# Student Geometry and Analysis Seminar

Hello and welcome to the Student Geometry and Analysis Seminar (SGAS) at the University of Toronto. This seminar’s wide range of topics focuses on the interplay between geometry and analysis and includes (but is not limited to):

• Analysis & PDEs : microlocal analysis, nonlinear PDEs, inverse problems, geometric measure theory, optimal transport, stochastic analysis
• Differential Geometry
• Riemannian-Lorentzian Geometry : quantitative geometry and metric geometry on Ricci and scalar curvature, general relativity
• Geometric Analysis : Ricci flow, (gradient and skew) mean curvature flows, minimal surfaces, Kähler-Einstein metric, CSCK metric
• Symplectic Geometry : Floer homology and geometric invariants, geometric fluids and Hamiltonian systems
• Dynamical systems : geodesic flow and other dynamical applications in geometry

# Talks

December 13: Jingze Zhu (Massachusetts Institute of Technology)
Bamler’s synthetic Ricci flow
In this talk, we will overview Bamler’s theory of synthetic Ricci flow. We will discuss the definition of metric flow and F-convergence of metric flows, then we will overview the compactness and partial regularity theory of metric flow. Finally we will discuss how the theory is applied to study singularity formation of Ricci flow in higher dimensions.

December 1: Ilia Kirillov (University of Toronto)
Geometric Hydrodynamics and Curvatures of Diffeomorphism Groups
In this talk I will explain the basics of geometric hydrodynamics (pioneered by V. Arnold) and its relation to sectional curvatures of L^2 metric on a group of volume-preserving diffeomorphisms.

November 25: Yang Yang (University of California Irvine)
The anisotropic Bernstein problem
The Bernstein problem asks whether entire minimal graphs in R^{n+1} are necessarily hyperplanes. It is known through spectacular work of Bernstein, Fleming, De Giorgi, Almgren, Simons, and Bombieri-De Giorgi-Giusti that the answer is positive if and only if n < 8. The anisotropic Bernstein problem asks the same question about minimizers of parametric elliptic functionals, which are natural generalizations of the area functional that both arise in many applications, and offer important technical challenges. We will discuss the recent solution of this problem (the answer is positive if and only if n < 4). This is joint work with C. Mooney

November 21: Tracey Balehowsky (University of Calgary)
An Inverse Problem for the Relativistic Boltzmann Equation
In this talk, we consider the following problem: Given the source-to-solution map for a relativistic Boltzmann equation on a neighbourhood $V$ of an observer in a Lorentzian spacetime $(M,g)$ and knowledge of $g|_V$, can we determine (up to diffeomorphism) the spacetime metric $g$ on the domain of causal influence for the set $V$?

We will show that the answer is yes. The problem we consider is a so-called inverse problem. We will first introduce the notion of an inverse problem. Then, we will present the motivation for our inverse problem. Next, we introduce the relativistic Boltzmann equation and comment on the existence of solutions to this PDE given some initial data. We then will comment on a key idea of the proof of our result.

The key point presented is that the nonlinear term in the relativistic Boltzmann equation which describes the behaviour of particle collisions captures information about a source-to-solution map for a related linearized problem. We use this relationship together with an analysis of the behaviour of particle collisions by classical microlocal techniques to determine the set of locations in $V$ where we first receive light particle signals from collisions in the unknown domain. From this data we are able to parametrize the unknown region and determine the metric.

The results presented in this talk are joint work with Antti Kujanpää, Matti Lassas, and Tony Liimatainen.

November 17: Damien Tageddine (McGill University)
Noncommutative differential geometry on infinitesimal spaces
Noncommutative differential geometry is a most prominent realization of noncommutative geometry. In this talk, we will show how the noncommutative analogue of spin geometry in terms of generalized Dirac operators $D$ acting on a representation space of an algebra $A$ can extend classical differential geometry to nonmanifold like spaces. We start with a brief review of inverse limit of posets as an approximation of a topological space. We then show how to associate a $C^*$-algebra over a poset, giving it a piecewise-linear structure. Furthermore, we explain how dually the algebra of continuous function $C(M)$ over a manifold $M$ can be approximated by a direct limit of $C^*$-algebras over posets. Finally, we prove how classical differential calculus is recovered as the spectrum of the commutator with a Dirac operator.

November 10: Fan Ye (Harvard University)
SU(2) representations of small Dehn surgeries on knots
Kronheimer-Mrowka proved that the fundamental group of r-surgery on a nontrivial knot for |r|<=2 always has an irreducible SU(2) representation, which answered the property P conjecture affirmly. They asked the case of r=3 and 4. The case r=4 was solved by Baldwin-Sivek and the case r=3 was solved by Baldwin-Li-Sivek-Ye. In this talk, I will describe the strategy of the proofs using instanton Floer homology.

October 25: Gabriel Rioux (Cornell University)
Aspects of Ricci curvature bounds on metric measure spaces
This talk aims to highlight the role that optimal transport, information theory, and gradient flows have played in the study of geometry in metric measure spaces. To this effect, the seminal work of J. Lott, C. Villani, and K.-T. Sturm on extending the standard curvature-dimensions conditions on Riemannian manifolds to general metric measure spaces will be reviewed and some extensions of this work will be discussed.

The driving force behind these developments is the interplay between the geometry of the optimal transport of probability measures on a metric measure space $(\mathcal X, d, \mu)$ and the geometric properties of $(\mathcal X, d, \mu)$, especially as it pertains to the properties of information functionals along geodesics in the space of probabilities.

If time permits, an application of this theory to the construction of heat flows in nonsmooth spaces will be provided.

October 10: Jintian Zhu (Peking University & Beijing International Center for Mathematical Research)
Recent developments in positive mass theorem and scalar curvature problems
The classical Riemannian positive mass theorem states that an asymptotically flat manifold (asymptotic to the Euclidean space) with nonnegative scalar curvature has nonnegative ADM mass. In this talk, we shall make a discussion on its further generalizations for those manifolds, whose ends approach quotient spaces of Euclidean space, under an additional but necessary incompressible condition.

October 5: Wenkui Du (University of Toronto)
Ancient solutions as singularity models of mean curvature flows
Mean curvature flow in extrinsic geometry is the twin of Ricci flow in intrinsic geometry. MCF is expected to have many applications in geometry and topology as the sucessful applications of Ricci flow such as Poincare conjecture, diffeomorphism sphere theorem, smale conjecture. To have applications, the key is to understand the singularity structure. Ancient solutions appear as the blowup limit of singularities of mean curvature flow for capturing the behavior of flow in a spacetime neighborhood of the singularities. The most important singularities are cylindrical singularities which are expected to be generic singularity under initial data perturbation. The ancient asymptotically cylindrical flows or ancient noncollapsed flows are singularity models perturbed from the cylindrical flows. I will survey the recent developments of classification of these ancient solutions of mean curvature flows.