All talks will be in HN1.33 (ground floor of the Hanna Neumann building).

Welcome reception will be at the BBQ area outside of the Hanna Neumann building.

There are a number of cafés and restaurants on campus. There are more options towards the city, and a café (Pollen) at the nearby botanic gardens (foot of black mountain).

Early mornings and evenings are a good time to spot wildlife around campus or on black mountain.

Schedule

Monday 17 February

1700-1800: Welcome Reception

Tuesday 18 February

1000-1100: Bourni (minicourse)

Title: Solving the Plateau problem.

Abstract:

In this mini-course, we will explore the classical Plateau problem, which seeks to find a surface in space that minimizes area while maintaining a fixed boundary. First proposed by Lagrange in 1760 and solved independently by Douglas and Rado in 1930, this problem has a rich history. We will present a solution to the Plateau problem in any dimension (codimension 1) by employing the theory of sets of finite perimeter—sets that possess stronger compactness properties than traditional smooth sets. Through a minimization technique, we will demonstrate how to derive a solution to the problem. We will also discuss the smoothness properties of the solution and provide estimates for the size of its singular set. 

1100-1130: Coffee

1130-1230: Flook

Abstract:

1230-1430: Lunch

1430-1530: Wei

Title: A Heintze-Karcher-type inequality in a hyperbolic half-space

Abstract: We prove a Heintze-Karcher-type inequality for compact embedded capillary hypersurfaces in a hyperbolic half-space. The proof utilizes a geodesic normal map flow with respect to a Finsler metric of Randers-type, induced by a special Zermelo navigation data. As an application, we derive an Alexandrov-type theorem for compact embedded capillary hypersurfaces of constant higher-order mean curvature in a hyperbolic half-space. This is joint work with Yingxiang Hu, Chao Xia, and Tailong Zhou.

1530-1600: Break

1600-1700: Stanfield

Title: Homogeneous generalized Ricci flows:

Abstract: 

The generalized Ricci flow , or GRF (first studied by Callan--Friedan--Martine--Perry, and later by Streets--Garcia-Fernandez) is an analogue of the Ricci flow in the setting of Hitchin's generalized geometry. It is a certain super-solution of Ricci flow coupled with the heat flow on 3-forms and subsumes several geometric flows, including the Ricci flow, the generalized Kähler-Ricci flow, and the pluriclosed flow.

In this talk, we discuss the behavior of the GRF on (discrete quotients of) Lie groups. We establish the global existence of the flow on Lie groups diffeomorphic to Euclidean space—a result that is new even for the pluriclosed flow. We also define a notion of generalized Ricci soliton that allows for non-trivial expanding examples. On nilmanifolds, we show that these solitons arise as rescaled limits of the GRF under certain circumstances. Our main tool is an adaptation of Lauret's "bracket flow" to the GRF, and a new formula for the generalized Ricci curvature in terms of the moment map for the action of a real-reductive Lie group on the space of generalized Lie brackets. This is based on joint work with Elia Fusi (Università di Parma) and Ramiro Lafuente (The University of Queensland).

Wednesday 19 February

1000-1100: Bourni (minicourse)

Title: Solving the Plateau problem.

Abstract:

In this mini-course, we will explore the classical Plateau problem, which seeks to find a surface in space that minimizes area while maintaining a fixed boundary. First proposed by Lagrange in 1760 and solved independently by Douglas and Rado in 1930, this problem has a rich history. We will present a solution to the Plateau problem in any dimension (codimension 1) by employing the theory of sets of finite perimeter—sets that possess stronger compactness properties than traditional smooth sets. Through a minimization technique, we will demonstrate how to derive a solution to the problem. We will also discuss the smoothness properties of the solution and provide estimates for the size of its singular set. 

1100-1130: Coffee

1130-1230: Buchanan

Abstract:

The Gauss–Bonnet theorem is a beautiful result that connects the topology and geometry of a manifold. It turns out that this result admits a proof through the analysis of differential forms under heat flow. More broadly, heat equation methods provide a powerful framework for extracting both geometric and topological information from the spectrum of the Laplacian on differential forms.

1230-1430: Lunch

1430-1530: Bush walk

1530-1600

1600-1700

1700-1800

1800 onwards: Dinner @ Civic Pub

Thursday 20 February

1000-1100: Bourni (minicourse)

Title: Solving the Plateau problem.

Abstract:

In this mini-course, we will explore the classical Plateau problem, which seeks to find a surface in space that minimizes area while maintaining a fixed boundary. First proposed by Lagrange in 1760 and solved independently by Douglas and Rado in 1930, this problem has a rich history. We will present a solution to the Plateau problem in any dimension (codimension 1) by employing the theory of sets of finite perimeter—sets that possess stronger compactness properties than traditional smooth sets. Through a minimization technique, we will demonstrate how to derive a solution to the problem. We will also discuss the smoothness properties of the solution and provide estimates for the size of its singular set. 

