# Geometric Analysis and PDE Seminar

Prior to 2013, PDE seminars were part of the broader IMIA seminar program, which is available here http://imia.uow.edu.au/seminars/index.html

In 2013 our group launched its own seminar series within IMIA, joining with our applied mathematics colleagues working on modelling using partial differential equations.

24 November 2014

**Prof Raseelo Joel Moitsheki (University of the Witwatersrand)**

Title: Exact and approximate analytical solutions for reaction-diffusion equations

Abstract: We determine exact and approximate analytical solutions for reaction-diffusion equations of the form u_t = (k(x; u)u_x)_x + q(u). Here we employ the classical Lie point symmetry methods, the non-classical symmetry techniques and the differential transform methods.

20 November 2014

**Prof Rodney Nillsen (University of Wollongong)**

Title: Generalised differences and multiplier operators in classical Fourier analysis

Abstract: Any zeros in the multiplier of an operator impose a condition on a function for it to be in the range of the operator. But if each function in a certain family of functions satisfies such a condition, when is this family the same as the range? We consider a case where the family of functions is formed by taking finite sums of "generalised differences", whose precise form derives from the operator. Although more general results are possible, the main case considered here is when the operator acts on a second order Sobolev space of the circle group $[0,2pi)$ and takes the form D^2+a^2I, where a is an integer. The techniques involve estimating integrals in higher dimensions over products of sets in a partition associated with the zeros of the multiplier and, perhaps, illustrates in a different context the "curse of dimension" referred to by Professor Sloan in his Colloquium last week.

30 October 2014

**Dr. Matthew Cooper (University of New England)**

Title: Biharmonic maps and their heat flow

Abstract: Biharmonic maps are natural generalizations of harmonic maps. The fact that the Euler equations are fourth-order (as opposed to second order for harmonic maps) presents a number of analytic challenges. In this talk, I will discuss $O(d)$-equivariant biharmonic maps and their associated heat flow. Among other results, I will present a blowup result for the biharmonic map heat flow from $B^4$ into $S^4$.

14 October 2014

**Prof Yihong Du (University of New England)**

Title: Reaction-diffusion equations and spreading of species

**Dr. Stephen McCormick (University of New England)**

Title: The Phase Space for the Einstein-Yang-Mills Equations and the First Law of Black Hole Thermodynamics

**Scott Parkins (University of Wollongong)**

Title: The Generalised Polyharmonic Curve Flow of Closed Planar Curves.

Abstract: We consider a family of higher order curvature flows on closed planar curves (which includes the curve shortening flow and curve diffusion flow). We look at a natural energy called the ``normalised oscillation of curvature'' that measures how far a closed curve is from being an embedded circle (in an averaged $L^2$ sense not unlike the Willmore energy for closed surfaces). We then show that under any of these flows, closed curves suitably close to a circle (i.e. with a small normalised oscillation of curvature, as well as satisfying a suitable isoperimetric condition) exist for all time and converge exponentially fast to a round embedded circle.

**Dr. Carla Cederbaum (University of Tuebingen)**

Title: On the definition of mass and center of mass of isolated systems in Newtonian gravity and general relativity

**Dr. Glen Wheeler (University of Wollongong)**

Title: Unstable Willmore Surfaces

Abstract: In this talk I will describe some recent work with Anna Dall’Acqua (Ulm) and Klaus Deckelnick (OvGU Magdeburg) that established the existence of compact Willmore surfaces with boundary that are unstable. The natural approach of using Palais-Smale and Mountain Pass doesn’t (really) work. I’ll explain why this is the case. We overcame this problem by using a completely different (and new) approach. I will finish by describing some open questions arising from the work.

**Dr. Yann Bernard (ETH Zurich)**

Title: Complete Minimal Surfaces in Asymptotically Flat Manifolds via the Willmore Energy

Abstract: We show how techniques and results which were recently developed in the context of Willmore surfaces may be applied to describe the asymptotic behavior of complete minimal surfaces in manifolds that are asymptotically Schwartszchild. The method is new and, unlike its predecessors, is particularly suited in higher codimension. This is part of an ongoing project with Tristan Rivière from the ETH-Zurich.

11 June 2014

**Assoc Prof Adam Rennie (University of Wollongong)**

Title: Zeta functions on pointy things

Abstract: Once upon a time I looked at a simple example of a Dirac operator on a cone. I was trying to build a zeta function with nasty behaviour, but the results were mixed. Much later, but still long long ago in a university far far away, I considered the question of nasty zeta functions on cones in more detail, but never satisfactorily finished the project (joint with Steve Rosenberg). So for some explicit examples and open questions, come along and enjoy a walk down memory lane.

