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.
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
Mircea Petrache (PUC Chile)
Sharp isoperimetric inequality, discrete PDEs and semidiscrete optimal transport
Abstract
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]
Yeli Niu (U of Alberta)
Discretization of integrals on compact metric measure spaces
Abstract
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 |
Organizers
Ryan Matzke | U of Minnesota | matzk053@umn.edu |
Tetiana Stepaniuk | Universität zu Lübeck | stepaniuk@math.uni-luebeck.de |
Alex Vlasiuk | Florida State | ovlasiuk@fsu.edu |
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Previously: Damir Ferizović | TU Graz | damir.ferizovic@math.tugraz.at |