Indiana University-Purdue University Indianapolis (IUPUI) / We study the Gaudin model associated to Lie algebras of classical types. First, we derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple. Second, we define a new stratification of the Grassmannian of N planes. We introduce a new subvariety of Grassmannian, called self-dual Grassmannian, using the connections between self-dual spaces and Gaudin model associated to Lie algebras of types B and C. Then we obtain a stratification of self-dual Grassmannian.

In this thesis we examine singularities of surfaces and affine Schubert varieties in the affine Grassmannian $mathcal{G}/mathcal{P}$ of type $A^{(1)}$, by considering the action of a particular torus $widehat{T}$ on $mathcal{G}/mathcal{P}$. Let $Sigma$ be an irreducible $widehat{T}$-stable surface in $mathcal{G}/mathcal{P}$ and let $u$ be an attractive $widehat{T}$-fixed point with $widehat{T}$-stable affine neighborhood $Sigma_u$. We give a description of the $widehat{T}$-weights of the tangent space $T_u(Sigma)$ of $Sigma$ at $u$, give some conditions under which $Sigma$ is nonsingular at $u$, and provide some explicit criteria for $Sigma_u$ to be normal in terms of the weights of $T_u(Sigma)$. We will also prove a conjecture regarding the singular locus of an affine Schubert variety in $mathcal{G}/mathcal{P}$. / Mathematics

This thesis discusses intersections of the Schubert varieties in the flag variety associated to a vector space of dimension n. The Schubert number is the number of irreducible components of an intersection of Schubert varieties. Our main result gives the lower bound on the maximum of Schubert numbers. This lower bound varies quadratically with n. The known lower bound varied only linearly with n. We also establish a few technical results of independent interest in the combinatorics of strong Bruhat orders.

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<div>We study the Gaudin model associated to Lie algebras of classical types.</div><div><br></div><div>First, we derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple.</div><div><br></div><div>Second, we define a new stratification of the Grassmannian of N planes. We introduce a new subvariety of Grassmannian, called self-dual Grassmannian, using the connections between self-dual spaces and Gaudin model associated to Lie algebras of types B and C. Then we obtain a stratification of self-dual Grassmannian. </div>