This thesis consists of two parts. The first half concerns various foundational aspects ofthe theory of enriched ∞-categories. We develop the theory of adjunctions and weighted limits and colimits in enriched ∞-categories. We introduce theories of enriched ∞-props and operads, which provide a framework for the study of higher algebra in the enriched context. Finally, we study the theory of monads and monadic adjunctions in enriched (∞, 2)-categories, and prove an enriched generalization of the Barr-Beck-Lurie monadicity theorem. The second half of this thesis applies the results of the first half to the study of higher categorical sheaf theory in derived algebraic geometry. We introduce and study a theory of quasicoherent sheaves of presentable stable (∞, n)-categories on prestacks, generalizing the case n = 1 studied in [Gai15]. We prove a universal property for the (∞, n + 1)-category of correspondences, generalizing and providing a new approach to the case n = 1 from [GR17], and use it to show that our higher quasicoherent sheaves give rise to representations of the higher categories of correspondences of prestacks. We also introduce a notion of n-affineness for prestacks and provide a simple inductive criterion for checking n-affineness, which allows one to reduce affineness questions to the case n = 1 studied in [Gai15].

We study the algebraic geometry and combinatorics of the affine Grassmannian and affine flag variety, which are infinite-dimensional analogs of the ordinary Grassmannian and flag variety. In particular, we analyze the intersections of Iwahori orbits and semi-infinite orbits in the affine Grassmannian and affine flag variety. These intersections have interesting geometric and topological properties, and are related to representation theory.

Moreover, we study the central degeneration (the degeneration that shows up in local models of Shimura varieties and Gaitsgory's central sheaves) of semi-infinite orbits, Mirkovic-Vilonen (MV) Cycles, and Iwahori orbits in the affine Grassmannian of type A, by considering their moment polytopes. We describe the special fiber limits of semi-infinite orbits in the affine Grassmannian by studying the action of a global group scheme. Moreover, we give some bounds for the number of irreducible components for the special fiber limits of Iwahori orbits and MV cycles in the affine Grassmannian. Our results are connected to Gaitsgory's central sheaves, affine Schubert calculus and affine Deligne-Lusztig varieties in number theory.

In this thesis we discuss the theory of vector bundles with real structure on the projective line. This extends classical work by Grothendieck classifying complex vector bundles on the projective line. In particular, we show that vector bundles with real structure can be classified in terms of the coroot lattice of GL(n), similarly to the complex case. In addition, we provide a comparison of a certain K-group of sheaves on the moduli space of vector bundles to a K- group of sheaves on the moduli space of local systems, a kind of Langlands duality statement for real bundles, and give a uniformization of the moduli space.

We study various topics lying in the crossroads of symplectic topology and geometric representation theory, with an emphasis on understanding central objects in geometric representation theory via approaches using Lagrangian branes and symplectomorphism groups.

The first part of the dissertation focuses on a natural link between perverse sheaves and holomorphic Lagrangian branes.

For a compact complex manifold $X$, let $D_c^b(X)$ be the bounded derived category of constructible sheaves on $X$, and $Fuk(T^*X)$ be the Fukaya category of $T^*X$. A Lagrangian brane in $Fuk(T^*X)$ is holomorphic if the underlying Lagrangian submanifold is complex analytic in $T^*X_{\mathbb{C}}$, the holomorphic cotangent bundle of $X$. We prove that under the quasi-equivalence between $D^b_c(X)$ and $DFuk(T^*X)$ established by Nadler and Zaslow, holomorphic Lagrangian branes with appropriate grading correspond to perverse sheaves.

The second part is motivated from general features of the braid group actions on derived category of constructible sheaves. For a semisimple Lie group $G_\mathbb{C}$ over $\mathbb{C}$, we study the homotopy type of the symplectomorphism group of the cotangent bundle of the flag variety and its relation to the braid group. We prove a homotopy equivalence between the two groups in the case of $G_\mathbb{C}=SL_3(\mathbb{C})$, under the $SU(3)$-equivariancy condition on symplectomorphisms.

This thesis studies the fundamental automorphic function theory associated to a markedgenus zero curve over a finite field. Following insights from topological field theory, one expects this theory is deeply related to the unramified automorphic representation theory of general function fields. There are two main contributions. First I present an explicit description of the action of Hecke operators for the groups PGL2 and SL3. Second, I give a conjecture, along with some evidence, that characterizes the action of Hecke operators on Eisenstein series for any group.

