Seminars, for informal dissemination of research results, exploratory work by research teams, outreach activities, etc., constitute the simplest form of meetings at a Mathematics research centre.
CAMGSD has recorded the calendar of its seminars for a long time, this page serving both as a means of public announcement of forthcoming activities but also as a historic record.
In the last decades, there has been fascinating progress in the variational theory for the area functional – that is, the codimension 1 volume – using tools from PDEs and Geometric Measure Theory, and in connection with the problem of finding prescribed mean curvature (PMC) hypersurfaces.
In this talk, we describe some recent contributions from joint work with Jared Marx-Kuo (Rice University) in which we construct infinitely many PMCs for a large class of prescribing functions in a compact Riemannian manifold containing a strictly stable minimal hypersurface.
The Q-curvature equation, a fourth-order elliptic partial differential equation with a critical exponent, is a prominent class of conformal equations, largely due to its connection with a natural concept of curvature. In light of the significant advances in the existence theory for the Q-curvature equation, in parallel with the Yamabe problem, this talk discusses the existence theory in both the compact and non-compact cases. We will also provide several interesting constructions based on techniques such as gluing and Lyapunov–Schmidt reduction, which shed light on the solution set of this equation.
The set of (smooth) metrics that can be placed on a Riemannian manifold defines an infinite-dimensional "superspace" that, remarkably, can itself be imbued with the structure of a (Fréchet) manifold. The subspace pertaining to (spatially-sliced) Einstein metrics was explored in detail by Wheeler and collaborators back in the late 50s, as it provides a means to describe a collection of spacetimes purely in terms of geometry through the famous words "mass without mass; charge without charge". At least in some restricted contexts, a natural basis relates to multipole moments which provide a tool to decompose a spacetime into a set of numbers. I will describe the construction of such superspaces, how to define inner products and (weak) Riemannian metrics there, and how they may be useful to provide astrophysical intuition. For instance, geodesics can be computed on Met(M) which allows one to define a single number that tells you how "distant" two spacetimes (e.g., two Kerr black holes) are from one another.
Existing machine learning frameworks operate over the field of real numbers ($\mathbb{R}$) and learn representations in real (Euclidean or Hilbert) vector spaces (e.g., $\mathbb{R}^d$). Their underlying geometric properties align well with intuitive concepts such as linear separability, minimum enclosing balls, and subspace projection; and basic calculus provides a toolbox for learning through gradient-based optimization.
But is this the only possible choice? In this seminar, we study the suitability of a radically different field as an alternative to $\mathbb{R}$ — the ultrametric and non-archimedean space of $p$-adic numbers, $\mathbb{Q}_p$. The hierarchical structure of the $p$-adics and their interpretation as infinite strings make them an appealing tool for code theory and hierarchical representation learning. Our exploratory theoretical work establishes the building blocks for classification, regression, and representation learning with the $p$-adics, providing learning models and algorithms. We illustrate how simple Quillian semantic networks can be represented as a compact $p$-adic linear network, a construction which is not possible with the field of reals. We finish by discussing open problems and opportunities for future research enabled by this new framework.
