs. In the theory of invariants, this connection gives analogues of the BKK theorem for horospherical varieties and some other varieties with the action of a reductive group. In abstract algebra, this relationship allows us to introduce a broad class of graded algebras, the Hilbert functions of which are not necessarily polynomials for large argument values, but have polynomial asymptotics. In my presentation, I will introduce these results in a way that is accessible to a general mathematical audience.

The slides from the talk are available.

**About the speaker:**
Askold Khovanskii is a professor of mathematics at the University of Toronto.

Wednesday 31 May 2023 (4-5pm, Carslaw Lecture Theatre 173)

**Speaker:** Dr. Alan Stapledon
Sydney Mathematics Research Institute

**Title:** The local motivic monodromy conjecture holds generically.

**Abstract:** (click to expand)

This is a combinatorics talk hiding in geometric clothing. The clothing consists of the local motivic monodromy conjecture, an analogue of the Weil conjectures. Given a polynomial f with integer coefficients, it predicts a remarkable relationship between arithmetic properties (number of solutions to f = 0 modulo an integer) and topological properties (eigenvalues of monodromy of the Milnor fibre of the complex hypersurface defined by f). The conjecture remains wide open in general. Surprisingly, it remains open for "generic" choices of f, even though there are well known and relatively simple combinatorial formulas for all the relevant quantities. This brings us to the heart of the talk: for "generic" f, we relate the local motivic monodromy conjecture to a long-standing open question of Stanley on triangulations of simplices. Progress towards the latter question then helps resolve the local motivic monodromy conjecture. This is joint work with Matt Larson and Sam Payne.

The slides from the talk are available.

**About the speaker:**
Alan Stapledon recently returned to mathematics after a decade of working for a hedge fund. His current interests are
combinatorics and algebraic geometry.

Wednesday 26 April 2023 (4-5pm, Carslaw Lecture Theatre 175)

**Speaker:** Professor Ciprian Manolescu
(Stanford University)

**Title:** Khovanov homology and four-dimensional topology

**Abstract:** (click to expand)

Over the last forty years, most progress in four-dimensional topology came from gauge theory and related invariants. Khovanov homology is an invariant of knots in R^3 of a different kind: its construction is combinatorial, and connected to ideas from representation theory. There is hope that it can tell us more about smooth 4-manifolds; for example, Freedman, Gompf, Morrison and Walker suggested a strategy to disprove the 4D Poincare conjecture using Rasmussen’s invariant from Khovanov homology. It is yet unclear whether their strategy can work. I will explain several recent results in this direction and some of the challenges that appear. A key problem is to certify when a knot is slice (bounds a disk in four-dimensional half-space), which can be tackled with machine learning. The talk is based on joint work with Sergei Gukov, Jim Halverson, Marco Marengon, Lisa Piccirillo, Fabian Ruehle, Mike Willis, and Sucharit Sarkar.

The slides from the talk are available.

**About the speaker:**
Ciprian Manolescu is a Professor of Mathematics at Stanford in the Topology Group
and in the Geometry Group

Wednesday 29 March 2023 (4-5pm, Carslaw 173)

**Speaker:** Professor Christof Melcher
(RWTH Aachen University)

**Title:** Topological solitons in variational models of condensed matter

**Abstract:** (click to expand)

Topological solitons occur in nonlinear field theories from various areas of mathematical physics. Originally, they served as particle models in certain quantum field theories. Recently, there has been a strong trend towards topological solitons in variational models of condensed matter. The recent experimental discovery of so-called "chiral magnetic skyrmions" has raised hopes to use topology as a concept for future information technologies. In this talk I will present some fundamental mathematical questions, advances and open problems in this exciting field at the interface of analysis, geometry and physics.

The slides from the talk are available.

**About the speaker:**
Christof Melcher is a Professor of Mathematics at Aachen and leads a group in Applied Analysis.

Thursday 23 February 2023 (4-5pm, Carslaw Lecture Room 373)

**Speaker:** Dr Nezhla Aghaee
(IMADA, University of Southern Denmark)

**Title:** Super Pentagon relation

**Abstract:** (click to expand)

The pentagon relation is an interesting relation in mathematics. I will discuss how to get the super OSP (1|2) generalization of this relation and motivate it from different topic in Mathematical Physics such as, AGT correspondence, Teichmueller TQFT, and Integrability.

The slides from the talk are available.

**About the speaker:**
Dr Nezhla Aghaee is a Marie Curie postdoc between the Quantum Mathematics center Odense and the University of Geneva.

Thursday 9 February 2023 (4-5pm, Carslaw Seminar Room 350)

**Speaker:** Professor Alexandru Suciu
(Northeastern University, USA)

**Title:** Topological invariants of groups and tropical geometry

**Abstract:** (click to expand)

There are several topological invariants that one may associate to a finitely generated group G -- the characteristic varieties, the resonance varieties, and the Bieri–Neumann–Strebel invariants -- which keep track of various finiteness properties of certain subgroups of G. These invariants are interconnected in ways that makes them both more computable and more informative. I will describe one such connection, made possible by tropical geometry, and I will provide examples and applications pertaining to complex geometry and low-dimensional topology.

The slides from the talk are available.

**About the speaker:**
Alex Suciu
is a Professor of Mathematics at Northeastern University, in Boston,
Massachusetts, USA. His main research interests are in topology,
and how it relates to algebra, geometry, and combinatorics.

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.)

The slides from the talk are available, as a pdf file or as the original power-point file.

**About the speaker:**
Geordie Williamson
is the Director of the Sydney Mathematical Research Institute and Professor of Mathematics at the University of Sydney.

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.

**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.

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**