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", to appear on
Advances in Nonlinear Analysis- "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)

- "Existence of traveling-waves solutions to the half-wave equation and orbital stability of the ground state" (with Vladimir Georgiev)
- "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)

- "On the compactness of minimizing sequences of an energy functional arising from a system of non-linear Klein-Gordon equations” at the 2014 KMS Spring Meeting, Gangreung University (Gangreung), on April 25, 2014 (slides)
- ”Traveling wave solutions to the half-wave equation” at the 2013 KMS Fall Meeting, Seoul University (Seoul), on October 26, 2013 (slides)
- ”Standing-waves with a small energy/charge ratio” at the KSIAM 2013 Spring Conference, Yonsei University (Seoul), on May 28, 2013 (slides)
- ”Orbital Stability by means of symmetric rearrangement” at the 2013 Joint Mathematics Meetings, San Diego Convention Center (San Diego), on January 10, 2013
- ”Standing-waves solutions to a system of non-linear Klein-Gordon equations with a sub-critical growth non-linearity” at 2012 Joint Mathematics Meetings, Hynes Convention Center (Boston), on January 7, 2012
- ”On the spectral flow for Banach spaces” at the 2011 KMS Fall Meeting, Gyeongbuk National University (Daegu), on October 22, 2011 (slides)
- ”Orbitally stable coupled standing waves for a coupled non-linear Klein-Gordon equation” at The 4th MSJ-SI Nonlinear dynamics in Partial Differential Equations, Kyushu University (Fukuoka), on September 14, 2011 (slides)
- ”Lyapunov functions for standing-wave solutions to a coupled NLKG equation” at The 3rd Kyushu University-POSTECH Joint Workshop, POSTECH (Pohang), on June 17, 2011
- ”Orbitally stable coupled standing waves” at the Korean Mathematical Society conference, POSTECH, on October 22, 2010
- ”A splitting property for two variables functions” at the Young Researcher in Mathematics 2010, Center for Mathematical Sciences (Cambridge), on March 26, 2010
- ”Non-linear elliptic systems and stability” at the Joint Meeting KMS-AMS, Ewha University (Seoul), on December 19, 2009
- ”On the spectral flow for Banach spaces” at Beyond Part III, Center for Mathematical Sciences (Cambridge), on April 16, 2009

Teaching

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

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).