The Galois group of a Schubert problem is a subtle invariant that encodes intrinsic structure of its set of solutions. These geometric invariants are difficult to determine in general. However, based on a special position argument due to Schubert and a combinatorial criterion due to Vakil, we show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. The result follows from a particular inequality of Schubert intersection numbers which are Kostka numbers of two-rowed tableaux. In most cases, the inequality follows from a combinatorial injection. For the remaining cases, we use that these Kostka numbers appear in the tensor product decomposition of sl2C-modules. Interpreting the tensor product as the action of certain Toeplitz matrices and using spectral analysis, the inequality can be rewritten as an integral. We establish the inequality by estimating this integral using only elementary Calculus.

We construct a strong filtered graph $\Gamma_s(\Lambda)$ dependent on the dominant weight $\Lambda$, and a weak filtered graph $\Gamma_w(\Kcen)$ dependent on the canonical central element $\Kcen$ for an arbitrary Kac-Moody algebra $g$. In our construction, both graphs $(\Gamma_s(\Lambda), \Gamma_w(\Kcen))$ have the vertex set as the Weyl group of $g$, with the grading given by the length function. The edges of the graph $\Gamma_s(\La)$ are labeled versions of the $\lambda$-chain model of K-Chevalley rules for Kac-Moody flag manifolds as developed by Lenart and Shimozono, originally defined by Lenart and Postnikov. Meanwhile, the labels on $\Gamma_w(\Kcen)$ come from the dual multiplication map of K-cohomology of affine Grassmannian $Gr_G$. We conjecture that the strong filtered graph and weak filtered graph are dual, which means we get an identity when we apply the up and down operators on the vertices. We proved this identity except one case that where we call the chain is $j$-present. Our identity is similar to the Möbius construction of the dual filtered graph, as previously studied by Patrias and Pylyavskyy, and in fact, in the limit $n\rightarrow \infty$ of the $A^{(1)}_{n-1}$, our construction recovers their identity. We also expect to recover their combinatorics of Möbius deformation of the shifted Young's lattice in type $C^{(1)}_n$ as $n$ approaches infinity. / Doctor of Philosophy / In this thesis, we introduce a pair of graphs $(\Gamma_s(\La),\Gamma_w(\Kcen))$ motivated by the study of affine Schubert calculus. Affine Schubert calculus emerges as an extension and generalization of classical Schubert calculus, which involves questions such as determining the number of lines intersecting four lines in three-dimensional space. This type of questions can often be translated into computations aimed at finding the structure constants for the Schubert basis in the K-(co)homology of the flag varieties such as affine Grassmannian. These structure constants represent the coefficients of the Schubert basis in the product of the other two Schubert bases, all indexed by the Weyl group of the affine Lie algebra $g$. We define up and down operators on the vertices of graphs $(\Gamma_s(\Lambda), \Gamma_w(\Kcen))$, which are elements in Weyl group of $g$, utilizing the structure constants as essential components. We conjecture that, in general, and prove in certain cases, this approach yields new identities for these operators, leading us to define this pair of graphs as a dual filtered graph.

The Mukhin-Tarasov-Varchenko Theorem (previously the Shapiro Conjecture) asserts that a Schubert problem has all solutions distinct and real if the Schubert varieties involved osculate a rational normal curve at real points. When conjectured, it sparked interest in real osculating Schubert calculus, and computations played a large role in developing the surrounding theory. Our purpose is to uncover generalizations of the Mukhin-Tarasov-Varchenko Theorem, proving them when possible. We also improve the state of the art of computationally solving Schubert problems, allowing us to more effectively study ill-understood phenomena in Schubert calculus. We use supercomputers to methodically solve real osculating instances of Schubert problems. By studying over 300 million instances of over 700 Schubert problems, we amass data significant enough to reveal generalizations of the Mukhin-Tarasov- Varchenko Theorem and compelling enough to support our conjectures. Combining algebraic geometry and combinatorics, we prove some of these conjectures. To improve the efficiency of solving Schubert problems, we reformulate an instance of a Schubert problem as the solution set to a square system of equations in a higher- dimensional space. During our investigation, we found the number of real solutions to an instance of a symmetrically defined Schubert problem is congruent modulo four to the number of complex solutions. We proved this congruence, giving a generalization of the Mukhin-Tarasov-Varchenko Theorem and a new invariant in enumerative real algebraic geometry. We also discovered a family of Schubert problems whose number of real solutions to a real osculating instance has a lower bound depending only on the number of defining flags with real osculation points. We conclude that our method of computational investigation is effective for uncovering phenomena in enumerative real algebraic geometry. Furthermore, we point out that our square formulation for instances of Schubert problems may facilitate future experimentation by allowing one to solve instances using certifiable numerical methods in lieu of more computationally complex symbolic methods. Additionally, the methods we use for proving the congruence modulo four and for producing an

