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

Neste trabalho aprimorou-se um método para aproximar soluções de uma classe de equações diferenciais algébricas (DAEs), conhecida como sistemas semi-implícitos quadrados. O método, chamado aqui de MII, fundamenta-se na teoria geométrica de desacoplamento para sistemas não lineares, aliada a técnicas eficientes de análise numérica. Ele usa uma estratégia mista com cálculos simbólicos e numéricos para construir um sistema explícito, cujas soluções convergem exponencialmente para as soluções do sistema implícito original. Duas versões do método são apresentadas. Com a primeira, chamada de MIIcond, procura-se obter matrizes numericamente estáveis, através de balanceamentos. E a segunda, MIIproj, aproveita uma interpretação geométrica para o campo vetorial obtido. As implementações foram desenvolvidas em Matlab/simulink com o pacote de computação simbólica. Através dos benchmarks, realizando inclusive comparações com outros métodos atualmente disponíveis, constatou-se que o MIIcond foi inviável em alguns casos, devido ao tempo de processamento muito extenso. Por outro lado, o MIIproj mostrou-se uma boa alternativa para esta classe de problemas, em especial para sistemas de alto índex. / This work improves a method to approximate solutions for a class of differential algebraic equations (DAEs), known as systems semi-implicit square. The method, called here MII, is based on geometric theory of decoupling for nonlinear systems combined with efficient techniques numerical analysis. It uses an algorithum that mixes symbolic and numerical calculations to build an explicit system, whose solutions converge exponentially to solutions of the original implicit system. Two versions of the method are given. The first one is called MIIcond, trying to obtain numerically stable matrices through balancing. The second one is the MIIproj, taking advantage of a geometricinterpretation of the vector field there obtained. The implementations were developed in Matlab/Simulink with the symbolic toolbox. Through benchmarks, including performing comparisons with other methods currently available, it was found that the MIIcond was not feasible in some cases, due to processing time too long. On the other hand, the MIIproj presented itself as good alternative to this class of problems, especially for systems of high index.

Neste trabalho aprimorou-se um método para aproximar soluções de uma classe de equações diferenciais algébricas (DAEs), conhecida como sistemas semi-implícitos quadrados. O método, chamado aqui de MII, fundamenta-se na teoria geométrica de desacoplamento para sistemas não lineares, aliada a técnicas eficientes de análise numérica. Ele usa uma estratégia mista com cálculos simbólicos e numéricos para construir um sistema explícito, cujas soluções convergem exponencialmente para as soluções do sistema implícito original. Duas versões do método são apresentadas. Com a primeira, chamada de MIIcond, procura-se obter matrizes numericamente estáveis, através de balanceamentos. E a segunda, MIIproj, aproveita uma interpretação geométrica para o campo vetorial obtido. As implementações foram desenvolvidas em Matlab/simulink com o pacote de computação simbólica. Através dos benchmarks, realizando inclusive comparações com outros métodos atualmente disponíveis, constatou-se que o MIIcond foi inviável em alguns casos, devido ao tempo de processamento muito extenso. Por outro lado, o MIIproj mostrou-se uma boa alternativa para esta classe de problemas, em especial para sistemas de alto índex. / This work improves a method to approximate solutions for a class of differential algebraic equations (DAEs), known as systems semi-implicit square. The method, called here MII, is based on geometric theory of decoupling for nonlinear systems combined with efficient techniques numerical analysis. It uses an algorithum that mixes symbolic and numerical calculations to build an explicit system, whose solutions converge exponentially to solutions of the original implicit system. Two versions of the method are given. The first one is called MIIcond, trying to obtain numerically stable matrices through balancing. The second one is the MIIproj, taking advantage of a geometricinterpretation of the vector field there obtained. The implementations were developed in Matlab/Simulink with the symbolic toolbox. Through benchmarks, including performing comparisons with other methods currently available, it was found that the MIIcond was not feasible in some cases, due to processing time too long. On the other hand, the MIIproj presented itself as good alternative to this class of problems, especially for systems of high index.