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 edgeperimeter of X, denoted \partial
X.
If the graph is periodic and locally finite, any X satisfies an
inequality of the form X^{d1} \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 pathconnected
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 finitedimensional 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 Garcı́aZelada  AixMarseille 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 WeylHeisenberg 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.uniluebeck.de 
Alex Vlasiuk  Florida State  ovlasiuk@fsu.edu 


Previously: Damir Ferizović  TU Graz  damir.ferizovic@math.tugraz.at 