Congratulations to Michael!

]]>Congratulations to PhD student in Pure Mathematics, Hao Guo (Supervisors Varghese, Hang Wang) who has been offered an NSF funded postdoc at Texas A&M which he will accept!

]]>**Abstract**: Let $G$ be a locally compact group, $\Gamma$ a discrete subgroup and $C_{G}(\Gamma)$ the commensurator of $\Gamma$ in $G$. The cohomology of $\Gamma$ is a module over the Shimura Hecke ring of the pair $(\Gamma,C_G(\Gamma))$. This construction recovers the action of the Hecke operators on modular forms for $SL(2,\mathbb{Z})$ as a particular case. In this talk I will discuss how the Shimura Hecke ring of a pair $(\Gamma, C_{G}(\Gamma))$ maps into the $KK$-ring associated to an arbitrary $\Gamma$-C*-algebra. From this we obtain a variety of $K$-theoretic Hecke modules. In the case of manifolds the Chern character provides a Hecke equivariant transformation into cohomology, which is an isomorphism in low dimensions. We discuss Hecke equivariant exact sequences arising from possibly noncommutative compactifications of $\Gamma$-spaces. Examples include the Borel-Serre and geodesic compactifications of the universal cover of an arithmetic manifold, and the totally disconnected boundary of the Bruhat-Tits tree of $SL(2,\mathbb{Z})$. This is joint work with M.H. Sengun (Sheffield).

**Title:** Radial Toeplitz operators on bounded symmetric domains

**When: **Friday, 9 March 2018 at 1:10pm in Lower Napier LG11

**Speaker**: Raul Quiroga-Barranco (CIMAT, Guanajuato, Mexico)

**Abstract:** The Bergman spaces on a complex domain are defined as the space of holomorphic square-integrable functions on the domain. These carry interesting structures both for analysis and representation theory in the case of bounded symmetric domains. On the other hand, these spaces have some bounded operators obtained as the composition of a multiplier operator and a projection. These operators are highly noncommuting between each other. However, there exist large commutative C*-algebras generated by some of these Toeplitz operators very much related to Lie groups. I will construct an example of such C*-algebras and provide a fairly explicit simultaneous diagonalization of the generating Toeplitz operators.

**Title:** Quantum Airy structures and topological recursion

**When: **Wednesday, 14 March 2018 at 1:10pm in Ingkarni Wardli B17

**Speaker:** Gaetan Borot

**Abstract:** Quantum Airy structures are Lie algebras of quadratic differential operators — their classical limit describes Lagrangian subvarieties in symplectic vector spaces which are tangent to the zero section and cut out by quadratic equations. Their partition function — which is the function annihilated by the collection of differential operators — can be computed by the topological recursion. I will explain how to obtain quantum Airy structures from spectral curves, and explain how we can retrieve from them correlation functions of semi-simple cohomological field theories, by exploiting the symmetries. This is based on joint work with Andersen, Chekhov and Orantin.

**Title:** Family gauge theory and characteristic classes of bundles of 4-manifolds

**When: **Friday, 16 March 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

**Speaker**: Hokuto Konno (University of Tokyo)

**Abstract:** I will define a non-trivial characteristic class of bundles of 4-manifolds using families of Seiberg-Witten equations. The basic idea of the construction is to consider an infinite dimensional analogue of the Euler class used in the usual theory of characteristic classes. I will also explain how to prove the non-triviality of this characteristic class. If time permits, I will mention a relation between our characteristic class and positive scalar curvature metrics.

**Title:** Computing trisections of 4-manifolds

**When: **Friday, 23 March 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

**Speaker:** Stephen Tillmann

**Abstract:** Gay and Kirby recently generalised Heegaard splittings of 3-manifolds to trisections of 4-manifolds. A trisection describes a 4-dimensional manifold as a union of three 4–dimensional handlebodies. The complexity of the 4–manifold is captured in a collection of curves on a surface, which guide the gluing of the handelbodies. The minimal genus of such a surface is the trisection genus of the 4-manifold. After defining trisections and giving key examples and applications, I will describe an algorithm to compute trisections of 4–manifolds using arbitrary triangulations as input. This results in the first explicit complexity bounds for the trisection genus of a 4–manifold in terms of the number of pentachora (4–simplices) in a triangulation. This is joint work with Mark Bell, Joel Hass and Hyam Rubinstein. I will also describe joint work with Jonathan Spreer that determines the trisection genus for each of the standard simply connected PL 4-manifolds.