28 May 2014

**Dr. Daniel Hauer (University of Sydney)**

Title: Uniform convergence of solutions to elliptic equations on domains with shrinking holes

Abstract: In this talk I want to present new results which I established during my postdoc year at the University of Sydney under the supervision of EN Dancer and D Daners. I consider solutions of the Poisson equation on a family of domains with holes shrinking to a point. This kind of domain convergence is very singular. Assuming Robin or Neumann boundary conditions on the boundary of the holes, I show that the solution converges uniformly to the solution of the Poisson equation on the domain without the holes. This is in contrast to Dirichlet boundary conditions where there is no uniform convergence. The results substantially improve earlier results on $L^p$-convergence.

14 May 2014

**Dr. Lashi Bandara (The Australian National University)**

Title: Rough metrics and quadratic estimates

Abstract: : A consequence of the Kato square root problem on manifolds is a certain "stability" it provides in terms of perturbation at the level of the metric. This is best examined by considering certain quadratic estimates associated with the problem. Here, I will talk about a class of Riemann-like metrics on manifolds, which are permitted to be both of low regularity and degenerate, under which these quadratic estimates remain invariant. This metric perturbation technique also allows us to capture Lipschitz transformations of spaces in terms of pullback metrics. Furthermore, we use the perturbation mechanism to show that the Kato square root problem can be solved for metrics with zero injectivity radius bounds.

30 May 2014

**Prof. Miles Simon (Otto-von-Guericke Universitaet)**

Title: Local estimates for Ricci-flow in two dimensions

Abstract: In all dimensions G. Perelman proved the following. If an open ball contained in a manifold is 'almost Euclidean', then one can prove estimates on how compactly contained sub-regions of this ball evolve under Ricci flow. We generalise this result in two dimensions to regions which are not necessarily almost Euclidean. The estimates we obtain depend on the infimum of the curvature within the ball at time zero and the volume of the ball at time zero.

27 March 2014

**Dr Peter Kim (University of Sydney) **

Title: Modelling dynamics of a localised immune response against a tumour.

Abstract: The next generation approach to cancer therapy envisions using stimulating a person's immune system to destroy incipient or residual tumours well below the size of clinical detection. This approach requires us to understand the dynamics of a localised anti-cancer immune response in the small environment surrounding an undetectable tumour (approximately 1 cubic millimetre).

22 October 2013**Dr. Yann Bernard (Freiburg University)**Title: Energy quantization for the Willmore functional

Abstract: We prove a bubble-neck decomposition and an energy quantization result for sequences of Willmore surfaces immersed into R^m (m>2) with uniformly bounded energy and non-degenerating conformal structure. We deduce the strong compactness (modulo the action of the Moebius group) of closed Willmore surfaces of a given genus below some energy threshold. This is joint-work with Tristan Rivière (ETH Zürich).

17 October 2013**Dr. Yann Bernard (Freiburg University)**Title: Conservation laws for the Willmore functional and associated results

Abstract: Using Noether's theorem, we will show that the Willmore equation can be recast in a system of equations in divergence form. Particularly suited for "critical" analysis, this system will in particular be used to understand the behavior of a Willmore immersion near a point-singularity. We will develop local asymptotic expansions for the immersion, its first and second derivatives in terms of residues computed as circulation integrals around the point singularities. We will deduce explicit "point removability" conditions ensuring the smoothness of the immersion.

15 October 2013**Prof. Song-Ping Zhu (University of Wollongong)**Title: American-style Parisian options and their pricing formulae

Abstract: In this talk, we shall first discuss the pricing of various American-style Parisian options under the Black-Scholes model. After pointing out the fundamental difference between American-style “in” and “out” Parisian options, we present an analytic solution for American-style up-and-in Parisian options, which does not explicitly involve a moving boundary as far as the “mother option” is concerned. The solution is worked out after combining the “moving window” technique developed in Zhu and Chen (2013) and the method of images, in order to simplify the solution procedure. Our final solution is written in the form of three double integrals, which can be easily computed numerically.

8 October 2013**Prof. Alan McIntosh (The Australian National University)**Title: Finite speed of propagation for first order systems, and Huygens' principle for hyperbolic equations

Abstract: I will present a new approach to proving finite propagation speed for first order systems, and show how this can be applied to obtain a weak Huygens' principle for certain second order hyperbolic equations. This is joint work with Andrew Morris.

