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In the bipartite case minimal partitions are closely related to eigenfunctions of the Laplacian, but in the non-bipartite case they are difficult to classify, even for simple domains like the square or the circle." /> <link rel="canonical" href="https://www.ias.edu/math/events/analysis-and-mathematical-physics-53" /> <link rel="shortlink" href="https://www.ias.edu/node/297451" /> <meta property="og:site_name" content="Institute for Advanced Study" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.ias.edu/math/events/analysis-and-mathematical-physics-53" /> <meta property="og:title" content="Analysis and Mathematical Physics" /> <meta property="og:description" content="A minimal partition is a decomposition of a manifold into disjoint sets that minimizes a spectral energy functional. 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<div class="two-col__top"> <div class="two-col__first two-col__column"> <figure class="media media--type-image media--view-mode-featured-two-column"> <div class="field field--name-field-media-image field--type-image field--label-hidden"> <div class="field__item"> <picture> <source srcset="/sites/default/files/styles/two_column_large/public/2019-09/sm_default.jpg?itok=KehPOEH- 1x" media="all and (min-width: 768px)" type="image/jpeg" width="768" height="480"/> <source srcset="/sites/default/files/styles/two_column_medium/public/2019-09/sm_default.jpg?itok=gMvWynkh 1x" media="all and (min-width: 524px)" type="image/jpeg" width="690" height="431"/> <source srcset="/sites/default/files/styles/two_column_medium/public/2019-09/sm_default.jpg?itok=gMvWynkh 1x" media="all and (max-width: 523px)" type="image/jpeg" width="690" height="431"/> <img loading="eager" width="690" height="431" src="/sites/default/files/styles/two_column_medium/public/2019-09/sm_default.jpg?itok=gMvWynkh" alt="School of Mathematics Event" /> </picture> </div> </div> <figcaption> </figcaption> </figure> <h1><span class="field field--name-title field--type-string field--label-hidden">Analysis and Mathematical Physics</span> </h1> <h2 class="field field--name-field-topic field--type-string field--label-hidden"> <span class="field__item">Geometry and Topology of Spectral Minimal Partitions</span> </h2> <div class="field field--name-body field--type-text-with-summary field--label-hidden"> <div class="field__item"><div class="tex2jax_process"><p>A minimal partition is a decomposition of a manifold into disjoint sets that minimizes a spectral energy functional. In the bipartite case minimal partitions are closely related to eigenfunctions of the Laplacian, but in the non-bipartite case they are difficult to classify, even for simple domains like the square or the circle.</p><p>I will present new results that imply, among other things, that a partition that minimizes energy locally is in fact a global minimum (in the bipartite case) and a minimum within a certain topological class of partitions in the non-bipartite case. Understanding this topological class of partitions requires the introduction of a modified Laplacian operator, or an equivalent Aharonov-Bohm Hamiltonian with singular magnetic potential. I will also explain how to construct energy-decreasing deformations of partitions that are critical but not minimal, giving insight into the geometric structure of the true minimum. This talk is based on joint work with Gregory Berkolaiko, Yaiza Canzani, Peter Kuchment and Jeremy Marzuola.<br> </p></div></div> </div> </div> <div class="two-col__second two-col__column"> <div class="field field--name-field-date field--type-daterange field--label-above"> <h2 class="field__label"> Date & Time </h2> <div class="field__items"> <div class="field__item">February 25, 2025 | 2:30pm – 3:30pm</div> </div> </div> <div class="field location"> <h3 class="field__label">Location</h3> Simonyi Hall 101 and Remote Access </div> <div class="field speaker"> <h3 class="field__label">Speakers</h3> <div class="field field--name-field-other-speaker field--type-entity-reference field--label-hidden"> <div class="field__items"> <div class="field__item">Graham Cox, Memorial University of Newfoundland</div> </div> </div> </div> <div class="field field--name-field-additional-info-url field--type-link field--label-above"> <h2 class="field__label"> Additional Info </h2> <div class="field__item"><a href="https://theias.zoom.us/j/85289078382?pwd=oC2AUoTvpaHkmvOfvqzfWSiPSQdPiR.1">https://theias.zoom.us/j/85289078382?pwd=oC2AUoTvpaHkmvOfvqzfWSiPSQdPiR.1</a></div> </div> <div class="field field--name-field-event-series field--type-entity-reference field--label-above"> <h2 class="field__label"> Event Series </h2> <div class="field__item"><a href="/event-series/analysis-and-mathematical-physics" hreflang="en">Analysis and Mathematical Physics</a></div> </div> <div class="field field--name-field-event-categories field--type-entity-reference field--label-above"> <h2 class="field__label"> Categories </h2> <div class="field__items"> <div class="field__item"><a href="/calendar?other_cat%5B251%5D=251" hreflang="en">Mathematics</a></div> <div class="field__item"><a href="/calendar?other_cat%5B17366%5D=17366" hreflang="en">Seminar</a></div> <div class="field__item"><a href="/calendar?cat%5B296%5D=296" hreflang="en">Academic</a></div> </div> </div> </div> </div> <div class="two-col__bottom"> 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