1100-1130: Coffee

1130-1230: Rajpal

Title: Affine Boosted Gauss Curvature Flow for sub-affine exponents

Abstract:

We study an affine boosted version of the $ \alpha $-Gauss curvature flow for sub-affine powers $ \alpha < \frac{1}{n+2} $. A new ingredient is the convexity of the associated Firey and Gaussian entropy in the affine orbit due to which we can do affine corrections. It can be proven that the new flow converges convex hypersurfaces to a point. The bounded monotonic entropies for affine boosted flow further suggest convergence to new self-similar solutions for sub-affine exponents. This is joint work with Ben Andrews.

1230-1330: Lunch

1330-1430: Nguyen

Title: Mean Curvature Flow in the Sphere

Abstract:

In this talk, I will discuss some recent results analysing singularities of the mean curvature flow and minimal submanifolds in the sphere in the high codimension case. We will see how the positive curvature of the background space allows a wider range of admissible limits and introduce some new curvature conditions for both the flow and the stationary case.

1430-1530: Kwong

Title: Alexandrov-Fenchel type inequalities with convex weight in space forms

Abstract:

In this talk, I will present new sharp weighted Alexandrov-Fenchel and Minkowski inequalities for smooth, closed hypersurfaces under various convexity conditions in space forms. These results extend and unify several classical unweighted inequalities, including the Alexandrov-Fenchel inequalities, the isoperimetric and Minkowski inequalities, and the Gauss-Bonnet formula, as well as more recent weighted generalisations such as weighted Alexandrov-Fenchel-type and Minkowski-type inequalities. For any convex, non-decreasing positive function, our approach provides a corresponding inequality, generating a broad family of geometric inequalities with significant flexibility. I will also discuss applications, including a proof of the conjectured Weinstock inequality for star-shaped mean-convex hypersurfaces in Euclidean space and a sharp upper bound for the first eigenvalue in the eigenvalue problem for an inhomogeneous membrane. This is joint work with Yong Wei.

1530-1600: Break

1600-1700: Kotschwar (Colloquium)

Title: Uniqueness problems for geometric heat flows

Abstract: 

One of the most basic questions one can ask of any evolutionary differential equation is that of uniqueness: whether two solutions which agree at some time must agree at all times. This question can be quite subtle, however, especially when the domain under consideration is noncompact. For the heat equation on Euclidean space, the answer is conditional. A famous example of Tychonoff demonstrates that there are smooth solutions to the heat equation on Euclidean space which vanish identically at t=0 but are nontrivial at all other times. However, uniqueness can be assured assuming, for example, that the solution is either bounded below or does not grow too quickly at infinity in either direction.

The question of uniqueness has added significance in the study of geometric heat flows such as the Ricci flow and mean curvature flow, where (in various guises) it governs the propagation of symmetry and other geometric structure along the flow. On noncompact spaces, however, solutions to these (nonlinear) equations may be influenced by their geometry at infinity in ways that the usual analogy with the linear heat equation fails to predict, and many basic questions regarding uniqueness and preservation of structure for such solutions remain open. In this talk, I will give an overview of some uniqueness results for these equations, highlighting some of the challenges in the geometric setting, and will discuss what is currently known and unknown about the uniqueness and propagation of structure along general complete solutions of potentially unbounded curvature.

Friday 21 February

1000-1100: Espinar

Title: Hypersurfaces in \mathbb{H}^{n+1} and Conformal metrics on \mathbb{S}^n

Abstract: 

In the first part, we will present a a correspondence between hypersurfaces in hyperbolic space and conformal metrics on the sphere, enriching the understanding of conformal invariants on the sphere and geometric PDEs in the hyperbolic space. 

Next, we will use such correspondence to obtain rigidity results in both sides, for example, the resolution of the Min-Oo conjecture for fully nonlinear conformally invariant equations, addressing rigidity results for compact, connected, locally conformally flat Riemannian manifolds with boundary

1100-1130: Coffee

1130-1230: Pediconi

Title: Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces

Abstract: 

Ricci flow solutions that exist for all negative times have special significance and are known as ancient solutions. In this talk, we describe the general structure of collapsed ancient solutions to the homogeneous Ricci flow on compact manifolds. Furthermore, we present a general existence theorem and show that, under certain algebraic assumptions, these solutions exhibit more symmetries than initially expected. This is joint work with S. Sbiti and A. Krishnan.

1230-1430: Lunch

Organising committee