25 September 2013**Mrs. Fatemah Mofarreh (University of Wollongong)**Title: Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions

Abstract: Inspired by earlier results on the quasilinear mean curvature flow, and recent investigations of fully nonlinear curvature flow of closed hypersurfaces which are not convex, we consider contraction of axially symmetric hypersurfaces by convex, degree-one homogeneous fully nonlinear functions of curvature. With a natural class of Neumann boundary conditions we show that evolving hypersurfaces exist for a finite maximal time. The maximal time is characterised by a curvature singularity at either boundary. Some results continue to hold in the cases of mixed Neumann-Dirichlet boundary conditions and more general curvature-dependent speeds.

25 September 2013**Mr. Scott Parkins (University of Wollongong)**Title: Gap Lemma for the Triharmonic Equation on Surfaces with small energy

Abstract: We use energy methods to establish a gap lemma for solutions to the Triharmonic flow on surfaces with small energy. We then talk about extension to high general Poly-Laplacian flows.

3 September 2013**Prof. Jerome Vetois (Nice Sophia Antipolis University)**Title: Instability of the Yamabe Equation under linear perturbations of the potential

Abstract: In this talk, I will present a joint work with P. Esposito and A. Pistoia on the instability of the set of solutions of the Yamabe Equation. We prove that a linear perturbation of the potential can generate the existence of blowing-up families of solutions. I will also present a joint work with F. Robert on the existence of blowing-up families of solutions which admit a non-isolated blow-up point. These results complete previous stability results and allow to have a sharp picture of the stability/instability of scalar curvature type equations on compact manifolds.

8 August 2013**Dr. Glen Wheeler (University of Wollongong)**Title: Gap phenomena for a class of fourth-order geometric differential operators on general surfaces with boundary

Abstract: Gap phenomena are a kind of geometric rigidity which prevent an associated tensor from being small in an appropriate norm: it is either larger than a universal constant, or identically zero. I will survey the idea behind classical gap theorems, providing a context for my own results (some joint with James McCoy). I aim to explain in detail the main results of my most recent preprint, which are a pair of gap theorems where the associated tensor is the second fundamental form or the trace-free second fundamental form. Key improvements over previous work are the generalisations to high codimension, surfaces with arbitrary topology and boundaries, and more general operators. The class of operators considered include the motivating examples of the Willmore operator, the Surface Diffusion operator (normal Laplacian) and the biharmonic or Chen’s operator (bilaplacian). Each of the gap theorems are new even for these model cases.

12 July 2013**Prof. Xiao Zhang (Chinese Academy of Sciences, Beijing China)**

Title: The positive energy theorem for spacetimes with the positive cosmological constant

Abstract: The positive energy theorem plays a fundamental role in general relativity. It was firstly proved by Schoen-Yau using minimal surfaces and PDEs, and then by Witten using spinors in the case of zero cosmological constant. In this talk, we shall discuss the theorem when the cosmological constant is positive. This case seems to gain much more importance as the recent cosmological observations indicate that our universe has a positive cosmological constant.

21 June 2013**Assoc Prof. Adam Rennie (University of Wollongong)**

Title: The topology of manifolds via elliptic PDEs

Abstract: : I will give a not-too-formal view of K-homology and its origins in index theory of elliptic operators.

12 June 2013**Dr. Sanjiban Santra (University of Sydney)**

Title: On the perturbed $Q$-curvature problem on $\mathbb{S}^N$.

Abstract: Let $g_0$ denote the standard metric on $\mathbb{S}^4$ and

$P_{g_0}=\De^2_{g_{0}}- 2 \De_{g_{0}}$ denote the corresponding Panietz operator. In this talk, we discuss the following fourth order elliptic problem with exponential nonlinearity :

$$ P_{g_{0}} u + 6 = 2Q(x)e^{4u} \mbox{ on } \mathbb{S}^4. $$ Here $Q$ is a prescribed smooth function on $\mathbb{S}^4$ which is assumed to be a perturbation of a constant. We prove existence results to the above problem under assumptions only on the ``shape'' of $Q$ near its critical points. These are more general than the non-degeneracy conditions on $Q$ assumed so far by Malchiodi-Struwe and Wei-Xu. We also prove uniqueness and exact multiplicity results for this problem. I will also discuss the case when $N\geq 5.$

7 May 2013**Mr. Lashi Bandara (The Australian National University)**

Title: Geometry and the Kato square root problem

Abstract: Since the resolution of the Kato square root problem on Euclidean space in 2002, there have been a number of developments in the understanding of versions this problem on manifolds and Lie groups. There have been a number of positive results for uniformly elliptic divergence form operators, for perturbations of the inhomogeneous Hodge-Dirac operator, and in the setting of Lie groups, uniformly sub-elliptic operators. I will talk about these results, their connection to PDE, and in particular, possible connections to geometric flows.