**Title:** Chaos in higher-dimensional complex dynamics

**When: **Friday, 20 April 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

**Speaker:** Finnur Larusson

**Abstract:** I will report on new joint work with Leandro Arosio (University of Rome, Tor Vergata). Complex manifolds can be thought of as laid out across a spectrum characterised by rigidity at one end and flexibility at the other. On the rigid side, Kobayashi-hyperbolic manifolds have at most a finite-dimensional group of symmetries. On the flexible side, there are manifolds with an extremely large group of holomorphic automorphisms, the prototypes being the affine spaces $\mathbb C^n$ for $n \geq 2$. From a dynamical point of view, hyperbolicity does not permit chaos. An endomorphism of a Kobayashi-hyperbolic manifold is non-expansive with respect to the Kobayashi distance, so every family of endomorphisms is equicontinuous. We show that not only does flexibility allow chaos: under a strong anti-hyperbolicity assumption, chaotic automorphisms are generic. A special case of our main result is that if $G$ is a connected complex linear algebraic group of dimension at least 2, not semisimple, then chaotic automorphisms are generic among all holomorphic automorphisms of $G$ that preserve a left- or right-invariant Haar form. For $G=\mathbb C^n$, this result was proved (although not explicitly stated) some 20 years ago by Fornaess and Sibony. Our generalisation follows their approach. I will give plenty of context and background, as well as some details of the proof of the main result.

**Title:** Index of Equivariant Callias-Type Operators

**When: **Friday, 27 April 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

**Speaker:** Hao Guo (University of Adelaide)

**Abstract:** Suppose M is a smooth Riemannian manifold on which a Lie group G acts properly and isometrically. In this talk I will explore properties of a particular class of G-invariant operators on M, called G-Callias-type operators. These are Dirac operators that have been given an additional Z_2-grading and a perturbation so as to be “invertible outside of a cocompact set in M”. It turns out that G-Callias-type operators are equivariantly Fredholm and so have an index in the K-theory of the maximal group C*-algebra of G. This index can be expressed as a KK-product of a class in K-homology and a class in the K-theory of the Higson G-corona. In fact, one can show that the K-theory of the Higson G-corona is highly non-trivial, and thus the index theory of G-Callias-type operators is not obviously trivial. As an application of the index theory of G-Callias-type operators, I will mention an obstruction to the existence of G-invariant metrics of positive scalar curvature on M.

**Title:** Index of Equivariant Callias-Type Operators

**When: **Friday, 4 May 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

**Speaker:** Tony Licata (Australian National University)

**Abstract: **The Artin braid group arise in a number of different parts of mathematics. The goal of this talk will be to explain how basic group-theoretic questions about the Artin braid group can be answered using some modern tools of linear and homological algebra, with an eye toward proving some open conjectures about other groups.

**Title:** Hitchin’s Projectively Flat Connection for the Moduli Space of Higgs Bundles

**When: **Friday, 15 June 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

**Speaker:** John McCarthy (University of Adelaide)

**Abstract: **In this talk I will discuss the problem of geometrically quantizing the moduli space of Higgs bundles on a compact Riemann surface using Kahler polarisations. I will begin by introducing geometric quantization via Kahler polarisations for compact manifolds, leading up to the definition of a Hitchin connection as stated by Andersen. I will then describe the moduli spaces of stable bundles and Higgs bundles over a compact Riemann surface, and discuss their properties. The problem of geometrically quantizing the moduli space of stables bundles, a compact space, was solved independently by Hitchin and Axelrod, Del PIetra, and Witten. The Higgs moduli space is non-compact and therefore the techniques used do not apply, but carries an action of C*. I will finish the talk by discussing the problem of finding a Hitchin connection that preserves this C* action. Such a connection exists in the case of Higgs line bundles, and I will comment on the difficulties in higher rank.

**T itle: **Comparison Theorems under Weak Assumptions

**Abstract: **The classical volume comparison states that under a lower bound on the Ricci curvature, the volume of the geodesic ball is bounded from above by that of the geodesic ball with the same radius in the model space. On the other hand, counterexamples show the assumption on the Ricci curvature cannot be weakened to a lower bound on the scalar curvature, which is the average of the Ricci curvature. In this talk, I will show that a lower bound on a weighted average of the Ricci curvature is sufficient to ensure volume comparison. In the course I will also prove a sharp volume estimate and an integral version of the Laplacian comparison theorem. If time allows, I will also present the Kahler version of the theorem.