We investigate a collection of posets- combinatorial arboreal singularities- which are the strata posets of the arboreal singularities constructed by David Nadler. Nadler demonstrated that any Lagrangian skeleton admits a non-characteristic deformation into a skeleton with only arboreal singularities, suggesting that arboreal singularities form the basis for a combinatorial theory of Lagrangian skeleta.

In this document, we introduce a form of combinatorial data called a `cyclic structure', which is essentially codimension-one data with compatibility conditions in codimensions two and three. We develop a comprehensive theory of isomorphisms of combinatorial arboreal singularities and cyclic arboreal singularities (singularities equipped with a cyclic structure). We show that a cyclic structure determines (up to quasi-equivalence) a sheaf of dg-categories on a combinatorial arboreal singularity, and investigate combinatorial properties of this sheaf. We describe a class of `arboreal moves', which are local mutations of combinatorial arboreal spaces preserving this sheaf of categories. Finally, we discuss how this combinatorial picture is related to the geometric understanding of Lagrangian embeddings of arboreal singularities.

Let $G$ be a connected reductive complex algebraic group, and $E$ a complex elliptic curve. Let $G_E$ denote the connected component of the trivial bundle in the stack of semistable $G$-bundles on $E$. We introduce a complex analytic uniformization of $G_E$ by adjoint quotients of reductive subgroups of the loop group of $G$. This can be viewed as a nonabelian version of the classical complex analytic uniformization $ E \simeq \mathbb{C}^*/q^{\mathbb{Z}}$. We similarly construct a complex analytic uniformization of $G$ itself via the exponential map, providing a nonabelian version of the standard isomorphism $\C^* \simeq \C/\Z$, and a complex analytic uniformization of $G_E$ generalizing the standard presentation $E = \C/(\Z \oplus \Z \tau )$. Finally, we apply these results to the study of sheaves with nilpotent singular support.

The space of principally regular matrices $\PR_n$ is the space of real symmetric matrices with every principal minor being non-zero. This space plays a vital role in algebraic geometry and algebraic statistics. On one side, it corresponds to the intersection of open cells in the Lagrangian Grassmannian $\text{LGr}(n,2n)$. On the other side, it has applications in conditional independence and determinantal point processes in statistics. Understanding the topology of $\PR_n$ will allow us to explore the new connections between geometry, statistics, and combinatorics. In this work, we discuss the topology of $\PR_n$ for small $n$ where every connected component is realized as an open cell in the complement to conics in $\R^2$ for $n=3$ and quadrics in $\R^3$ for $n=4$. Also, we show that the lower bound of the number of connected components of $\PR_n$ is given as the number of orientations on the uniform Oriented Lagrangian Matroid. Finally, we formulate our main conjecture that every connected component of $\PR_n$ is contractible.

Let k be an algebraically closed field and Z an effective Cartier divisor in the projective over k with complement U. When k = C, a local system on the analytification of U is said to be physically rigid when it is determined by the conjugacy classes of its monodromy operators around the points of Z. Katz proves a convenient cohomological characterization of irreducible physically rigid local systems. Roughly, it arises from the observation that irreducible physically rigid local systems are smooth isolated points in the moduli of local systems on U with fixed local monodromy data along Z.

In this dissertation, we consider the situation where char(k) > 0 and local systems are replaced with overconvergent isocrystals on U. The "moduli of overconvergent isocrystals" is an elusive object, but we establish some results about the formal deformation theory of overconvergent isocrystals with fixed "local monodromy" along Z. These results bear strong resemblances to facts about the infinitesimal structure of the moduli of local systems with fixed monodromy.

En route, we establish a general result which shows that a Hochschild cochain complex governs deformations of a module over an arbitrary associate algebra. We also relate this Hochschild cochain complex to a de Rham complex in order to understand the deformations of a differential module over a differential ring.

Mikio Sato’s fundamental idea of viewing objects, a priori defined on a space, as living on the cotangent bundle of that space led to the birth of the subject of microlocal analysis and spread to other fields of mathematics. It has been applied to and greatly enriched the theories of D-modules and constructible sheaves in the real or complex analytic context, with important applications to geometric representation theory and much more. In this dissertation, we study étale sheaves in positive characteristic from the microlocal point of view. The main results are: i) generically on a smooth surface, the vanishing cycle form a local system with respect to the variation of transverse test functions in high enough order terms; ii) the vanishing cycle of a tame simple normal crossing sheaf has the same stability as in the complex constructible case; iii) for a monodromic sheaf on a finite dimensional vector space, its characteristic cycle is canonically identified with that of the Fourier transform of the sheaf. In the Introduction, we also discuss the implications of these results in a broader context and an application of iii) to the study of character sheaves in positive characteristic.