Abstract: In this talk, we explore explicit cross-sections to the horocycle and geodesic flows on $\operatorname{SL}(2, \mathbb{R})/G_q$, with $q \geq 3$. Our approach relies on extending properties of the primitive integers $\mathbb{Z}_\text{prim}^2 := \{(a, b) \in \mathbb{Z}^2 \mid \gcd(a, b) = 1\}$ to the discrete orbits $\Lambda_q := G_q (1, 0)^T$ of the linear action of $G_q$ on the plane $\mathbb{R}^2$. We present an algorithm for generating the elements of $\Lambda_q$ that extends the classical Stern-Brocot process, and from that derive another algorithm for generating the elements of $\Lambda_q$ in planar strips in increasing order of slope. We parametrize those two algorithms using what we refer to as the \emph{symmetric $G_q$-Farey map}, and \emph{$G_q$-BCZ map}, and demonstrate that they are the first return maps of the geodesic and horocycle flows resp. on $\operatorname{SL}(2, \mathbb{R})/G_q$ to particular cross-sections. Using homogeneous dynamics, we then show how to extend several classical results on the statistics of the Farey fractions, and the symbolic dynamics of the geodesic flow on the modular surface to our setting using the $G_q$-BCZ and symmetric $G_q$-Farey maps. This talk is self-contained and does not assume any prior knowledge of Hecke triangle groups or homogeneous dynamics.
Abstract: We develop a correspondence between Borel equivalence relations induced by closed subgroups of $S_\infty$ and symmetric models of set theory without choice, and apply it to prove a conjecture of Hjorth-Kechris-Louveau (1998).
For example, we show that the equivalence relation $\cong^\ast_{\omega+1,0}$ is strictly below $\cong^\ast_{\omega+1,
Abstract: We present a novel algorithm to handle both equality and inequality constraints in
infinite dimensional optimization problems. The inequality constraints are tackled
via a nonstandard penalty. On the other hand, the equality constraints are handled
using trust region methods. The latter permits inexact PDE solves. As applications,
we consider PDE constrained optimization (PDECO) problems with contact type
constraints and topology optimization problems.
We will also introduce novel optimal control concepts within the realm of PDECO
problems with fractional/nonlocal PDEs as constraints and discuss their applications
in geophysics and imaging sciences. We will further illustrate the role of fractional
operators as a regularizer in machine learning.
We conclude this talk by introducing a general framework based on Gibbs posterior
to update the belief distributions for inverse problems governed by PDEs. Contrary to
traditional Bayesian analysis, noise model is not assumed to be known.
Abstract: We consider transport equations on graphs, where mass is distributed over vertices and is transported along the edges. The first part of the talk will deal with the graph analogue of the Wasserstein distance, in the particular case where the notion of density along edges is inspired by the upwind numerical schemes. This natural notion of interpolation however leads to the fact that Wasserstein distance is only a quasi-metric. In the second part of the talk we will interpret the nonlocal-interaction equation equations on graphs as gradient flows with respect to the graph-Wasserstain quasi-metric of the nonlocal-interaction energy. We show that for graphs representing data sampled from a manifold, the solutions of the nonlocal-interaction equations on graphs converge to solutions of an integral equation on the manifold. We also show that the limiting equation is a gradient flow of the nonlocal-interaction energy with respect to a nonlocal analogue of the Wasserstein metric.
Abstract: Arithmetic Transfer conjectures are the analogues of Wei
Zhang's Arithmetic Fundamental Lemma conjecture in the presence of
ramification. Some such conjectures were given in two papers by
Smithling, Zhang and myself. I will present more such conjectures. This
is joint work in progress with S. Kudla, B. Smithling and W. Zhang.
Abstract: This will be a lecture/demonstration on the latest developments
in Mathematica and Wolfram|Alpha, and how they can be of use to
mathematicians.
Abstract: Network science is a rapidly expanding field, with a large and growing body of work on network-based dynamical processes. Most theoretical results in this area rely on the so-called "locally tree-like approximation" (which assumes that one can ignore small loops in a network). This is, however, usually an `uncontrolled' approximation, in the sense that the magnitudes of the error are typically unknown, although numerical results show that this error is often surprisingly small. In our work, we place this approximation on more rigorous footing by calculating the magnitude of deviations away from tree-based theories in the context of network cascades (i.e., a network dynamical process describing the spread of activity through a network). For this widely applicable problem, we discuss the conditions under which tree-like approximations give good results, and also explain the reasons for deviation from this approximation. More specifically, we show that these deviations are negligible for networks with a large number of network links, justifying why tree-based theories appear to work well for most real-world networks.
Abstract: We present recent results on time-scales separation in fluid
mechanics. The fundamental mechanism to detect in a precise
quantitative manner is commonly referred to as fluid mixing. Its
interaction with advection, diffusion and nonlocal effects produces a
variety of time-scales which explain many experimental and numerical
results related to hydrodynamic stability and turbulence theory.
Abstract: Graph embeddings, a class of dimensionality reduction techniques designed for relational data, have proven useful in exploring and modeling network structure. Most dimensionality reduction methods allow out-of-sample extensions, by which an embedding can be applied to observations not present in the training set. Applied to graphs, the out-of-sample extension problem concerns how to compute the embedding of a vertex that is added to the graph after an embedding has already been computed. In this talk, we will consider the out-of-sample extension problem for two graph embedding procedures: the adjacency spectral embedding and the Laplacian spectral embedding. In both cases, we prove that when the underlying graph is generated according to a latent space model called the random dot product graph, which includes the popular stochastic block model as a special case, an out-of-sample extension based on a least-squares objective obeys a central limit theorem. Our results also yield a convenient framework in which to analyze trade-offs between estimation accuracy and computational expenses, which we will explore briefly.
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