A considerable amount of evidence has shown that for every prime p &neq; N observed, a simultaneous eigenvector v_0 of Hecke operators T(l,i), i=1,2, in H^3(Γ_0(N),F(0,0,0)) has a “lift” v in H^3(Γ_0(N),F(p−1,0,0)) — i.e., a simultaneous eigenvector v of Hecke operators having the same system of eigenvalues that v_0 has. For each prime p>3 and N=11 and 17, we construct a vector v that is in the cohomology group H^3(Γ_0(N),F(p−1,0,0)). This is the first construction of an element of infinitely many different cohomology groups, other than modulo p reductions of characteristic zero objects. We proceed to show that v is an eigenvector of the Hecke operators T(2,1) and T(2,2) for p>3. Furthermore, we demonstrate that in many cases, v is a simultaneous eigenvector of all the Hecke operators.

In the 1970s and 1980s, Jean-Pierre Serre formulated a conjecture connecting two-dimensional Galois representations and modular forms. The conjecture came to be known as Serre's modularity conjecture. It was recently proved by Khare and Wintenberger in 2008. Serre's conjecture has various important consequences in number theory. Most notably, it played a key role in the proof of Fermat's last theorem. A natural question is, what is the analogue of Serre's conjecture for higher dimensional Galois representations? In 2002, Ash, Doud and Pollack formulated a precise statement for a higher dimensional analogue of Serre's conjecture. They also provided numerous computational examples as evidence for this generalized conjecture. We consider the three-dimensional version of the Ash-Doud-Pollack conjecture. We find specific examples of three-dimensional Galois representations and computationally verify the generalized conjecture in all these examples.

FJRW-theory is a recent advancement in singularity theory arising from physics. The FJRW-theory is a graded vector space constructed from a quasihomogeneous weighted polynomial and symmetry group, but it has been conjectured that the theory only depends on the weights of the polynomial and the group. In this thesis, I prove this conjecture using Poincaré polynomials and Koszul complexes. By constructing the Koszul complex of the state space, we have found an expression for the Poincaré polynomial of the state space for a given polynomial and associated group. This Poincaré polynomial is defined over the representation ring of a group in order for us to take G-invariants. It turns out that the construction of the Koszul complex is independent of the choice of polynomial, which proves our conjecture that two different polynomials with the same weights will have isomorphic FJRW rings as long as the associated groups are the same.

The goal of this thesis is to make Fröberg's conjecture more accessible to the average math graduate student by building up the necessary background material to understand specific examples where Fröberg's conjecture is true.

Hadwiger's Conjecture from 1943 states that every graph with no Kt minor is (t-1)-colorable; it remains wide open for t ≥ 7. For positive integers t and s, let Kt-s denote the family of graphs obtained from the complete graph Kt by removing s edges. We say that a graph has no Kt-s minor if it has no H minor for every H in Kt-s. In 1971, Jakobsen proved that every graph with no K7-2 minor is 6-colorable. In this dissertation, we first study the extremal functions for K8-4 minors, K9-6 minors, and K10-12 minors. We show that every graph on n ≥ 9 vertices with at least 4.5n-12 edges has a K8-4 minor, every graph on n ≥ 9 vertices with at least 5n-14 edges has a K9-6 minor, and every graph on n ≥ 10 vertices with at least 5.5n-17.5 edges has a K10-12 minor. We then prove that every graph with no K8-4 minor is 7-colorable, every graph with no K9-6 minor is 8-colorable, and every graph with no K10-12 minor is 9-colorable. The proofs use the extremal functions as well as generalized Kempe chains of contraction-critical graphs obtained by Rolek and Song and a method for finding minors from three different clique subgraphs, originally developed by Robertson, Seymour, and Thomas in 1993 to prove Hadwiger's Conjecture for t = 6. Our main results imply that H-Hadwiger's Conjecture is true for each graph H on 8 vertices that is a subgraph of every graph in K8-4, each graph H on 9 vertices that is a subgraph of every graph in K9-6, and each graph H on 10 vertices that is a subgraph of every graph in K10-12.

