Abstract:
Let $kE$ denote the group algebra of an elementary abelian $p$-group of rank $r$ over an algebraically closed field of characteristic $p$. We investigate the functors $\mathcal{F}_i$ from $kE$-modules of constant Jordan type to vector bundles on $\mathbb{P}^{r-1}(k)$, constructed by Benson and Pevtsova. For a $kE$-module $M$ of constant Jordan type, we show that restricting the sheaf $\mathcal{F}_i(M)$ to a dimension $s-1$ linear subvariety of $\mathbb{P}^{r-1}(k)$ is equivalent to restricting $M$ along a corresponding rank $s$ shifted subgroup of $kE$ and then applying $\mathcal{F}_i$. In the case $r=2$, we examine the generic kernel filtration of $M$ in order to show that $\mathcal{F}_i(M)$ may be computed on certain subquotients of $M$ whose Loewy lengths are bounded in terms of $i$. More precise information is obtained by applying similar techniques to the $n$th power generic kernel filtration of $M$. The latter approach also allows us to generalise our results to higher ranks $r$.

Abstract:
We introduce the class of modules of constant Jordan type for a finite group scheme $G$ over a field $k$ of characteristic $p > 0$. This class is closed under taking direct sums, tensor products, duals, Heller shifts and direct summands, and includes endotrivial modules. It contains all modules in an Auslander-Reiten component which has at least one module in the class. Highly non-trivial examples are constructed using cohomological techniques. We offer conjectures suggesting that there are strong conditions on a partition to be the Jordan type associated to a module of constant Jordan type.

Abstract:
The theories of $\pi$-points and modules of constant Jordan type have been a topic of much recent interest in the field of finite group scheme representation theory. These theories allow for a finite group scheme module $M$ to be restricted down and considered as a module over a space of small subgroups whose representation theory is completely understood, but still provide powerful global information about the original representation of $M$. This paper provides an extension of these ideas and techniques to study finite dimensional supermodules over a classical Lie superalgebra $\mathfrak{g} = \mathfrak{g}_{\overline{0}} \oplus \mathfrak{g}_{\overline{1}}$. Definitions and examples of $\mathfrak{g}$-modules of constant super Jordan type are given along with proofs of some properties of these modules. Additionally, endotrivial modules (a specific case of modules of constant Jordan type) are studied. The case when $\mathfrak{g}$ is a detecting subalgebra, denoted $\mathfrak{f}_r$, of a stable Lie superalgebra is considered in detail and used to construct super vector bundles over projective space $\mathbb{P}^{r-1}$. Finally, a complete classification of supermodules of constant super Jordan type are given for $\mathfrak{f}_1 = \mathfrak{sl}(1|1)$.

Abstract:
Inspired by the work of Benson, Carlson, Friedlander, Pevtsova, and Suslin on modules of constant Jordan type for finite group schemes, we introduce in this paper the class of representations of constant Jordan type for an acyclic quiver $Q$. We do this by first assigning to an arbitrary finite-dimensional representation of $Q$ a sequence of coherent sheaves on moduli spaces of thin representations. Next, we show that our quiver representations of constant Jordan type are precisely those representations for which the corresponding sheaves are locally free. We also construct representations of constant Jordan type with desirable homological properties. Finally, we show that any element of $\mathbb{Z}^L$, where $L$ is the Loewy length of the path algebra of $Q$, can be realized as the Jordan type of a virtual representation of $Q$ of relative constant Jordan type.

Abstract:
We study (i) asymptotic behaviour of wild harmonic bundles, (ii) the relation between semisimple meromorphic flat connections and wild harmonic bundles, (iii) the relation between wild harmonic bundles and polarized wild pure twistor $D$-modules. As an application, we show the hard Lefschetz theorem for algebraic semisimple holonomic $D$-modules, conjectured by M. Kashiwara.

Abstract:
To study the Lawson-Osserman's counterexample to the Bernstein problem for minimal submanifolds of higher codimension, a new geometric concept, submanifolds in Euclidean space with constant Jordan angles(CJA), is introduced. By exploring the second fundamental form of submanifolds with CJA, we can characterize the Lawson-Osserman's cone from the viewpoint of Jordan angles.

Abstract:
The aim of this paper is to initiate a study of the jet bundles on the grassmannian $X$ over a field of characteristic zero using higher direct images of $G$-linearized sheaves, Lie theoretic methods, enveloping algebra theoretic methods and generalized Verma modules. We calculate the $P$-module of the dual jet bundle $J^l(L)^*$ and prove it equals the $l$'th piece of the canonical filtration for $H^0(X,L)^*$. We use the results obtained to prove the discriminant of any linear system on any grassmannian is irreducible.

Abstract:
This survey presents some recent results of G.-M.Greuel and the author on vector bundles over algebraic curves and on Cohen-Macaulay modules over surface singularities. It is mainly devoted to the classification problems, especially to the tame-wild dichotomy and to the description of vector bundles and Cohen-Macaulay modules in the tame cases.

Abstract:
We show that the category of affine bundles over a smooth manifold M is equivalent to the category of affine spaces modelled on projective finitely generated C^\infty(M)-modules. Using this equivalence of categories, we are able to give an alternate proof of the main result of [13], showing that the characterization of vector bundles by means of their Lie algebras of homogeneous differential operators also holds for vector bundles of rank 1 and over any base manifolds.