The next PNGS

University of British Columbia, Vancouver, BC, May 10-11, 2025.

To register for this event, please visit: PIMS website and click Register.  (You will need to create an account first)

Invited speakers:

• Richard Bamler, (UC Berkeley)

• Tristan Collins, (U Toronto)

• Laura Fredrickson (U Oregon)

• Yi Lai (Stanford)

• Tin Yau Tseng (UBC)

• Sergio Zamora (Oregon State)

• Jonathan Zhu (U Washington)

Location:

Talks: Earth Sciences Building (ESB) 2012

Coffee and snacks: Outside ESB 2012

Schedule:

Saturday May 10

• 8:30-9:00: Coffee and snacks

• 9:00-10:00:  Richard Bamler: On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces
  
  

The Mean Curvature Flow describes the evolution of a family of embedded surfaces in Euclidean space that move in the direction of the mean curvature vector. It is the gradient flow of the area functional and a natural analog of the heat equation for an evolving surface. Initially, this flow tends to smooth out geometries over brief time-intervals. However, due to its inherent non-linearity, the Mean Curvature Flow equation frequently leads to the formation of singularities. The analysis of such singularities is a central goal in the field.

A long-standing conjecture addressing this goal has been the Multiplicity One Conjecture. Roughly speaking, the conjecture asserts that singularities along the flow cannot form by an "accumulation of several parallel sheets”. In recent joint work with Bruce Kleiner, we resolved this conjecture for surfaces in R^3. This had several applications. First, combining our work with previous results, we obtain that the problem of evolving embedded 2-spheres via the Mean Curvature Flow equation is well-posed within a natural class of singular solutions. Second, we remove an additional condition in recent work of Chodosh-Choi-Mantoulidis-Schulze to show that the Mean Curvature Flow starting from any generically chosen embedded surface only incurs cylindrical or spherical singularities. Third, our approach offers a new regularity theory for solutions of general Mean Curvature Flows that flow through singularities.

This talk is based on joint work with Bruce Kleiner.

• 10:15-11:15: Laura Fredrickson: The asymptotics of hyperkahler metrics from Gaiotto--Moore--Neitzke's conjecture
  
  Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichm\"uller theory, and the geometric Langlands correspondence. The Hitchin moduli space carries a natural hyperk\"ahler metric, a rich and rigid geometric structure.  An intricate conjectural description of its asymptotic structure appears in the work of Gaiotto--Moore--Neitzke.  I will discuss some recent and less-recent results using tools coming out of geometric analysis which are well-suited for verifying these extremely delicate conjectures. This strategy often stretches the limits of what can currently be done via geometric analysis, and simultaneously leads to new insights into these conjectures.

This is based on works joint with R. Mazzeo, J. Swoboda, H. Weiss, M. Zimet.

• 11:30-12:30: Sergio Zamora: Torus stability and lower curvature bounds

Given a Gromov-Hausdorff convergent sequence of spaces $X_n$ that share a lower (sectional or Ricci) curvature bound, the problem of understanding the topology of the limit space in terms of the topology of the spaces $X_n$ has been studied extensively. I will talk about variants of this problem, specializing to the case when the spaces $X_n$ are tori.

• LUNCH BREAK
  
  

• 3:15-4:15 Tristan Collins: Free-boundary Monge-Ampere equations and geometric applications

I will discuss a general class of free-boundary Monge-Ampere equations, with a particular emphasis on geometric applications, including the existence of complete Calabi-Yau metrics on some quasi-projective varieties, and hemispheres with constant Gauss curvature and specified Gauss map.  This talk is based on joint works with Y. Li, F. Tong, S.-T. Yau and B. Firester.

• 4:30-5:30 Jonathan Zhu: Uniqueness results for sphere-ish minimal surfaces

Minimal surfaces in the round 3-sphere have prompted several influential and attractive uniqueness problems, particularly for low genus. Sweeping progress has been made in extending these results, by analogy, to other settings, such as free boundary minimal surfaces in the Euclidean ball, and mean curvature flow shrinkers. We will propose perspectives by which these other settings may be analysed in a continuous family starting with the round 3-sphere. In this talk, we’ll discuss (half-space) intersection properties, and also connections with capillary minimal surfaces. This is based on joint work (including some in progress) with Keaton Naff.

• CONFERENCE DINNER

Sunday, May 11

• 8:30-9:00: Coffee and snacks

• 9:00-10:00 Yi Lai: Convergence of Ricci flow and long-time existence of Harmonic map heat flow

For an ancient Ricci flow asymptotic to a compact integrable shrinker, or a Ricci flow developing a finite-time singularity modelled on the shrinker, we establish the long-time existence of a harmonic map heat flow between the Ricci flow and the shrinker for all times. This provides a global parabolic gauge for the Ricci flow and implies the uniqueness of the tangent flow without modulo any diffeomorphisms.

We present two main applications: First, we construct and classify all ancient Ricci flows asymptotic to any compact integrable shrinker, showing that they converge exponentially. Second, we obtain the optimal convergence rate at singularities modelled on the shrinker, characterized by the first negative eigenvalue of the stability operator for the entropy. This is joint work with K. Choi.

• 10:15-11:15 Tin Yau Tseng: Mass for the large and for the small 

Seeking a meaningful geometric (physical) invariant to describe a spacetime has sparked researches in both mathematics and physics. Interestingly, comparison geometry and conservation of energy are aligned. In this talk, we will first review the fundamental progress made by Schoen-Yau and Witten on ADM (Arnowitt-Deser-Misner) mass and scalar curvature. Then, we will see some recent progress on quasilocal masses, particularly the cases with the presence of apparent horizons.