{"id":376,"date":"2025-07-11T11:08:49","date_gmt":"2025-07-11T10:08:49","guid":{"rendered":"https:\/\/mat.uab.cat\/web\/tfg\/?p=376"},"modified":"2025-07-11T11:20:21","modified_gmt":"2025-07-11T10:20:21","slug":"analisi-harmonica-i-geometria-fractal","status":"publish","type":"post","link":"https:\/\/mat.uab.cat\/web\/tfg\/analisi-harmonica-i-geometria-fractal\/","title":{"rendered":"An\u00e0lisi harm\u00f2nica i geometria fractal"},"content":{"rendered":"\n<p>Una manera de generalitzar la noci\u00f3 de &#8220;dimensi\u00f3&#8221; als subconjunts de \\(\\mathbb{R}^n\\) que no s\u00f3n necess\u00e0riament subvarietats \u00e9s mitjan\u00e7ant la <em>dimensi\u00f3 de Hausdorff<\/em>. Els conjunts amb dimensi\u00f3 de Hausdorff no entera sovint es diuen <em>fractals<\/em>; un exemple cl\u00e0ssic \u00e9s el conjunt de Cantor a la l\u00ednia real. Determinar la dimensi\u00f3 de Hausdorff d&#8217;un conjunt pot ser desafiant en molts casos. Un m\u00e8tode clau per fer-ho implica <em>m\u00e8todes de transformada de Fourier<\/em>, i en particular l&#8217;estudi de la transformada de Fourier de mesures suportades en el fractal.<\/p>\n<p>Aquestes t\u00e8cniques es poden aplicar en diversos contextos geom\u00e8trics (per exemple, teoremes de projecci\u00f3 de tipus Marstrand, el problema del conjunt de dist\u00e0ncies de Falconer, problemes de tipus Kakeya). Un possible projecte de TFG podria implicar triar una aplicaci\u00f3 dels m\u00e8todes de transformada de Fourier en la geometria fractal i entendre-la.<\/p>\n<p><strong>Les\/els estudiants interessats en aquests temes poden contactar personalment amb <\/strong><strong>mi <\/strong>(carmelo.puliatti@uab.cat)<strong> i podem acordar un projecte despr\u00e9s d\u2019una reuni\u00f3.<\/strong><\/p>\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Una manera de generalitzar la noci\u00f3 de &#8220;dimensi\u00f3&#8221; als subconjunts de \\(\\mathbb{R}^n\\) que no s\u00f3n necess\u00e0riament subvarietats \u00e9s mitjan\u00e7ant la dimensi\u00f3 de Hausdorff. Els conjunts amb dimensi\u00f3 de Hausdorff no entera sovint es diuen fractals; un exemple cl\u00e0ssic \u00e9s el conjunt de Cantor a la l\u00ednia real. Determinar la dimensi\u00f3 de Hausdorff d&#8217;un conjunt pot [&hellip;]<\/p>\n","protected":false},"author":68,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29],"tags":[42],"class_list":["post-376","post","type-post","status-publish","format-standard","hentry","category-analisi-matematica","tag-carmelo-puliatti"],"_links":{"self":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/376","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/users\/68"}],"replies":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/comments?post=376"}],"version-history":[{"count":4,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/376\/revisions"}],"predecessor-version":[{"id":388,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/376\/revisions\/388"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/media?parent=376"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/categories?post=376"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/tags?post=376"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}