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.
Let $L = (L(t))_{t\geq 0}$ be a multivariate L\'evy process with L\'evy measure $\nu(dy) = \exp(-f(|y|)) dy$ for a smoothly regularly varying function $f$ of index $\alpha>1$. The process $L$ is renormalized as $X^\epsilon(t) = \epsilon L(r_\epsilon t)$, $t\in [0, T]$, for a scaling parameter $r_\epsilon = o(\epsilon^{-1})$, as $\epsilon \to 0$. We study the behavior of the bridge $Y^{\epsilon, x}$ of the renormalized process $X^\epsilon$ conditioned on the event $X^\epsilon(T) = x$ for a given end point $x\neq 0$ and end time $T>0$ in the regime of small $\epsilon$. Our main result is a sample path large deviations principle (LDP) for $Y^{x, \epsilon}$ with a specific speed function $S(\epsilon)$ and an entropy-type rate function $I_{x}$ on the Skorokhod space in the limit $\epsilon \to 0+$. We show that the asymptotic energy minimizing path of $Y^{\epsilon, x}$ is the linear parametrization of the straight line between $0$ and $x$, while all paths leaving this set are exponentially negligible. Since on these short time scales ($r_\epsilon = o(\epsilon^{-1})$) direct LDP methods cannot be adapted we use an alternative direct approach based on convolution density estimates of the marginals $X^{\epsilon}(t)$, $t\in [0, T]$, for which we solve a specific nonlinear functional equation.
I will review two results pertaining to 3-dimensional Reshetikhin–Turaev TQFTs, defined from modular tensor categories M. These theories were not constructed as “fully local” TQFTs (in the framework of Lurie’s Cobordism Hypothesis): no algebraic structures were assigned to points. (The obstruction was the Witt class of M.) Kevin Walker solved the locality problem in the setting of anomalous theories. A ‘no-go’ theorem (joint with Dan Freed) showed that, if localized as linear theories, these RT theories did not admit local topological boundary conditions, and could therefore not be generated from a point by this method. (The group-like case had been addressed by Kapustin and Saulina.) In recent work with Freed and Claudia Scheimbauer, we displayed a fully local realization of these theories, by objects in a target 3-category which enlarges that of fusion categories. This allowed us to settle some conjectures relating orientations and spherical structures.
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.
The Malmquist-Takenaka (MT) system is a complete orthonormal system in $H^2(T)$ generated by an arbitrary sequence of points in the unit disk that do not approach the boundary very fast. The nth point of the sequence is responsible for multiplying the nth and subsequent terms of the system by a Möbius transform taking the point to 0. One can recover the classical trigonometric system, its perturbations or conformal transformations, as particular examples of the MT system. However, for many interesting choices of the generating sequence, the MT system is less understood. We prove almost everywhere convergence of the MT series for three different classes of generating sequences.
