Schedule and speakers Archive Organizers

This weekly seminar will cover topics related to point distributions, potential and discrepancy theory, and adjacent areas. The target audience is young researchers working in the listed directions, but everyone is welcome! Please share this information with anyone interested.

We will meet on Wednesdays, at 10am CST/11am EST/5pm CET. To join the seminar via Zoom, follow the link. It becomes active half an hour before the meeting. To join our mailing list, go here.

Previous talks are available in the Archive.

Coming up

  • Mar 3
    Mircea Petrache (PUC Chile)
    Sharp isoperimetric inequality, discrete PDEs and semidiscrete optimal transport
    Consider the following basic model of finite crystal cluster formation: in a periodic graph G with vertices in R^d (representing possible molecular bonds) a subset (of atoms) must be chosen, so that the total number of bonds between a point in X and one outside X is minimized. These bonds form the edge-perimeter of X, denoted \partial X. If the graph is periodic and locally finite, any X satisfies an inequality of the form |X|^{d-1} \leq C |partial X|^d, where the optimal C depends on the graph. How can we determine the structure of sets X realizing equality in the above, based on the geometry and of G? If we take the continuum limit of G, then the classical Wulff shape theory describes optimal limit shapes, and at least two proofs of isoperimetric inequality apply, one based on PDE and calibration ideas, and the other based on Optimal Transport ideas. We focus on using the heuristic coming from the continuum analogue, to answer the above question in some cases, in the discrete case. This approach highlights the tight connection between discrete PDEs and semidiscrete Optimal Transport, and a link to the Minkowski theorem for convex polyhedra.

    Paper: [arXiv1] [arXiv2]
    Slides: [pdf]

  • Mar 10
    Yeli Niu (U of Alberta)
    Discretization of integrals on compact metric measure spaces
    Let $\mu$ be a Borel probability measure on a compact path-connected metric space $(X, \rho)$ for which there exist constants $c,\beta\ge 1$ such that $\mu(B) \ge c r^{\beta}$ for every open ball $B\subset X$ of radius $r>0$. For a class of Lipschitz functions $\Phi:[0,\infty)\to\mathbb R$ that are piecewise within a finite-dimensional subspace of continuous functions, we prove under certain mild conditions on the metric $\rho$ and the measure $\mu$ that for each positive integer $N\ge 2$, and each $g\in L^\infty(X, d\mu)$ with $\|g\|_\infty=1$, there exist points $y_1, \ldots, y_{ N }\in X$ and real numbers $\lambda_1, \ldots, \lambda_{ N }$ such that for any $x\in X$, \begin{align*} & \left| \int_X \Phi (\rho (x, y)) g(y) \,\text{d} \mu (y) - \sum_{j = 1}^{ N } \lambda_j \Phi (\rho (x, y_j)) \right| \leqslant C N^{- \frac{1}{2} - \frac{3}{2\beta}} \sqrt{\log N}, \end{align*} where the constant $C>0$ is independent of $N$ and $g$. In the case when $X$ is the unit sphere $\mathbb{S}^{d}$ of $\mathbb R^{d+1}$ with the ususal geodesic distance, we also prove that the constant $C$ here is independent of the dimension $d$. Our estimates are better than those obtained from the standard Monte Carlo methods, which typically yield a weaker upper bound $N^{-\frac 12}\sqrt{\log N}$.

  • Schedule and speakers

    The schedule is given in the local time for CST (Chicago) / EST (New York) / CET (Paris, Berlin) time zones; during daylight saving time it remains 10am/11am/5pm, respectively.

    Date Time Speaker Affiliation Title
    Feb 3 10am CST/11am EST/5pm CET David Garcı́a-Zelada Aix-Marseille U A large deviation principle for empirical measures
    Feb 10 10am CST/11am EST/5pm CET Arno Kuijlaars KU Leuven The spherical ensemble with external sources
    Feb 17 10am CST/11am EST/5pm CET Alex Iosevich U of Rochester Finite point configurations and frame theory
    Feb 24 10am CST/11am EST/5pm CET Kasso Okoudjou Tufts U Completeness of Weyl-Heisenberg POVMs
    Mar 3 10am CST/11am EST/5pm CET Mircea Petrache PUC Chile Sharp isoperimetric inequality, discrete PDEs and semidiscrete optimal transport
    Mar 10 *9:30am CST/10:30am EST/4:30pm CET Yeli Niu U of Alberta Discretization of integrals on compact metric measure spaces
    Daylight Saving Time in the US
    Mar 17 *12pm CDT/1pm EDT/6pm CET Xuemei Chen UNC Wilmington TBA
    Mar 24 10am CDT/11am EDT/4pm CET Ruiwen Shu U of Maryland TBA
    Daylight Saving Time in Europe
    Mar 31 10am CDT/11am EDT/5pm CEST Oleg Musin U of Texas Rio Grande Valley TBA
    Apr 7 10am CDT/11am EDT/5pm CEST Woden Kusner U of Georgia TBA
    Apr 14 10am CDT/11am EDT/5pm CEST Peter Dragnev Purdue Fort Wayne TBA
    Apr 28 *1pm CDT/2pm EDT/8pm CEST Shujie Kang UT Arlington TBA
    May 5 10am CDT/11am EDT/5pm CEST Mario Ullrich JKU Linz TBA
    May 12 TBA William Chen Macquarie U TBA
    May 19 10am CDT/11am EDT/5pm CEST Johann Brauchart TU Graz TBA


    Ryan Matzke     U of Minnesota
    Tetiana Stepaniuk     Universität zu Lübeck
    Alex Vlasiuk     Florida State

    Previously: Damir Ferizović     TU Graz