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Cris Negron, UNC Chapel Hill
I will discuss the notion of a noncommutative complete intersection algebra (ncci), and relations with support theory. I will provide a number of examples of such ncci’s, and explain how they appear naturally in several Hopf algebraic contexts. For a number of Hopfy examples, I will explain how this notion can be leveraged to provide a classification thick ideals in the corresponding derived category of representations. More specifically, I will explain how a certain geometry which one associates to a ncci can be used to provide the proposed classification. This is joint work with Julia Pevtsova.
Elmar Schrohe, Leibniz University
Let $X$ be a manifold with boundary and bounded geometry. On $X$ we consider a uniformly strongly elliptic second order operator $A$ that locally is of the form
$A=-\sum_{j,k} a_{jk} \partial_{x_j}\partial_{x_k}+ \sum_{j} b_j\partial_{x_j} +c. $
$A$ is endowed with a boundary operator $T$ of the form
$T=\varphi_0\gamma_0 + \varphi_1\gamma_1,$
where $\gamma_0$ and $\gamma_1$ denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary, and $\varphi_0$, $\varphi_1$ are non-negative $C^\infty_b$ functions on the boundary with $\varphi_0+\varphi_1\ge c_0>0$. This problem is not elliptic in the sense of Lopatinskij and Shapiro, unless either $\varphi_1\not=0$ everywhere or $\varphi_1=0$ everywhere.
We show that the realization $A_T$ of $A$ in $L^p(\Omega)$ has a bounded $H^\infty$-calculus of arbitrarily small angle whenever the $a_{jk}$ are H\"older continuous and $b_j$ as well as $c$ are $L^\infty$.
For the proof we first treat the operator with smooth coefficients on $\mathbb R^n_+$. Here we rely on an extension of Boutet de Monvel's calculus to operator-valued symbols of H\"ormander type $(1,\delta)$. We then use $H^\infty$-perturbation techniques in order to treat the non-smooth case.
The existence of a bounded $H^\infty$-calculus allows us to apply maximal regularity techniques. We show how a theorem of Cl\'ement and Li can be used to establish the existence of a short time solution to the porous medium equation on $X$ with boundary condition $T$.
(Joint work with Thorben Krietenstein, Hannover)
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