On first sight, a comparison between restitution for Nazi victims in Germany West and East does not seem to leave ample space for interpretation: While the Federal Republic at least in principle accepted their obligation to compensate former Nazi victims and paid huge amounts for that purpose over the last 50 years, the GDR only offered elaborated social security for the tiny faction of Nazi victims who decided to live in the GDR after 1949. As a consequence, while restitution in the West has been a predominantly Jewish affair, restitution in the East was chiefly a communist matter. However, in my talk I will not focus on a comparison of material payments. Rather, I am interested in the different structure of the answers of two German societies to the same problem: the persecution and killing of millions of people by the Nazi regime. This implies three sets of questions. First: On which perception of the events between 1933 and 1945 were the respective attempts at rehabilitation and compensation for Nazi victims in the two German societies based? Second: What relation between former Nazi victims and German post war societies underpinned the respective attempts at restitution? And third: What consequences did German reunification have for this process?

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This thesis develops a new approach to computing the quantum cohomology of symplectic reductions of partial flag varieties $X$; such symplectic reductions are known as weight varieties. Motivated by a conjecture of Teleman \cite{Tel14}, we use a mirror family Landau-Ginzburg model $(M_{P},f_{P})$ of $X$ introduced by Rietsch \cite{Rie08} to give a conjectural explicit description of the quantum cohomology of weight varieties.

We specialise to the class of polygon spaces $\mcP_{r,n}$, these are symplectic reductions of the complex Grassmannian of $2$-planes $\Gr_{\C}(2,n)$ by the maximal torus action. Polygon spaces in low rank have been classified and the quantum cohomology of these varieties is known. As a result, we are able to verify our conjectural description explicitly.

In addition, we investigate the appearance of representation-theoretic combinatorial structures in the mirror symmetry of complete flag varieties. We show that, on the $B$-model side, the extended string cone $\underline{C}_{\rexi}$ introduced by Caldero \cite{Cal02} to define toric degenerations on the $A$-model can be recovered via a discretisation process known as tropicalisation. Specifically, using a non-standard parameterisation of $M_{B}$, tropicalisation recovers the precise inequalities defining $\underline{C}_{\rexi}$. This provides an explicit approach to results previously obtained by Berenstein-Kazhdan \cite{BK07}. We conclude with a description of a conjectural program relating these combinatorial structures on the $B$-model with hierarchies of integrable systems on the $A$-model.

We develop some aspects of the theory of D-modules on schemes and indschemes of pro-finite type. These notions are used to define D-modules on (algebraic) loop groups and, consequently, actions of loop groups on DG categories. We also extend the Fourier-Deligne transform to Tate vector spaces.

Let N be the maximal unipotent subgroup of a reductive group G. For a non-degenerate character c of N((t)), and a category C acted upon by N((t)), there are two possible notions of the category of (N((t)),c)-objects: the invariant category and the coinvariant category. These are the Whittaker categories of C, which are in general not equiva- lent.

However, there is always a natural functor T from the coinvariant category to the invariant category. We conjecture that T is an equivalence, provided that the N((t))-action on C is the restriction of a G((t))-action.

We prove this conjecture for G=GLn and show that the Whittaker categories can be obtained by taking invariants of C with respect to a very explicit pro-unipotent group subscheme (not indscheme) of G((t)).

For manifolds $M$ with a specific rational homotopy type, I study a non-commutative Landau-Ginzburg model whose underlying ring is the differential-graded algebra (dga) $B=C_*(\Omega(M))$, that is chains on the based loop space with Pontryagin product and with potential $W$ in $B$. For $M=\mathbb{C}P^{n_1}\times \mathbb{C}P^{n_2} \times \ldots \mathbb{C}P^{n_k}$ or $S^{n_1} \times S^{n_2} \times \ldots S^{n_k}$, we explain how the field theories we define have a Fukaya category interpretation.

ACCEM (Asociación Comisión Católica Española de Migración) es una organización muy interesante por sus esfuerzos de sensibilización en España. Ellos utilizan los programas *Refugiados en el cómic *y *Refugiados en el cine *para promover empatía e inclusión. En esta entrevista exploro su uso del cómic, el cine y la gastronomía para facilitar el encuentro entre inmigrantes y nativos. Además de estos temas, tocamos brevemente sobre la xenofobia y racismo en España y Europa para contextualizar la necesidad tener ONGs como ACCEM.

Moduli problems have become a central area of interest in a wide range of mathematical fields such as representation theory and topology but particularly in the geometries (differential, complex, symplectic, algebraic). In addition, studying moduli problems often requires utilizing tools from other mathematical fields and creates unexpected bridges within mathematics and between mathematics and other fields.

A notable example came in 1991 when the mathematical physicists Edward Witten made a conjecture connecting the partition function for quantum gravity in two dimensions with numbers associated to the cohomology of the moduli space of stable curves, a space that was already of independent interest to algebraic geometers.

