We prove that the tree-like Deligne-Mumford operad is a homotopical model for the trivialization of the circle in the higher-genus framed little discs operad. Our proof is based on a geometric argument involving nodal annuli. We use as a model for the higher-genus framed little discs an operad of Riemann surfaces with analytically parametrized boundary. We develop the formalism of topological moduli problems as a framework to accommodate the orbifold nature of the Deligne-Mumford operad.

We construct a canonical chain of formality quasiisomorphisms for the operad of chains on framed little disks and the operad of chains on little disks. The construction is done in terms of logarithmic algebraic geometry and is remarkable for being rational (and indeed definable integrally) in de Rham cohomology.

We introduce framed formal curves, which are formal algebraic curves with boundary components parametrized by the punctured formal disk. We study the moduli space of nodal framed formal curves, which we endow with a logarithmic structure. We show that this moduli space is a smooth formal logarithmic stack. The remarkable property of our construction is that framed formal curves admit a natural operation of "gluing along the boundary" which works well in families and preserves smoothness (both in a formal and in a logarithmic sense), and this induces gluing maps on the level of moduli. Using moduli spaces of framed formal curves we enhance the operad E2 of little disks (as well as its cousin, the framed little disks operad) to a fully log motivic operad. We use this structure to obtain a purely algebro-geometric proof of the formality of chains on these classical operads (initially proven for little disks by Tamarkin using analytic methods). We also recover and extend the known Galois action on the l-adic cohomology of framed and unframed little disks, and on Drinfeld associators, and extend it to the action on integral chains of a larger group scheme: the logarithmic motivic Galois group. Our methods generalize to a higher genus context, giving new "motivic" enrichments (for example, action by the absolute Galois group and by the log motivic Galois group) on the operad of chains in the oriented geometric bordism operad of Ayala and Lurie, which encodes the algebraic structure on the Hochschild cochains of any fully dualizable DG category.

Given an elliptic curve E over a number fieldK, the l-torsion points E[l] of E define a Galois representation Gal(K/K) → GL2(Fl). A famous theorem of Serre [9] states that as long as E has no Complex Multiplication (CM), the map Gal(K/K)→ GL2(Fl) is surjective for all but finitely many l. We say that a prime number l is exceptional (relative to the pair (E,K)) if this map is not surjective. Here we give a new bound on the largest exceptional prime, as well as on the product of all exceptional primes of E. We show in particular that conditionally on the Generalized Riemann Hypothesis (GRH), the largest exceptional prime of an elliptic curve E without CM is no larger than a constant (depending on K) times logNE , where NE is the absolute value of the norm of the conductor. This answers affirmatively a question of Serre in [10].

It is expected that the periodic cyclic homology of a DG algebra over the field of complex numbers (and, more generally, the periodic cyclic homology of a DG category) carries a lot of additional structure similar to the mixed Hodge structure on the de Rham cohomology of algebraic varieties. Whereas a construction of such a structure seems to be out of reach at the moment its counterpart in finite characteristic is much better understood thanks to recent groundbreaking works of Kaledin. In particular, it is proven by Kaledin that under some assumptions on a DG algebra $A$ over a perfect field $k$ of characteristic $p$, a lifting of $A$ over the ring of second Witt vectors $W_2(k)$ specifies the structure of a Fontaine-Laffaille module on the periodic cyclic homology of $A$. The purpose of this paper is to develop a relative version of Kaledin's theory for DG algebras over a base $k$-algebra $R$ incorporating in the picture the Gauss-Manin connection on the relative periodic cyclic homology constructed by Getzler. Our main result asserts that, under some assumptions on $A$, the Gauss-Manin connection on its periodic cyclic homology can be recovered from the Hochschild homology of $A$ equipped with the action of the Kodaira-Spencer operator as the inverse Cartier transform (in the sense of Ogus-Vologodsky). As an application, we prove, using the reduction modulo $p$ technique, that, for a smooth and proper DG algebra over a complex punctured disk, the monodromy of the Gauss-Manin connection on its periodic cyclic homology is quasi-unipotent.

