Monday, October 4, 2021
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All day |
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10am |
10/04/2021 - 10:30am Abstract: We discuss two statistical problems that arise in the course of physical modeling. First, we consider an experiment where a low-dimensional dynamical system is observed as a high-dimensional signal, e.g., a video of a chaotic pendulums system. Assuming that we know the dynamical model, but do not know the observation function (the experimental design) - can we estimate the true parameters from the experiment? The key information lies in the temporal inter-dependencies between the signal and the model, and we exploit this information using a kernel-based score. In the second part of the talk, we turn to uncertainty propagation; in many scientific areas, the parameters of deterministic models are uncertain or noisy. A comprehensive model should therefore provide a statistical description of the quantity of interest. Underlying this computational problem is a fundamental question - if two “similar” functions push-forward the same measure, would the new resulting measures be close, and if so, in what sense? Through optimal transport theory, a Wasserstein-distance formulation of our problem yields a simple and applicable theory. Location:
https://yale.zoom.us/j/2188028533
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4pm |
10/04/2021 - 4:00pm We consider the action of the diagonal subgroup $\{a(t)=(t^{n-1},t^{-1},\ldots,t^{-1})\}\subset G=\mathrm{SL}(n,\mathbb{R})$ on $X=G/\Gamma$, where $\Gamma=\mathrm{SL}(n,\mathbb{Z})$. Let $C$ be a finite piece of an analytic curve on the expanding horophere ($\cong{\mathbb R}^{n-1}$) of $\{a(t)\}_{{t>1}}$ in $G$ . Let $\mu_{C}$ be a smooth probability measure on the trajectory $C[\Gamma]$ on $X$. We provide necessary and sufficient conditions on the smallest affine subspace containing $C$ in terms of Diophantine approximation and algebraic number fields so that the measures $a(t)\mu_{C}$ get equidistributed in $X$ as $t\to\infty$. This result generalizes the speaker’s earlier work showing equidistribution of translates of curves, which are not contained in proper affine subspaces. The result answers a question of Davenport and Schmidt on non-improvability of Dirichlet’s approximation. The case of $n=3$ is a joint work D. Kleinbock, N. de Saxcé, and P. Yang; and the general case is a joint work with Pengyu Yang. Location:
Zoom
10/04/2021 - 4:30pm Abstract: I will talk about the positive part of a certain affine Springer fiber studied by Goresky, Kottwitz, and MacPherson, and a certain interesting open subvariety. The Hilbert series of their Borel-Moore homology turn out to be related to reproducing kernels of the Bergeron-Garsia nabla operator. This operator is easy to define in the basis of modified Macdonald polynomials, but producing explicit combinatorial evaluations of this operator is usually difficult and (conjecturally) relates to interesting Hilbert series associated to various moduli spaces. Our work is motivated by the nabla positivity conjecture of Bergeron, Garsia, Haiman, and Tesler that predicts that nabla evaluated on a Schur function is sometimes positive, sometimes negative. We categorify this conjecture and reduce it to a vanishing conjecture for the interesting open variety. It turns out, each irreducible S_n representation mysteriously prefers to live in certain degrees and weights in the cohomology. This is a joint work with Erik Carlsson. Zoom link: https://yale.zoom.us/j/99305994163, contact the organizers (Gurbir Dhillon and Junliang Shen) for the passcode. Location:
Zoom
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