In modern algebraic geometry solutions of polynomial equations are studied from a qualitative point of view using highly sophisticated tools such as cohomology, \(D\)-modules and Hodge structures. The latter have been unified in Saito’s far-reaching theory of mixed Hodge modules, that has shown striking applications including vanishing theorems for cohomology. A mixed Hodge module can be seen as a special type of filtered \(D\)-module, which is an algebraic counterpart of a system of linear differential equations. We present the first algorithmic approach to Saito’s theory. To this end, we develop a Gröbner basis theory for a new class of algebras generalizing PBW-algebras. The category of mixed Hodge modules satisfies Grothendieck’s six-functor formalism. In part these functors rely on an additional natural filtration, the so-called \(V\)-filtration. A key result of this thesis is an algorithm to compute the \(V\)-filtration in the filtered setting. We derive from this algorithm methods for the computation of (extraordinary) direct image functors under open embeddings of complements of pure codimension one subvarieties. As side results we show how to compute vanishing and nearby cycle functors and a quasi-inverse of Kashiwara’s equivalence for mixed Hodge modules. Describing these functors in terms of local coordinates and taking local sections, we reduce the corresponding computations to algorithms over certain bifiltered algebras. It leads us to introduce the class of so-called PBW-reduction-algebras, a generalization of the class of PBW-algebras. We establish a comprehensive Gröbner basis framework for this generalization representing the involved filtrations by weight vectors.

Composite materials are used in many modern tools and engineering applications and consist of two or more materials that are intermixed. Features like inclusions in a matrix material are often very small compared to the overall structure. Volume elements that are characteristic for the microstructure can be simulated and their elastic properties are then used as a homogeneous material on the macroscopic scale. Moulinec and Suquet [2] solve the so-called Lippmann-Schwinger equation, a reformulation of the equations of elasticity in periodic homogenization, using truncated trigonometric polynomials on a tensor product grid as ansatz functions. In this thesis, we generalize their approach to anisotropic lattices and extend it to anisotropic translation invariant spaces. We discretize the partial differential equation on these spaces and prove the convergence rate. The speed of convergence depends on the smoothness of the coefficients and the regularity of the ansatz space. The spaces of translates unify the ansatz of Moulinec and Suquet with de la Vallée Poussin means and periodic Box splines, including the constant finite element discretization of Brisard and Dormieux [1]. For finely resolved images, sampling on a coarser lattice reduces the computational effort. We introduce mixing rules as the means to transfer fine-grid information to the smaller lattice. Finally, we show the effect of the anisotropic pattern, the space of translates, and the convergence of the method, and mixing rules on two- and three-dimensional examples. References [1] S. Brisard and L. Dormieux. “FFT-based methods for the mechanics of composites: A general variational framework”. In: Computational Materials Science 49.3 (2010), pp. 663–671. doi: 10.1016/j.commatsci.2010.06.009. [2] H. Moulinec and P. Suquet. “A numerical method for computing the overall response of nonlinear composites with complex microstructure”. In: Computer Methods in Applied Mechanics and Engineering 157.1-2 (1998), pp. 69–94. doi: 10.1016/s00457825(97)00218-1.

Using valuation theory we associate to a one-dimensional equidimensional semilocal Cohen-Macaulay ring \(R\) its semigroup of values, and to a fractional ideal of \(R\) we associate its value semigroup ideal. For a class of curve singularities (here called admissible rings) including algebroid curves the semigroups of values, respectively the value semigroup ideals, satisfy combinatorial properties defining good semigroups, respectively good semigroup ideals. Notably, the class of good semigroups strictly contains the class of value semigroups of admissible rings. On good semigroups we establish combinatorial versions of algebraic concepts on admissible rings which are compatible with their prototypes under taking values. Primarily we examine duality and quasihomogeneity. We give a definition for canonical semigroup ideals of good semigroups which characterizes canonical fractional ideals of an admissible ring in terms of their value semigroup ideals. Moreover, a canonical semigroup ideal induces a duality on the set of good semigroup ideals of a good semigroup. This duality is compatible with the Cohen-Macaulay duality on fractional ideals under taking values. The properties of the semigroup of values of a quasihomogeneous curve singularity lead to a notion of quasihomogeneity on good semigroups which is compatible with its algebraic prototype. We give a combinatorial criterion which allows to construct from a quasihomogeneous semigroup \(S\) a quasihomogeneous curve singularity having \(S\) as semigroup of values. As an application we use the semigroup of values to compute endomorphism rings of maximal ideals of algebroid curves. This yields an explicit description of the intermediate rings in an algorithmic normalization of plane central arrangements of smooth curves based on a criterion by Grauert and Remmert. Applying this result to hyperplane arrangements we determine the number of steps needed to compute the normalization of a the arrangement in terms of its Möbius function.

A fast numerical method for an advanced electro-chemo-mechanical model is developed which is able to capture phase separation processes in porous materials. This method is applied to simulate lithium-ion battery cells, where the complex microstructure of the electrodes is fully resolved. The intercalation of ions into the popular cathode material LFP leads to a separation into lithium-rich and lithium-poor phases. The large concentration gradients result in high mechanical stresses. A phase-field method applying the Cahn-Hilliard equation is used to describe the diffusion. For the sake of simplicity, the linear elastic case is considered. Numerical tests for fully resolved three-dimensional granular microstructures are discussed in detail.

In this thesis, we deal with the finite group of Lie type \(F_4(2^n)\). The aim is to find information on the \(l\)-decomposition numbers of \(F_4(2^n)\) on unipotent blocks for \(l\neq2\) and \(n\in \mathbb{N}\) arbitrary and on the irreducible characters of the Sylow \(2\)-subgroup of \(F_4(2^n)\). S. M. Goodwin, T. Le, K. Magaard and A. Paolini have found a parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), a Sylow \(2\)-subgroup of \(F_4(q)\), of \(F_4(p^n)\), \(p\) a prime, for the case \(p\neq2\). We managed to adapt their methods for the parametrization of the irreducible characters of the Sylow \(2\)-subgroup for the case \(p=2\) for the group \(F_4(q)\), \(q=p^n\). This gives a nearly complete parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), namely of all irreducible characters of \(U\) arising from so-called abelian cores. The general strategy we have applied to obtain information about the \(l\)-decomposition numbers on unipotent blocks is to induce characters of the unipotent subgroup \(U\) of \(F_4(q)\) and Harish-Chandra induce projective characters of proper Levi subgroups of \(F_4(q)\) to obtain projective characters of \(F_4(q)\). Via Brauer reciprocity, the multiplicities of the ordinary irreducible unipotent characters in these projective characters give us information on the \(l\)-decomposition numbers of the unipotent characters of \(F_4(q)\). Sadly, the projective characters of \(F_4(q)\) we obtained were not sufficient to give the shape of the entire decomposition matrix.