Home Page of Daniele Garrisi
My current research is on non-linear PDEs, in particular on the orbital stability of standing-wave or traveling-wave solutions to several evolution equations (as the non-linear Schrödinger and the non-linear Klein-Gordon equations). My interests include problems and variational techniques strictly connected with the orbital stability, as the existence and the local uniqueness of dispersive equations, symmetries and multiplicities of solutions to non-linear elliptic equations, symmetric rearrangements and concentration-compactness theorems. My Ph.D. thesis is on the spectral flow for paths of bounded operators on a Banach space. In infinite-dimensional Hilbert spaces the spectral flow is a group isomorphism. The main result of this work is highlighting a natural connection between the spectral flow and properties of the Banach space environment. Such relation can be illustrated in the space product $ \small E\times E $: a loop of index $ k $ exists if and only if $ \small E $ is isomorphic to a closed subspace of co-dimension $ k $; if the linear group of $ \small E $ is simply connected, then the spectral flow is injective.
- "Standing-wave solutions to a system of non-linear Klein-Gordon equations with a small energy/charge ratio"
- "Orbitally stable standing-wave solutions to a coupled non-linear Klein-Gordon equation" (Proc. of the 4th MSJ-SI conference on Nonlinear Dynamics in PDEs) to appear on Advanced Studies in Pure Mathematics, Vol. 64, 2013
- "On the orbital stability of standing-waves pair solutions of a coupled non-linear Klein-Gordon equation" on Advanced Nonlinear Studies, Vol. 12 N. 3, 639-658, August 2012 (arXiv:1009.2281, pdf )
- "On the spectral flow for continuous path of essentially hyperbolic operators on a Banach space" on Topological Methods in Nonlinear Analysis, Vol. 36 N. 2, 353-381, December 2010 (arXiv:0806.4094)
- "On the connected components of the space of projectors" (arXiv:0809.1176)
- "Ordinary differential equations in Banach spaces and the spectral flow" (Ph.D. dissertation, arXiv:0803.1685)
Since Spring 2012 I am working as a Teaching Professor at the College of Mathematics Education of Inha University (Incheon, South Korea) where I am teaching two classes a semester
On Spring 2014, I am teaching Set Theory and Measure Theory
Fall 2013: Set Theory and Applied Analysis
Spring 2013: Ordinary Differential Equations
Fall 2012: Set Theory (EP135) and Applied Analysis (EP226)
Spring 2012: Ordinary Differential Equations (EP128) and Measure Theory (EP425) .
From Spring 2007 to Fall 2008 I worked as a Teaching Assistant at Università degli Studi di Pisa. All of these were Italian language lectures
Fall 2008: Analisi Matematica III at the Faculty of Physics (with Professor Carlo Viola), Matematica and Matematica con elementi di statistica medica at the Faculty of Pharmacy (with Professor Adele Manes)
Fall 2007 and Spring 2008: Matematica II at the Faculty of Engineering (with Professor Lina Conti). This course lasted one academic year
Fall 2007: Matematica and Matematica con elementi di statistica medica at the Faculty of Pharmacy (with Professor Adele Manes)
Spring 2007: Matematica II at the Faculty of Engineering (with Professor Lina Conti).