29 May 2013 **Dr. Marianito Rodrigo (University of Wollongong) **

Title: Continuous dynamical systems, Laplace transforms and second-degree polynomial equations

Abstract: A solution trajectory of a linear system of first-order ODEs can be visualised as a curve in R^n (the "time domain"). Its Laplace transform can also be viewed as a curve in R^n (the "frequency domain"). The curve in the time domain can be quite complicated. How does the corresponding curve in the frequency domain look like? In R^2 the curve always lies on a conic section. In R^3 it always lies on the intersection of two quadric surfaces. More generally, in R^n the curve lies on the intersection of second-degree polynomial equations. For nonlinear systems of ODEs, the solution trajectory can be made to lie close to such an intersection of second-degree polynomial equations provided the frequency variable is large enough. I will also show some numerical simulations. This talk is aimed at a general audience

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22 May 2013**Dr. Jiakun Liu (University of Wollongong) **

Title: : An introduction to optimal mass transport

Abstract: Optimal mass transport is a very active research area nowadays, which deals with the redistribution of materials in the most economical way. There are numerous practical applications of optimal mass transport, such as in optics, engineering and economics. On theoretical side, it is closely related to PDE, differential geometry and functional analysis. In this talk, we will give a general introduction of the mass transport problem and will study the existence, uniqueness and smoothness of the optimal mapping.

17 May 2013**Dr. Glen Wheeler(University of Wollongong) **

Title: : An introduction to Chen's conjecture and submanifolds of finite type

Abstract: : Chen's conjecture is a curiosity which arose in the course of his grand scheme to classify all submanifolds of Euclidean space. While this lofty program is nowhere near completion, work on the conjecture in its original form is almost at an end. In this talk I will take Chen's original path to his conjecture, which begins with jealousy of the Order from Algerbraic Geometry, goes through an unusual application of spectral resolution of the Laplacian from Functional Analysis, to end up with a beautiful nonlinear system of partial differential equations. After arriving at the conjecture, I will quickly overview progress on it in its original form, including recent solutions (joint work with V.-M. Wheeler and Y. Bernard). My talk is designed to be accessible and elementary, and will not require anything difficult, scary or intimidating to understand.

1 May 2013**Dr. Zhou Zhang (University of Sydney) **

Title: : Finite Time Singularities of Kahler-Ricci Flows

Abstract: Kahler-Ricci flow is the parabolic version of the complex Monge-Ampere equation. The understanding of the (possible) finite time singularities is of great importance from both PDE and geometry points of view. In this talk, we start by setting up of the Kahler-Ricci flow in general and then survey the recent results and describe some open problems for future study.

17 April 2013**Dr. James McCoy (University of Wollongong) **

Title: Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions

Abstract: Inspired by earlier results on the mean curvature flow, and recent investigations of fully nonlinear curvature flow of closed hypersurfaces which are not convex, we consider contraction of axially symmetric hypersurfaces by convex, degree-one homogeneous fully nonlinear functions of curvature. With a natural class of Neumann boundary conditions we show that evolving hypersurfaces exist for a finite maximal time. The maximal time is characterised by a curvature singularity at either boundary. Some results continue to hold in the cases of mixed Neumann-Dirichlet boundary conditions and more general curvature-dependent speeds. This is joint work with Fatemah Mofarreh and Graham Williams.

10 April 2013**Dr Bin Zhou (Australian National University)**

Title: The Bernstein theorem for a class of fourth order equations

Abstract: : In this talk, we concern on a class of fourth order equations of Monge-Ampere type. In particular, affine maximal surface equation from affine geometry and Abreu's equation arising from scalar curvature equation on toric Kahler manifolds are included. I will talk about the 2-dimensional Bernstein theorem for these equations, i.e., the entire solution to these equations must be quadratic polynomials. The main ingredients are interior estimates and the proof of strict convexity for the solutions.

3 April 2013**Dr. Jason Sharples (University of New South Wales Canberra) **

Title: Analysis of complex combustion processes: combustion wave stability and dynamic behaviour of bushfires.

