The present work is divided in two parts. The first is concerned with the dynamics on the Grassmann manifold of k-dimensional subvector spaces of an n dimensional real or complex vector space induced by a linear invertible transformation A of the vector space into itself. The Grassmann map GA sends p to Ap, and one asks, what are the dynamics of GA? In the second part, I consider dynamics induced by a linear cocycle covering a diffeomorphism of a compact manifold, acting on the Grassmann bundle of k-dimensional linear subspaces of TN. I prove a Kupka-Smale theorem for the space of cocycles covering diffeomorphisms of a compact manifold. The proof of this theorem implies the same type of results for derived cocycles parametrized in the space of diffeomorphisms. The results of the second part can be generalized without effort to cocycles covering endomorphisms of N.

The present work is divided in two parts. The first is concerned with the dynamics on the Grassmann manifold of k-dimensional subvector spaces of an n dimensional real or complex vector space induced by a linear invertible transformation A of the vector space into itself. The Grassmann map GA sends p to Ap, and one asks, what are the dynamics of GA? In the second part, I consider dynamics induced by a linear cocycle covering a diffeomorphism of a compact manifold, acting on the Grassmann bundle of k-dimensional linear subspaces of TN. I prove a Kupka-Smale theorem for the space of cocycles covering diffeomorphisms of a compact manifold. The proof of this theorem implies the same type of results for derived cocycles parametrized in the space of diffeomorphisms. The results of the second part can be generalized without effort to cocycles covering endomorphisms of N.

We compute the RO(C2)-graded Bredon cohomology of certain families of real and complex C2-equivariant Grassmannians.

Dimensionality reduction methods allow for the study of high-dimensional systems by producing low-dimensional descriptions that preserve the relevant structure and features of interest. For dynamical systems, attractors are particularly important examples of such features, as they govern the long-term dynamics of the system, and are typically low-dimensional even if the state space is high- or infinite-dimensional. Methods for reduction need to be able to determine a suitable reduced state space in which to describe the attractor, and to produce a reduced description of the corresponding dynamics. In the presence of a parameter space, a system can possess a family of attractors. Parameters are important quantities that represent aspects of the physical system not directly modelled in the dynamics, and may take different values in different instances of the system. Therefore, including the parameter dependence in the reduced system is desirable, in order to capture the model's full range of behaviour. Existing methods typically involve algebraically manipulating the original differential equation, either by applying a projection, or by making local approximations around a fixed-point. In this work, we take more of a geometric approach, both for the reduction process and for determining the dynamics in the reduced space. For the reduction, we make use of an existing secant-based projection method, which has properties that make it well-suited to the reduction of attractors. We also regard the system to be a manifold and vector field, consider the attractor's normal and tangent spaces, and the derivatives of the vector field, in order to determine the desired properties of the reduced system. We introduce a secant culling procedure that allows for the number of secants to be greatly reduced in the case that the generating set explores a low-dimensional space. This reduces the computational cost of the secant-based method without sacrificing the detail captured in the data set. This makes it feasible to use secant-based methods with larger examples. We investigate a geometric formulation of the problem of dimensionality reduction of attractors, and identify and resolve the complications that arise. The benefit of this approach is that it is compatible with a wider range of examples than conventional approaches, particularly those with angular state variables. In turn this allows for application to non-autonomous systems with periodic time-dependence. We also adapt secant-based projection for use in this more general setting, which provides a concrete method of reduction. We then extend the geometric approach to include a parameter space, resulting in a family of vector fields and a corresponding family of attractors. Both the secant-based projection and the reproduction of dynamics are extended to produce a reduced model that correctly responds to the parameter dependence. The method is compatible with multiple parameters within a given region of parameter space. This is illustrated by a variety of examples.

Schubert calculus has become an indispensable tool for enumerative geometry. It concerns the multiplication of Schubert classes in the cohomology of flag varieties, and is typically conducted using algebraic combinatorics by way of a polynomial ring presentation of the cohomology ring. The polynomials that represent the Schubert classes are called Schubert polynomials. An ongoing project in Schubert calculus has been to provide geometric foundations for the combinatorics. An example is the recovery by Knutson and Miller of the Schubert polynomials for finite flag varieties as the equivariant cohomology classes of matrix Schubert varieties. The present thesis is the start of a project to recover Schubert polynomials for the Borel-Moore homology of the (special linear) affine Grassmannian by an analogous process. This requires finitizing an affine Schubert variety to produce a matrix affine Schubert variety. This involves a choice of ``window'', so one must then identify a class representative that is independent of this choice. Examples lead us to conjecture that this representative is a k-Schur function. Concluding the discussion is a preliminary investigation into the combinatorics of Gröbner degenerations of matrix affine Schubert varieties, which should lead to a combinatorial proof of the conjecture. / Ph. D.

In this thesis, we will study the smooth linear section of the Grassmannian Gr(2, n). Explicitly, we give a criterion for the rationality of such linear section in terms of its codimension in the Plü ̈cker embedding in projective space. Moreover, to obtain a better understanding of the birational parametrization of these linear sections, we analyze their Hodge structures in the cases of even and odd codimensions. To be more precise, we provide numerous examples which suggest certain patterns of Hodge diamonds corresponding to even and odd cases and derive the proof of general patterns for codimension 3 smooth linear section of Gr(2, n) corresponding to odd and even n. / page 29 is missing from hardcopy

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

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Doctor of Philosophy / Department of Mathematics / Yan Soibelman / Given a quiver Q with/without potential, one can construct an algebra structure on the cohomology of the moduli stacks of representations of Q. The algebra is called Cohomological Hall algebra (COHA for short). One can also add a framed structure to quiver Q, and discuss the moduli space of the stable framed representations of Q. Through these geometric constructions, one can construct two representations of Cohomological Hall algebra of Q over the cohomology of moduli spaces of stable framed representations. One would get the double of the representations of Cohomological Hall algebras by putting these two representations together. This double construction implies that there are some relations between Cohomological Hall algebras and some other algebras. In this dissertation, we focus on the quiver without potential case. We first define Cohomological Hall algebras, and then the above construction is stated under some assumptions. We computed two examples in detail: A₁-quiver and Jordan quiver. It turns out that A₁-COHA and its double representations are related to the half infinite Clifford algebra, and Jordan-COHA and its double representations are related to the infinite Heisenberg algebra. Then by the fact that the underlying vector spaces of these two COHAs are isomorphic to each other, we get a COHA version of Boson-Fermion correspondence.