]]>

It is an outstanding achievement for our students that have won all of the available prizes against a field of students from across the country!

]]>More information: http://www.austms.org.au/The+Bernhard+Neumann+Prize”

]]>15:10

**Title**: Models, machine learning, and robotics: understanding biological networks

15:10 **Fri 16 Mar,** 2018 :: Horace Lamb 1022 :: Prof Steve Oliver :: University of Cambridge

**Abstract:** The availability of complete genome sequences has enabled the construction of computer models of metabolic networks that may be used to predict the impact of genetic mutations on growth and survival. Both logical and constraint-based models of the metabolic network of the model eukaryote, the ale yeast Saccharomyces cerevisiae, have been available for some time and are continually being improved by the research community. While such models are very successful at predicting the impact of deleting single genes, the prediction of the impact of higher order genetic interactions is a greater challenge. Initial studies of limited gene sets provided encouraging results. However, the availability of comprehensive experimental data for the interactions between genes involved in metabolism demonstrated that, while the models were able to predict the general properties of the genetic interaction network, their ability to predict interactions between specific pairs of metabolic genes was poor. I will examine the reasons for this poor performance and demonstrate ways of improving the accuracy of the models by exploiting the techniques of machine learning and robotics. The utility of these metabolic models rests on the firm foundations of genome sequencing data. However, there are two major problems with these kinds of network models – there is no dynamics, and they do not deal with the uncertain and incomplete nature of much biological data. To deal with these problems, we have developed the Flexible Nets (FNs) modelling formalism. FNs were inspired by Petri Nets and can deal with missing or uncertain data, incorporate both dynamics and regulation, and also have the potential for model predictive control of biotechnological processes.

**Title** Complexity of 3-Manifolds

15:10 **Fri 23 Mar**, 2018 :: Horace Lamb 1022 :: A/Prof Stephan Tlllmann :: University of Sydney

**Abstract:**In this talk, I will give a general introduction to complexity of 3-manifolds and explain the connections between combinatorics, algebra, geometry, and topology that arise in its study. The complexity of a 3-manifold is the minimum number of tetrahedra in a triangulation of the manifold. It was defined and first studied by Matveev in 1990. The complexity is generally difficult to compute, and various upper and lower bounds have been derived during the last decades using fundamental group, homology or hyperbolic volume. Effective bounds have only been found in joint work with Jaco, Rubinstein and, more recently, Spreer. Our bounds not only allowed us to determine the first infinite classes of minimal triangulations of closed 3-manifolds, but they also lead to a structure theory of minimal triangulations of 3-manifolds.

**Title** Knot homologies

15:10 **Fri 4 May,** 2018 :: Horace Lamb 1022 :: Dr Anthony Licata :: Australian National University

**Abstract**: The last twenty years have seen a lot of interaction between low-dimensional topology and representation theory. One facet of this interaction concerns “knot homologies,” which are homological invariants of knots; the most famous of these, Khovanov homology, comes from the higher representation theory of sl_2. The goal of this talk will be to give a gentle introduction to this subject to non-experts by telling you a bit about Khovanov homology.

**Title** Stability Through a Geometric Lens

15:10 Fri 18 May, 2018 :: Horace Lamb 1022 :: Dr Robby Marangell :: University of Sydney

**Abstract**: Focussing on the example of the Fisher/KPP equation, I will show how geometric information can be used to establish (in)stability results in some partial differential equations (PDEs). Viewing standing and travelling waves as fixed points of a flow in an infinite dimensional system, leads to a reduction of the linearised stability problem to a boundary value problem in a linear non-autonomous ordinary differential equation (ODE). Next, by exploiting the linearity of the system, one can use geometric ideas to reveal additional structure underlying the determination of stability. I will show how the Riccati equation can be used to produce a reasonably computable detector of eigenvalues and how such a detector is related to another, well-known eigenvalue detector, the Evans function. If there is time, I will try to expand on how to generalise these ideas to systems of PDEs.