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Ce travail comporte deux parties dont la première concerne l'ensemble asymptotique $S_F$ d'une application polynomiale $F: C^n to C^n$. Dans les année 90s, Jelonek a montré que cet ensemble est une variété algébrique complexe singulière de dimension (complexe) $n-1$. Nous donnons une méthode, appelée {it méthode des fa{c c}ons}, pour stratifier cet ensemble. Nous obtenons une stratification de Thom-Mather. Par ailleurs, il existe une stratification de Whitney de $S_F$ telle que l'ensemble des fa{c c}ons possibles soit constant sur chaque strate. En utilisant les fa{c c}ons, nous donnons un algorithme pour expliciter l'ensemble asymptotique d'une application quadratique dominante en trois variables. Nous obtenons aussi une liste des ensembles asymptotiques possibles dans ce cas. La deuxième partie concerne l'ensemble $V_F$ : En 2010, Anna et Guillaume Valette ont construit une pseudo-variété réelle $V_F subset R^{2n + p}$, où $p > 0$, associée à une application polynomiale $F: C^n to C^n$. Dans le cas $n = 2$, ils ont prouvé que si $F$ est une application polynomiale de déterminant jacobien partout non nul, alors $F$ n'est pas propre si et seulement si l'homologie d'intersection de $V_F$ n'est pas triviale en dimension 2. Nous donnons une généralisation de ce résultat, dans le cas d'une application polynomiale $F : C^n to C^n$ de jacobien partout non nul. Nous donnons aussi une méthode pour stratifier l'ensemble $V_F$. Comme applications, nous obtenons des stratifications de l'ensemble des valeurs critiques asymptotiques de $F$ et de l'ensemble des points de bifurcation de $F$. / There are two parts in the present work. The first part concerns the asymptotic set of a polynomial mapping $F: C^n to C^n$. In the 90s, Zbigniew Jelonek showed that this set is a $(n-1)$ - (complex) dimensional singular variety. We give a method, called {it m'ethode des fa{c c}ons}, for stratifying this set. We obtain a Thom-Mather stratification. Moreover, there exists a Whitney stratification such that the set of possible fa{c c}ons is constant on every stratum. By using the fa{c c}ons, we give an algorithm for expliciting the asymptotic sets of a dominant quadratic polynomial mapping in three variables. As a result, we have a complete list of the asymptotic sets in this case. The second part concerns the set called Valette set $V_F$. In 2010, Anna and Guillaume Valette constructed a real pseudomanifold $V_F subset R^{2n + p}$, where $p > 0$, associated to a polynomial mapping $F: C^n to C^n$. In the case $n = 2$, they proved that if $F$ is a polynomial mapping with nowhere vanishing Jacobian, then $F$ is not proper if and only if the homology (or intersection homology) of $V_F$ is not trivial in dimension 2. We give a generalization of this result, in the case of a polynomial mapping $F : C^n to C^n$ with nowhere vanishing Jacobian. We give also a method for stratifying the set $V_F$. As applications, we have the stratifications of the set of asymptotic critical values of $F$ and the set of bifurcation points of $F$.

Na presente dissertação, discutimos o Problema do Isomorfismo em anéis de grupo para grupos infinitos da forma G × C, apresentado no artigo de Mazur [14], que enuncia um teorema mostrando a equivalência para o Problema do Isomorfismo entre essa classe de grupos infinitos e grupos finitos que satisfaçam a Conjectura do Normalizador. Nossa ênfase concentra-se na relação entre a Conjectura do Isomorfismo e a Conjectura do Normalizador, primeiramente, observada nesse artigo. Em seguida, consideramos um teorema de estrutura para as unidades centrais em anéis de grupo comunicado, pela primeira vez, no artigo de Jespers-Parmenter-Sehgal [9], e generalizado por Polcino Milies-Sehgal em [17], e Jespers-Juriaans em [7]. Evidenciamos a importância desse teorema para a Teoria de Anéis de Grupo e apresentamos uma nova demonstração para o teorema de equivalência de Mazur, considerando, para tanto, uma apropriada unidade central e sua estrutura, caracterizada pelo teorema comunicado para as unidades centrais. Concluímos a dissertação, descrevendo a construção do grupo das unidades centrais para o anel de grupo ZA5 , um grupo livre finitamente gerado de posto 1, utilizando a construção dada no artigo de Aleev [1]. / In this dissertation, we discuss the Problem of the Isomorphism in group rings for infinite groups as G × C. This is presented in [14]. Such article states a theorem which shows an equivalence to the isomorphism problem between that infinite class group and finite groups verifying the Normalizer Conjecture. Our main purpose is the Normalizer Conjecture and the Isomorphism Conjecture relationship remarked in the cited article to the groups above. Following, we consider a group ring theorem to the central units subgroup firstly communicated in [9] and generalized in [17] and [7]. We point up the importance of such theorem to the Group Ring Theory and we give a short and a new demonstration to Mazurs equivalence theorem from using a suitable central unit altogether with its structure lightly by the Central Unit Theorem on focus. We conclude this work sketching the ZA5 central units subgroup on showing it is a free finitely generated group of rank 1 from the presenting construction in Aleevs article [1].