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Announcement
Carolyn Nguyen and Anna Oliva
University of Houston and Carnegie Vanguard HS
Evasion Paths in Hyperbolic Sensor Networks and Symmetry, Fixed Points, and Quantum Billiards: A Confluence of Ideas
April 18, 2024
4:00pm PGH 646
Abstract


Carolyn Nguyen:
In this talk we focus on how to effectively monitor a domain with a network of
mobile sensors in a coordinate free way. We equip these mobile sensors with
minimal sensing ability; they do not track their absolute positions in space
(i.e. no GPS data), but do know the identities and cyclic ordering of nearby
sensors and can sense evaders within a relatively small ball of given radius.
Minimal mobile sensing problems have received considerable attention over the
past 15 years. This field is part of applied topology, an emerging research
area that links theoretical concepts in topology to applied problems in data
analysis. Potential applications of our work include scanning a region to
detect forest fires, radiological or biological hazards, hidden mines, or a
specific individual in a crowd.
Our talk will evaluate the effectiveness of mobile sensor networks by studying
how long an evader can remain undetected. In particular, we are interested in
how the hyperbolicity of the collective sensor motion affects this detection
time. We will present results in which we introduce hyperbolicity through the
boundary geometry using the Bunimovich stadium.
Anna Oliva:
A mathematical billiard system is one composed of a planar or
multidimensional surface and a moving object whose trajectory is defined by
its initial position and speed vector, along with some reflection law. The
study of these systems has yielded applications in quantum computing and
physical modeling. The purpose of this research was to study a novel
reflection law for billiard systems of regular ngons in which an object
starting on one of the sides and moving with any given slope reflects from
a limited reflection towards the interior with a prescribed constant angle.
We constructed the map for the position of such an object as a function of
the starting point and the slope in the case of a regular triangle and
square. For the case of the square we proved that the object’s path
converges to a stable inner rectangle for all initial conditions and all
slopes. We showed that the same results is true for a triangular billiard
if the starting slope is less than 60 degrees. Otherwise, we showed that
for starting slopes between 60 and 90 there exists only a countable set of
starting points which lead to convergent trajectory. We have also
constructed a numerical model for the object’s trajectory and determined
equations yielding the speed of the object’s convergence to a stable path.
We have performed relevant simulations and discussed the resulting data in
the context of the proposed model. The analysis of these systems can be
used to develop independent control mechanisms for ground robots in
delivery missions working within contested environments or simpler and
smaller microchips in the form of quantum billiards with correlated
electrons without the necessity of perfectly elastic collisions, offering
an innovative way to encode information.
Pizza will be served.
Colloquium sponsored by the NSM Dean's office DUSEM grant
[click for poster]







