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| supervisor for each project. | supervisor for each project. | ||
| - | **- Phase 3** (June 20 to June 25, 2022): There will be a final one-week | + | **- Phase 3** (June 20 to June 24, 2022): There will be a final one-week |
| workshop in Salerno (Italy), where the project | workshop in Salerno (Italy), where the project | ||
| teams of Phase 2 will present their projects and additional lectures | teams of Phase 2 will present their projects and additional lectures | ||
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| ==== ISem Team 2021/2022 ==== | ==== ISem Team 2021/2022 ==== | ||
| - | === Virtual | + | === Virtual |
| * Angela Albanese | * Angela Albanese | ||
| Line 48: | Line 49: | ||
| * Davide Addona | * Davide Addona | ||
| + | * Loredana Caso | ||
| * Federica Gregorio | * Federica Gregorio | ||
| * Cristian Tacelli | * Cristian Tacelli | ||
| Line 56: | Line 58: | ||
| - | $$\sigma \left(T(t)\right)\setminus\{0\}=e^{t\sigma(A)}$$ | + | Starting from basic concepts of operator theory, the course concentrates on spectral theory for linear operators and strongly continuous semigroups on Hilbert and in general on Banach spaces. |
| + | |||
| + | The main task of the spectral theory of linear operators is the following: | ||
| + | |||
| + | Given a linear operator $A$ with domain $D(A)$ on a Banach space $X$, determine the set, called | ||
| + | //the resolvent set// | ||
| + | of $A$, | ||
| + | $$\rho(A): | ||
| + | its complement $\sigma(A)$ in $\mathbb C$ called //the spectrum// of $A$ and, consequently, | ||
| + | $(\lambda I-A)^{-1}$, if they exist. | ||
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| + | An important application of the spectral theory is the study of the long time behaviour | ||
| + | of the solution $u(t)=e^{tA}u_0$ to the initial value problem | ||
| + | $$ | ||
| + | \left\{\begin{array}{ll} | ||
| + | u'(t)=Au(t),\quad t\ge 0,\\ | ||
| + | u(0)=u_0\in D(A), | ||
| + | \end{array} | ||
| + | \right. | ||
| + | $$ | ||
| + | where the operators $e^{tA}$ form a so-called strongly continuous semigroup generated by $A$. | ||
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| + | |||
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| + | In this course we shall develop the basic tools from spectral theory for linear operators and apply them to strongly continuous semigroups. | ||
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| + | |||
| + | Topics to be covered include: | ||
| + | |||
| + | |||
| + | * Compact operators and Riesz-Schauder theory; | ||
| + | |||
| + | * Spectral representation theorem for bounded and unbounded operators; | ||
| + | |||
| + | * Strongly continuous and analytic semigroups; | ||
| + | |||
| + | * Spectral mapping theorems for semigroups; | ||
| + | |||
| + | * Asymptotic behaviour of strongly continuous and analytic semigroups; | ||
| + | |||
| + | * Long time behaviour of solutions to nonhomogeneous abstract Cauchy problems. | ||
| + | |||
| + | Some of these topics will be elaborated on in Phase 2, where the students will have the possibility to work on projects which are related to active research. | ||
| + | You can download the program and the poster here: | ||
| - | You can download the program here: | + | {{ :imagines: |
| - | {{ :imagines:isem25-program.pdf | ISem25 | + | {{ :imagines:poster.pdf | ISem25 |
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