Made available in DSpace on 2015-05-15T11:45:59Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 941118 bytes, checksum: f8ae9b800c3284a22de7188884029167 (MD5) Previous issue date: 2011-05-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this dissertation we expose the solve the four line problem in Schubert Calculus using the Plucker embedding, giving emphasis to the study of the relative position of the four given lines in P3, this allows us to obtain an explicit description of the solution's set as well as to give the precise meaning to the notion of general position. In chapter 1, we insert the notion of projective space and other related, which are the basic notions for addressing the problem that we treat. In chapter 2, we introduce the Plucker embedding, !, which allows us to identify the set of lines that meet a xed given line l0 with the intersection of the Plucker's quadric, Q, and the tangent space of Q at !(l0). We also give the description of all the linear varieties contained in the Plucker's quadric Q. Finally, in chapter 3 we demonstrate the Theorem 3.0.3 which is a key ingredient to and solutions for our problem. Moreover, we establish a relationship between the relative position of the four given lines and their solution's set. Finally, we conclude in the appendix with the Shapiro-Shapiro conjecture in the case of the four line problem in Schubert Calculus. / Neste trabalho expomos a resolução do problema das 4 retas do Cálculo de Schubert utilizando o mergulho de Plücker, com ênfase no estudo da posição relativa das 4 retas dadas em P3, o que nos permite obter uma descrição explícita do conjunto de soluções é dar sentido preciso à noção de posição geral. No capítulo 1 inserimos a noção de espaço projetivo e outras correlatas que servirão de base no estudo do problema a ser resolvido. No capítulo 2 introduzimos o Mergulho de Plücker, ω, o qual nos permite identificar o conjunto das retas que encontram uma reta fixa l0 com a interseção da quádrica de Plücker e o espaço tangente à mesma no ponto ω l0. Além disso damos a descrição das variedades lineares contidas na quádrica de Plücker. Porém, no capítulo 3 demonstramos o Teorema 3.0.3 que é a chave para resolução do nosso problema e fazemos a descrição do conjunto solução cada para posição relativa possível das 4 retas. Concluímos com um apêndice onde tratamos da conjectura de Shapiro-Shapiro no caso do problema das quatro retas do cálculo de Shubert.

The study of algebraic varieties originates from the study of smooth manifolds. One of the focal points is the theory of differential forms and de Rham cohomology. It’s algebraic counterparts are given by algebraic cycles and Chow groups. Linearizing and taking the pseudo-abelian envelope of the category of smooth projective varieties, one obtains the category of pure motives. In this thesis, we concentrate on studying the pure Chow motives of Severi-Brauer varieties. This has been a subject of intensive investigation for the past twenty years, with major contributions done by Karpenko, [Kar1], [Kar2], [Kar3], [Kar4]; Panin, [Pan1], [Pan2]; Brosnan, [Bro1], [Bro2]; Chernousov, Merkurjev, [Che1], [Che2]; Petrov, Semenov, Zainoulline, [Pet]; Calmès, [Cal]; Nikolenko, [Nik]; Nenashev, [Nen]; Smirnov, [Smi]; Auel, [Aue]; Krashen, [Kra]; and others. The main theorem of the thesis, presented in sections 4.3 and 4.4, extends the result of Zainoulline et al. in the paper [Cal] by providing new examples of motivic decompositions of generalized Severi-Brauer varieties.