On the existence of a Barenblatt solution for a nonlocal doubly nonlinear evolution equation

• Speaker: Daniel Hauer (Sydney)

Seminar room 1.33

Grayson’s theorem with free boundary

• Speaker: Mat Langford (ANU)

Abstract: We introduce a reflected chord-arc profile for curves with orthogonal boundary condition and obtain a chord-arc estimate for embedded free boundary curve shortening flows in convex planar domains. As a consequence, we are able to prove that any such flow either converges in infinite time to a "critical chord'', or contracts in finite time to a "round half-point'' on the boundary. This work is joint with Jonathan Zhu.

 Seminar room 1.33

Spectral Optimisation of the Robin Laplacian on Quadrilaterals

• Speaker: James Larsen-Scott (Monash)

Given a collection of domains of fixed volume, which domain maximises or minimises the first eigenvalue of the Laplacian?
A famous question of this type is Lord Rayleigh's conjecture that the drum shape that minimises the lowest frequency of vibration of the drum is the disk.
This conjecture would later be resolved by the proof of the Faber-Krahn inequality, the now well-known result that for Dirichlet boundary conditions the ball is the minimising domain.
Similarly, when one is restricted to triangular or quadrilateral domains, the equilateral triangle and the square are the minimisers, respectively.
We discuss the current state of knowledge with regards to these shape optimisation problems when one considers Robin boundary conditions instead.
Furthermore, we provide a new result that for quadrilateral domains, the square is a local minimiser when the Robin boundary parameter is negative, alongside some asymptotic results.

Seminar room 1.33

Symplectic Dirac Operators

• Speaker: Stepan Hudecek (UQ)

Abstract: Given a Riemannian manifold one can ask if there exists a 'square root' of the Laplacian, i.e., an order one operator that squares to the Laplacian. This leads to the construction of the (orthogonal) Dirac operator.  In the talk we will show how we can mimic this construction on symplectic manifolds to produce an analogue of the Dirac operator - the symplectic Dirac operator. On a homogeneous space, square of the Dirac operator differs from the Casimir operator by a constant (multiple of scalar curvature) we will present how the situation changes in the symplectic setting.

Seminar room 1.33

• Speaker: Haotian Wu (Sydney)

Seminar room 1.33

Higher order linear curvature flow

• Speaker: James McCoy (New Castle)

Curvature diffusion of planar curves with generalised Neumann boundary conditions inside cones

• Speaker: Mashniah Gazwani (Newcastle)

Abstract: The main goal of this paper is to study families of smooth immersed regular plane curves   $\alpha : [-1,1]\times [0,T )\to \mathbb{R}^{2}$ with  Neumann  boundary conditions on the cone evolving by the curve diffusion flow.  The evolving curves meet the boundary on each side perpendicularly and satisfy a no-curvature flux condition. On the normalised oscillation of curvature, we provide estimates and monotonicity, facilitating characterisation of long time behaviour of solutions. We show that such evolving curves exist for all time and converge exponentially in the $C^{\infty }$ topology to circular arcs.

Seminar room 1.37

Independence of Singularity Type for Numerically Effective Kähler-Ricci Flows

• Speaker: Hosea Wondo (Sydney)

Abstract: The study of long time singular solutions to the Kähler-Ricci flow has generated much interest due to its relevance to the final step of Song and Tian's analytic minimal model program.  We show that the singularity type of such solutions to the Kähler-Ricci flow  does not depend on the initial metric. More precisely, if X is a numerically effective manifold admitting a type III solution to the Kähler-Ricci flow, then any other solution starting from a different initial metric will also be Type III. This generalises a previous result by Y. Zhang for the semi-ample case.  This talk is based on joint work with Zhou Zhang.

 Seminar room 1.37

• Speaker: Tim Buttsworth (UQ)

Spectral gap for Schrödinger operators on metric trees

• Speaker: Julie Clutterbuck (Monash)

Tutorial room 1.58

Ricci solitons with solvable symmetry

• Speaker: Adam Thompson (UQ)

Abstract: There are many examples of Ricci solitons that are constructed using the following ansatz: the soliton admits a cohomogeneity one group action by a compact Lie group. On the other hand, there are very few examples of cohomogeneity one Ricci solitons where the group acting is non-compact. We will discuss our construction of new examples of complete cohomogeneity one gradient Ricci solitons where the group action is by a non-compact solvable Lie group. 

Seminar room 1.37

• Speaker: Artem Pulemotov (UQ)

Seminar room 1.37

Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds (Joint work with Alix Deruelle and Felix Schulze)

• Speaker: Miles Simon (Magdeburg)

Abstract:

• Speaker: William Trad (Sydney)

Tutorial room 1.58 

A family of three-term quermassintegrals inequalities and their relation to the isoperimetric inequality

• Speaker: Kwok-Kun Kwong (ANU)

Abstract: Quermassintegrals play a fundamental role in convex geometry, extending classical notions such as volume and surface areas. It has been relatively recent that mathematicians have found a way to prove inequalities involving quermassintegrals using inverse curvature flows (ICF). However, most of these results only compare two geometric quantities. In this talk, I will present a family of sharp geometric inequalities involving a weighted curvature integral and two quermassintegrals in space forms. A main ingredient is identifying monotone quantities involving two quantities along ICF. We establish a novel connection between a special case of these inequalities and the $L^2$ distance between a convex body and its Steiner ball, resulting in a stability result for both this inequality and the isoperimetric inequality. This research is a joint work with Yong Wei. 

Tutorial room 1.58