*Avviso Seminario di Matematica* Data:  Monday, September 26, 2022 Ora:   16:15 - 17:15 Luogo:  Aula Volterra, Normale Speaker:  Anna Weller                 University of Cologne Titolo: *A spectral Galerkin method for the solution of reaction-diffusion equations on metric graphs* Abstract: Weinvestigateaspectralsolutionapproachforreaction-diffusionequationsongraphs interpretedastopologicalspace(metricgraphs).Ofspecialinterestisthenumerical computationofeigenfunctionsofthenegativesecondorderderivativeactingoneach edge.Remarkably,itispossibletogiveanexplicitcharacterizationoftheseeigenfunctions andcorrespondingeigenvalues.Moreover,forequilateralgraphs,wewillshow howtoefficientlycomputeanarbitrarylowerpartofthespectrumusingaveryuseful relationshiptothegraphLaplacianmatrixoftheunderlyingcombinatorialgraph. Finally,wecanusethebasisofeigenfunctionsinaGalerkinapproachtosolvevarious PDEsonmetricgraphs,wereweherefocusonreaction-diffusionequations.This problemismotivatedbyarecentcollaborationwiththeInstituteofGeophysicsand MeteorologyoftheUniversityofCologneonthemodelingofproteindistributionin Alzheimer’sdiseasetogetherwiththeUniversityHospitalCologne1.Partofmynumerical resultsarejointworkwithProf.Dr.MarkAinsworth(BrownUniversity)and withChong-SonDroege(UniversityofCologne). References [1]M.Ainsworth,A.Weller,AspectralGalerkinmethodforthesolutionofpartialdifferential equationsonmetricgraphs,inpreparation. [2]M.Ainsworth,A.Weller,AspectralGalerkinmethodforthesolutionofreaction-diffusion equationsonmetricgraphs,OberwolfachReports,WorkshopReport36,2021. [3]A.Kunoth,A.Weller,T.Yilmaz,Acomputationalbrainspheremodelforthesimulationof Alzheimer’sDisease.SnapshotsofmodernmathematicsfromOberwolfach,inpreparation. [4]A.Weller,G.N.Bischof,P.Schl̈uter,N.Richter,J.Dronse,̈O.Onur,J.Kukolja,B.Neu- maier,A.Kunoth,Y.Shao,T.vanEimeren,A.Drzezga,Findingnewcommunities:A principleofneuronalnetworkreorganizationinAlzheimersdisease.BrainConnectivity, 11(3):225-238,2021.DOI:10.1089/brain.2020.0889 ThisworkwassupportedbytheExcellenceInitiativeoftheUniversityofCologne