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SUMMARY:Using numerical simulations for solving spectral geometry problems
  - Beniamin Bogosel (Aurel Vlaicu University)
DTSTART:20260420T101500Z
DTEND:20260420T110000Z
UID:TALK244909@talks.cam.ac.uk
DESCRIPTION:Numerical tools play an essential role nowadays in the study o
 f any mathematical phenomenon for which analytic tools are not available. 
 In spectral geometry\, the dependence of the eigenvalues of differential o
 perators on the geometry of the object studied is complex and often not ex
 plicit. The usage of numerical simulations provides valuable insights rega
 rding research problems.&nbsp\;\nIn this talk I will discuss how numerical
  simulations regarding the optimization of spectral quantities with respec
 t to the geometry of the domain can influence and guide the theoretical st
 udy. In some situations\, numerical simulations using guaranteed error bou
 nds can even provide computer assisted proofs of theoretical results. I wi
 ll present a recent result in this direction\, obtained in collaboration w
 ith Dorin Bucur\, regarding the local minimality of the regular n-gon when
  minimizing the fundamental eigenvalue of the Dirichlet-Laplace operator o
 n n-gons of fixed area.&nbsp\;\nOne complex aspect regarding numerical sha
 pe optimization is the computation of the sensitivity of the objective fun
 ction with respect to geometric parameters. Spectral quantities can be enc
 oded using neural networks\, providing a natural differentiable structure\
 , thus speeding the optimization process. I will present recent preliminar
 y results on this topic obtained in collaboration with Dorin Bucur and Ale
 xis de Villeroch&eacute\;.&nbsp\;\n&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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