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*

- Riemannian-Lorentzian Geometry :
- Dynamical systems :
*geodesic flow and other dynamical applications in geometry*

- Wenkui Du (wenkui.du@mail.utoronto.ca)
- David Knapik (david.knapik@mail.utoronto.ca)
- Xinze Li (xbryanli.li@mail.utoronto.ca)

**November 17: Yuchao Yi (UCSD)***Inverse problems for Lorentzian manifolds*

We use microlocal analysis method to detect the shape of far universe by sending waves and receiving the signals from their nonlinear interaction.

**November 10: Xinze Li (University of Toronto)***Positive Mass Theorem*

We discuss the proof of positive mass theorem which says the ADM mass is nonnegative in asymptotical flat 3 manifolds with positive scalar curvature.

**November 3: Jingze Zhu (Massachusetts Institute of Technology)***Bernstein Theorem of Stable Minimal Hypersurfaces in R^4*

This talk discusses the work of Chodosh-Li proof that embedded two sided complete smooth stable minimal hypersurfaces in R^4 must be hyperplanes.

**October 27: Ali Feizmohammadi (University of Toronto)***Inverse problems for wave equations*

We will give an introductory overview of inverse problems for linear and nonlinear wave equations. I will describe results in both stationary and non-stationary spacetimes. An example of inverse problems in stationary spacetimes is the imaging
of internal structure of the earth from surface measurements of seismic waves arising from earthquakes or artificial explosions. Here, the materialistic properties of the internal layers of the earth are generally assumed to be independent of
time. On the other hand, inverse problems for non-stationary spacetimes are inspired by the theory of general relativity as well as gravitational waves where waves follow paths that curve not only in space but also in time.

**October 16: Almut Burchard (University of Toronto)***On pointwise monotonicity of heat kernels*

The fact that the heat kernel K_t(x,y) on the standard sphere decreases with the distance between the points x and y has important consequences in Probability and Functional Analysis. Does the heat kernel on a general Riemannian manifold have similar monotonicity properties?

During his time as a postdoc in Toronto, Angel Martinez and I worked on metrics for which the heat kernel decreases monotonically as y moves along a minimal geodesic emanating from x. It turns out that such metrics are extremely rare - but we also found a surprising example!

Instead of having invited speakers give talks on their own research, the format for the Winter 2023 semester consisted of meetings where students gave lectures on a few topics (one topic per person) in order to provide the background information needed for those interested in pursuing research in the fields of geometry and analysis. Thank you to everyone who participated, and a special thanks to Wenkui for taking the reins as the primary organizer for the seminar this semester.

**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.

**September 30: Aram-Alexandre Pooladian (NYU)***Leveraging entropic regularization to approximate optimal transport maps*

Optimal transport and its variants have gained significant interest over the past decade, primarily due to computational advancements by way of entropic regularization. In this talk, I will introduce the notions of regularized and unregularized optimal transport (for the squared-Euclidean cost, specifically), and talk about my recent work on approximating the optimal transport map, or Monge map, on the basis of entropic regularization. This is a foundational aspect of statistical optimal transport, which I will only mention briefly for the sake of motivation (joint work with Jon Niles-Weed)

**September 21 & 23: Yushan Jiang (CUNY)***Dynamical Systems and Ergodic Theory: Some Topics Related to Subsurfaces in Three Manifolds*

Dynamical systems and ergodic theory have so many wonderful applications in different fields like geometry, topology, complex analysis, number theory and so on. One fantastic field which connects almost all such branches is hyperbolic geometry. In this talk, I will briefly introduce hyperbolic plane and space (2 & 3 dim) and some related dynamical systems (geodesic/frame flow and horocycle flow). Also I need to say a word about some terms in ergodic theory. Then I would like to talk (without proof) some famous applications in 3-manifolds theory (post-Thurston era). One is Surface Subgroup Conjecture (related to geodesic/frame flow and the famous Virtual Haken Conjecture). The other one is about the rigidity of the closures of immersed totally geodesic subsurfaces (related to unipotent group action and the famous Ratner theorems).

**September 16: Minghao Miao (Nanjing University & Beijing International Center for Mathematical Research)***Kahler-Einstein metric on Fano manifold*

In this talk, I will survey some recent progress of Yau-Tian-Donaldson conjecture, which relate certain algebro-geometric stability condition with the existence of Kahler-Einstein metric on Fano manifold. Another canonical metric, Kahler-Ricci soliton will also be discussed in a parallel setting.