Abstract: In this presentation I will discuss two loosely-related aspects of my research: stability of travelling wave solutions of PDE systems representing multi-step combustion reactions, and dynamic fire spread propagation arising in coupled fire-atmosphere models. Knowledge of unstable regimes in combustion reactions is important in a number of industrial applications. In the first part of the presentation I will discuss the oscillatory and chaotic behaviour of combustion waves and how the Hopf locus can be determined using Evans function techniques. In the second part of the presentation I will discuss coupled fire-atmosphere models, which are becoming an increasingly common tool in managing extreme bushfires. In particular, I will show how these models have been able to shed light on an extremely dangerous mode of bushfire propagation.

13 March 2013**Dr. Wenting Chen (University of Wollongong) **

Title: An inverse finite element method for pricing American options

Abstract: The pricing of American options has been widely acknowledged as “a much more intriguing” problem in financial engineering. In this paper, a “convergency-proved” IFE (inverse finite element) approach is introduced to the field of financial engineering to price American options for the first time. Without involving any linearization process at all, the current approach deals with the nonlinearity of the pricing problem through an “inverse” approach. Numerical results show that the IFE approach is quite accurate and efficient, and can be easily extended to multi-asset or stochastic volatility pricing problems. The key contribution of this paper to the literature is that we have managed to provide a comprehensive convergence analysis for the IFE approach, including not only an error estimate of the adopted discrete scheme but also the convergence of the adopted iterative scheme, which ensures that our numerical solution does indeed converge to the exact one of the original nonlinear system. This is joint work with Prof. Song-Ping Zhu.

28 February 2013**Prof. Philip Broadbridge (La Trobe University) **

Title: Shannon entropy as a diagnostic tool for PDEs in conservation form

Abstract: After normalization, an evolving real non-negative function may be viewed as a probability density. From this we may derive the corresponding evolution law for Shannon entropy. Parabolic equations, hyperbolic equations and fourth-order “diffusion” equations evolve information in quite different ways. Entropy and irreversibility can be introduced in a self-consistent manner and at an elementary level by reference to some simple evolution equations such as the linear heat equation. It is easily seen that the 2nd law of thermodynamics is equivalent to loss of Shannon information when temperature obeys a general nonlinear 2nd order diffusion equation. With fourth order diffusion terms, new problems arise. We know from applications such as thin film flow and surface diffusion, that fourth order diffusion terms may generate ripples and they do not satisfy the Second Law. Despite this, we can identify the class of fourth order quasilinear diffusion equations that increase the Shannon entropy.

21February 2013**Dr. Peter Kim (University of Sydney) **

Title: T cell state transitions and change detection

Abstract: Numerous immune cells exhibit transitions from inactive to activated states. We focus on T cells and develop a model of T cell activation, expansion, and contraction. Our study suggests that state transitions enable T cells to respond to changes in antigen levels, rather than simply the presence or absence of antigen. A key component that gives rise to this change detector is naïve T cell activation. The activation step separates the slow dynamics of naïve T cells from the fast dynamics of effector T cells. As a result, the T cell population responds to sudden shifts in antigen levels, even if the antigen were present prior to the change. This feature provides a mechanism for T cells to react to rapidly expanding sources of antigen, such as viruses, while maintaining tolerance to gradually changing sources, such as healthy tissue during growth.

14 February 2013**Dr. Bishnu Lamichane (University of Newcastle) **

Title: Some mixed finite element methods for the biharmonic problem

Abstract: Finite element methods provide a powerful tool in engineering analysis and numerical solutions of boundary value problems. These methods are based on variational principle and therefore have intrinsic mathematical beauty and can be applied to complicated and nonlinear problems. The other advantage of finite element methods is that when applied to differential equations they naturally fit into the concept of so-called weak solutions in a Sobolev space. There are many situations in solving differential equations where classical solution does not exist and one has to rely on the weak solution. In the first part of the talk, we give a brief introduction to the finite element methods including the concept of weak solutions. We mainly focus on the biharmonic equation. In the second part of the talk, we consider a mixed finite element method based on biorthogonal or quasi-biorthogonal systems for the biharmonic problem. We consider two approaches: one of them is based on the primal mixed finite element method due to Ciarlet and Raviart for the biharmonic equation. Using different finite element spaces for the stream function and vorticity, this approach leads to a formulation only based on the stream function. The second approach is based on using the gradient of the solution as an additional unknown. We prove optimal a priori estimates for both approaches.