**Title **Modelling phagocytosis

15:10 Fri 25 May, 2018 :: Horace Lamb 1022 :: Dr Ngamta (Natalie) Thamwattana :: University of Wollongong

**Abstract:** Phagocytosis refers to a process in which one cell type fully encloses and consumes unwanted cells, debris or particulate matter. It plays an important role in immune systems through the destruction of pathogens and the inhibiting of cancerous cells. In this study, we combine models on cell-cell adhesion and on predator-prey modelling to generate a new model for phagocytosis that is capable of relating the interaction between cells in both space and time. Numerical results are presented, demonstrating the behaviours of cells during the process of phagocytosis.

**Title** Quantifying language change

15:10 Fri 1 Jun, 2018 :: Horace Lamb 1022 :: A/Prof Eduardo Altmann :: University of Sydney

**Abstract:** Mathematical methods to study natural language are increasingly important because of the ubiquity of textual data in the Internet. In this talk I will discuss mathematical models and statistical methods to quantify the variability of language, with focus on two problems: (i) How the vocabulary of languages changed over the last centuries? (ii) How the language of scientific disciplines relate to each other and evolved in the last decades? One of the main challenges of these analyses stem from universal properties of word frequencies, which show high temporal variability and are fat-tailed distributed. The later feature dramatically affects the statistical properties of entropy-based estimators, which motivates us to compare vocabularies using a generalized Jenson-Shannon divergence (obtained from entropies of order alpha).

]]>

More information: http://www.anziam.org.au/The+EO+Tuck+Medal

Photo: Mark McGuinness

]]>* When:* Friday, 16 /2/18

Ingkarni Wardli 5.57 3:10-4pm

Speaker: Dr Guillermo Gomez, Centre for Cancer Biology, Uni SA

Title: Active mechanical relaxation of adherens junctions in the vicinity of apoptotic cells facilitates cell extrusion by promoting epithelial topological transitions.

Abstract: Cell extrusion allows the elimination of minorities of cells from the epithelium. Although this process entails active events that occur within the extruding cell, much less is known on the role of its neighbouring cells. Using apoptotic cell extrusion as a model, we found that as cell extrusion completes, the junctions on neighbouring cells, elongate and form multicellular junctions or “rosettes”. Computational modelling and experimentation show that active mechanical softening of junctions plays a key role during these junctional rearrangements that have all the characteristic of a topological transition. Junctional mechanotransduction is essential for epithelial topological transitions during extrusion as tension-sensitive junctional accumulation of cofilin-1 activates SFKs that are required for active junctional softening. We therefore propose that softening of the tissue plays a key role facilitating the topological transitions that favours extrusion.

**Seminar:**

* When:* Thursday, 22/2/18

Ingkarni Wardli 5.57 3:10-4pm

Speaker: Ms Lisa Reischmann, University of Augsberg

Title: A multiscale approximation of a Cahn-Larche system with phase separation on the microscale

Abstract:

We consider the process of phase separation of a binary system under the influence of mechanical deformation and we derive a mathematical multiscale model, which describes the evolving microstructure taking into account the elastic properties of the involved materials.

Motivated by phase-separation processes observed in lipid monolayers in film-balance experiments, the starting point of the model is the Cahn-Hilliard equation coupled with the equations of linear elasticity, the so-called Cahn-Larche system.

Owing to the fact that the mechanical deformation takes place on a macrosopic scale whereas the phase separation happens on a microscopic level, a multiscale approach is imperative.

We assume the pattern of the evolving microstructure to have an intrinsic length scale associated with it, which, after nondimensionalisation, leads to a scaled model involving a small parameter epsilon>0, which is suitable for periodic-homogenisation techniques.

For the full nonlinear problem the so-called homogenised problem is then obtained by letting epsilon tend to zero using the method of asymptotic expansion.

Furthermore, we present a linearised Cahn-Larche system and use the method of two-scale convergence to obtain the associated limit problem, which turns out to have the same structure as in the nonlinear case, in a mathematically rigorous way. Properties of the limit model will be discussed.

**Seminar:**

**When:** Friday 20 April

3:10pm, IW 5.58

**Speaker**: Dr Joe Giddings, University of Adelaide

**Title:** One Dimensional Models for Slugging in Channel Flow

**Abstract**:Gas-liquid pipe flows are extremely important in many industries, one of which is the oil/gas industry which is where the motivation for this work comes from. In subsea natural gas pipelines the gas is compressed before being pumped through the pipe at high pressure. As it flows through the pipe some of the gas condenses into a low density mixture of hydrocarbon liquids. When gas and liquid flow together there are several possible flow regimes that can occur depending on the velocity of the gas and liquid, one of which is slug flow where the liquid forms a series of plugs (slugs) separated by relatively large gas pockets. The occurrence of slug flow is a major concern in the oil and gas industry due to the difficulty of dealing with large changes in the oil and gas flow rates at the exit of the pipe.