We study a related moduli problem M_{G} of principal G-bundle on stable curves for G a simple algebraic group. A defect of M_{G} over singular curves is that it is not compact and thus more difficult to study. We focus specifically on nodal singularities and examine how to compactify M_{G} over nodal curves.

The approach we present relies on two main mathematical objects: the loop group and the wonderful compactification of a semisimple adjoint group. For an algebraic group G the loop group LG is the group of maps D^{x} → G where ^{x} is a punctured formal disk, see 2.2 for a precise definition. The connection between LG and M_{G} is that G-bundles can be described by transition functions and roughly speaking any such transition function comes from an element of LG.

The wonderful compactification is a particularly nice way of comapactifying a semi simple group. Then in a sentence, the aim this dissertation is to (1) extend the construction of the wonderful compactification for semi simple group to LG and (2) use this compactification to compactify M_{G} over nodal curves.

We give a brief introduction in Chapter 1. Chapter 2 addresses (1) and Chapter 3 addresses (2).

We begin in Chapter 2 with a discussion of the classical wonderful compactification of an adjoint group given by De Concini and Procesi in [DCP83]. Because the group LG is infinite dimensional many of the elements in De Concini and Procesi's construction do not immediately extend or have more than one possible generalization. The technical heart of the paper is developing the appropriate analogs of all the elements needed to make the construction possible for LG. Also building on work of Brion and Kumar we give an enhancement of the compactificaiton from schemes to stacks that we utilize in Chapter 3.

Chapter 3 returns to the problem of compactifying M_{G} over nodal curves. We begin by carefully studying the points in the boundary of the compactification of LG and relating them to moduli problems over nodal curves. The moduli problems that appear in this way are closely related to flag varieties for the loop group and can be identified as moduli of torsors for a particular group scheme determined by parabolic subgroups of the loop group.

We go on to show that the moduli problem of torsors on nodal curves is isomorphic to a moduli problem of G-bundles on *twisted* nodal curve; these are orbifolds that are isomorphic to the original nodal curve on the smooth locus. Finally, building on related work of Kausz [Kau00,Kau05a] and Thaddeus and Martens [MaT] and the results of Chapter 2 we introduce a larger moduli problem X_{G} of G bundles on twisted curves which compactifies M_{G}.

The strong Macdonald theorems state that, for L reductive and s an odd variable, the cohomology algebras H^{*}(L[z]/z^{N}) and H^{*}(L[z, s]) are freely generated, and describe the co-homological, s-, and z-degrees of the generators. The resulting identity for the z-weighted Euler characteristic is equivalent to Macdonald's constant term identity for a finite root system. The proof of the strong Macdonald theorems, due to Fishel, Grojnowski, and Teleman, uses a Laplacian calculation for the (continuous) cohomology of L[[z]] with coefficients in the symmetric algebra of the (continuous) dual of L[[z]].

Our main result is a generalization of this Laplacian calculation to the setting of a general parahoric p of a (possibly twisted) loop algebra g. As part of this result, we give a detailed exposition of one of the key ingredients in Fishel, Grojnowski, and Teleman's proof, a version of Nakano's identity for infinite-dimensional Lie algebras.

We apply this Laplacian result to prove new strong Macdonald theorems for H^{*}(p/z^{N}p) and H^{*}(p[s]), where p is a standard parahoric in a twisted loop algebra. We show that H(p/z^{N}p) contains a parabolic subalgebra of the coinvariant algebra of the fixed-point subgroup of the Weyl group of L, and thus is no longer free. We also prove a strong Macdonald theorem for H^{*}(b; S^{*} n^{*} ) and H(b /z^{N} n) when b and n are Iwahori and nilpotent subalgebras respectively of a twisted loop algebra. For each strong Macdonald theorem proved, taking z-weighted Euler characteristics gives an identity equivalent to Macdonald's constant term identity for the corresponding affine root system. As part of the proof, we study the regular adjoint orbits for the adjoint action of the twisted arc group associated to L, proving an analogue of the Kostant slice theorem.

Our Laplacian calculation can also be adapted to the case when g is a symmetrizable Kac-Moody algebra. In this case, the Laplacian calculation leads to a generalization of the Brylinski identity or affine Kac-Moody algebras. In the semisimple case, the Brylinski identity states that, at dominant weights, the q-analog of weight multiplicity is equal to the Poincare series of the principal nilpotent filtration of the weight space. This filtration is known as the Brylinski filtration. We show that this identity holds in the affine case, as long as the principal nilpotent filtration is replaced by the principal Heisenberg. We also give an example to show that the Poincare series of the principal nilpotent filtration is not always equal to the q-analog of weight multiplicity, and give some partial results for indefinite Kac-Moody algebras.