Session: Variational Analysis: Theory and Applications II
Chair: Walaa Moursi
Cluster: Nonsmooth Optimization
Talk 1: Faces of quasidensity
Speaker: Stephen Simons
Abstract: Let E be a nonzero real Banach space. Quasidensity is a concept that can be applied to subsets of E x E^*. Every closed quasidense monotone set is maximally monotone, but there exist maximally monotone sets that are not quasidense. We give eight conditions equivalent to the statement that a closed monotone set be quasidense. The most significant of these is that it be of type (NI). The graph of a subdifferential is quasidense.
We discuss the Gossez extension of a quasidense maximally monotone set and give an example of a maximally monotone set of type (NI) whose Gossez extension is maximally monotone but not of type (NI).
The closed quasidense sets satisfy a sum theorem and a dual sum theorem. These ideas also lead to generalizations of Rockafellar's surjectivity theorem and the Brezis--Browder theorem on adjoint linear subspaces to nonreflexive Banach spaces. The main tool that we will use is the (possibly exact) inf-convolution of certain functions on E x E^* and E^* x E^**. We will also use the weighted parallelogram law.
Talk 2: TBA
Speaker: Jon Vanderwerff
Abstract: TBA
Talk 3: Maps between bornological spaces
Speaker: Gerald Beer
Abstract: Let $(X,\mathcal{B})$ and $(Y,\mathcal{C})$ be bornological spaces, that is, nonempty sets equipped with families of subsets that contain the singletons, that are hereditary, and that are stable under finite union. A map $f:X \rightarrow Y$ is called bornological if $\forall B \in \mathcal{B}$, we have $f(B) \in \mathcal{C}$; it is called coercive if $\forall C \in \mathcal{C}$, we have $f^{-1}(C) \in \mathcal{B}$. We introduce natural bornologies on the space of bornological maps (resp. coercive maps) from $X$ to $Y$. Membership of $E$ to our bornology on the bornological maps means that the functions belonging to $E$ have a common expansion modulus. In the first case, evaluation is a bornological map, but in the second case, evaluation need not be coercive.
Separately, we characterize those bornological spaces that are the bornological (resp. coercive) images of a given bornological space $(X,\mathcal{B})$.