We develop a hydraulic theory to describe the occurrence and structure of slugging in two-layer-gas-liquid flow generated by prescribed, constant, upstream flow rates in each layer. We will investigate how small-amplitude disturbances affect the flow in order to study the stability of spatially uniform solutions. We will then consider the existence of periodic travelling wave solutions numerically in order to investigate the influencing factors that may lead to a transition from stratified flow to slug flow. We then solve the governing equations numerically as an initial value problem in order to improve our understanding of how and why slugs form and are able to compare our solutions to those predicted by the periodic travelling wave theory.

Finally, we investigate the effects of non-horizontal channels with small, slowly varying inclination on the development of slug flow by re-writing our equations in terms of a curvilinear co-ordinate system.From this we find that the height of the layer of liquid increases with the angle of the channel and our solutions are significantly different to those in the horizontal case.

*Seminar:*

**Title:** Obstructions to smooth group actions on 4-manifolds from families Seiberg-Witten theory

**When: **Friday, 25 May 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

**Speaker:** David Baraglia (University of Adelaide)

**Abstract: **Let X be a smooth, compact, oriented 4-manifold and consider the following problem. Let G be a group which acts on the second cohomology of X preserving the intersection form. Can this action of G on H^2(X) be lifted to an action of G on X by diffeomorphisms? We study a parametrised version of Seiberg-Witten theory for smooth families of 4-manifolds and obtain obstructions to the existence of such lifts. For example, we construct compact simply-connected 4-manifolds X and involutions on H^2(X) that can be realised by a continuous involution on X, or by a diffeomorphism, but not by an involutive diffeomorphism for any smooth structure on X.

**Seminar:**

**Title:** The mass of Riemannian manifolds

**When: **Friday, 1 June 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

**Speaker:** Matthias Ludewig (MPIM Bonn)

**Abstract: **We will define the mass of differential operators L on compact Riemannian manifolds. In odd dimensions, if L is a conformally covariant differential operator, then its mass is also conformally covariant, while in even dimensions, one has a more complicated transformation rule. In the special case that L is the Yamabe operator, its mass is related to the ADM mass of an associated asymptotically flat spacetime. In particular, one expects positive mass theorems in various settings. Here we highlight some recent results.

—

]]>When: Monday 20 November, 11:10am

Where: Engineering North N132

Abstract:

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Urban systems are complex in nature and comprise of a large number of individuals that act according to utility, a measure of net benefit pertaining to preferences. The actions of individuals give rise to an emergent behaviour, creating the so-called urban structure that we observe. In this talk, I develop a stochastic model of urban structure to formally account for uncertainty arising from the complex behaviour. We further use this stochastic model to infer the components of a utility function from observed urban structure. This is a more powerful modelling framework in comparison to the ubiquitous discrete choice models that are of limited use for complex systems, in which the overall preferences of individuals are difficult to ascertain. We model urban structure as a realization of a Boltzmann distribution that is the invariant distribution of a related stochastic differential equation (SDE) that describes the dynamics of the urban system. Our specification of Boltzmann distribution assigns higher probability to stable configurations, in the sense that consumer surplus (demand) is balanced with running costs (supply), as characterized by a potential function. We specify a Bayesian hierarchical model to infer the components of a utility function from observed structure. Our model is doubly-intractable and poses significant computational challenges that we overcome using recent advances in Markov chain Monte Carlo (MCMC) methods. We demonstrate our methodology with case studies on the London retail system and airports in England.

]]>Where: Ingkarni Wardli B17

Presented by Dr Sophie Hautphenne, University of Melbourne

Abstract:

Markovian binary trees form a general and tractable class of continuous-time branching processes, which makes them well-suited for real-world applications. Thanks to their appealing probabilistic and computational features, these processes have proven to be an excellent modelling tool for applications in population biology. Typical performance measures of these models include the extinction probability of a population, the distribution of the population size at a given time, the total progeny size until extinction, and the asymptotic population composition. Besides giving an overview of the main performance measures and the techniques involved to compute them, we discuss recently developed statistical methods to estimate the model parameters, depending on the accuracy of the available data. We illustrate our results in human demography and in conservation biology

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