We write down a new "logarithmic" quasicoherent category $\operatorname{Qcoh}_{log}(U, X, D)$ attached to a smooth open algebraic variety $U$ with toroidal compactification $X$ and boundary divisor $D$. This is a (large) symmetric monoidal Abelian category, which we argue can be thought of as the categorical substrate for logarithmic Hodge theory of $U$. We show that its Hochschild homology theory coincides with the theory of log-forms on $X$ with logarithmic structure induced by $D$, and in particular, that the noncommutative Hodge-to de Rham sequence on $\operatorname{Qcoh}_{log}(U, X, D)$ recovers known log Hodge structure on the de Rham cohomology of the open variety $U$. As an application, we compute the Hochschild homology of the category of coherent sheaves on the infinite root stack of Talpo and Vistoli in the toroidal setting. We prove a derived invariance result for this theory: namely, that strictly toroidal changes of compactification do not change the derived category of $\operatorname{Qcoh}_{log}(U, X, D)$. The definition is motivated by the coherent object appearing in the author's microlocal mirror symmetry result [20]. In this paper, the first in a series, we work over an algebraically closed field of characteristic zero. The next installment will develop the characteristic p and mixed-characteristic theories.

We write down a new “logarithmic” quasicoherent category Qcohlog(U,X,D) attached to a smooth open algebraic variety U with toroidal compactification X and boundary divisor D. This is a (large) symmetric monoidal Abelian category, which we argue can be thought of as the categorical substrate for logarithmic Hodge theory of U . We show that its Hochschild homology theory coincides with the theory of log-forms on X with logarithmic structure induced by D, and in particular, that the noncommutative Hodge-to de Rham sequence on Qcohlog(U,X,D) recovers known log Hodge structure on the de Rham cohomology of the open variety U . As an application, we compute the Hochschild homology of the category of coherent sheaves on the infinite root stack of Talpo and Vistoli in the toroidal setting. We prove a derived invariance result for this theory: namely, that strictly toroidal changes of compactification do not change the derived category of Qcohlog(U,X,D). The definition is motivated by the coherent object appearing in the author’s microlocal mirror symmetry result [20]. In this paper, the first in a series, we work over an algebraically closed field of characteristic zero. The next installment will develop the characteristic p and mixed-characteristic theories.

Recall that homological mirror symmetry, as conjectured by Kontsevich [K], relates the derived category of coherent sheaves on a (nice) algebraic variety to the Fukaya category of the mirror symplectic manifold. Since Kontsevich’s formulation, many variations on the theme of homological mirror symmetry have emerged, in which the categories compared keep track of additional structures such as equivariance, potential functionals, or boundary conditions on one or both of the manifolds in the mirror pair. The sequence of papers [NZ], [N], [FLTZ1], [FLTZ2], [FLTZ3] and [T] works out such a (modified) mirror symmetry statement for toric varieties. The correspondence is broken up into two parts. The first step is Nadler’s result [N] (based on work with Zaslow [NZ]) that, for S a topological space, the (triangulated envelope of the) Fukaya category on the cotangent space T ∗S is equivalent to the derived category of complexes of (nice) constructible topological sheaves on S. The second part is Bondal’s observation [Bo] that, for X an n-dimensional toric variety, the derived category of perfect coherent complexes of sheaves on X embeds into the derived category of constructible sheaves on the topological torus (S). These two results then compose to give an embedding of triangulated categories D Perf(X)→ DFuk((S)). Treumann [T] (building up from joint work in [FLTZ1] and [FLTZ2], with Fang, Liu and Zaslow) further shows that the functor D Perf(X) → D Constr(X) factors through the full (triangulated) subcategory D ConstrΛ(X) of constructible complexes with a certain singular support condition (determined by the fan of X). On the Fukaya side, this corresponds (via the equivalence of [N]) to a certain asymptotic condition on the branes in the Fukaya category. All of this is surveyed (with another construction of the direct functor D Perf(X) → DFuk((S))) in [FLTZ3]. The functor D Perf(X) → D ConstrΛ((S)) is called the coherent-constructible correspondence by Treumann et al, and it is conjectured e.g. in [FLTZ1] that it is an equivalence: they prove that a related constructible model for torusequivariant sheaves on X is equivalent to the category of perfect sheaves on X. The paper [SS] by Sibilla and Scherotzke proves such a statement for certain special classes of toric varieties (including Fano surfaces). In the present paper we prove the coherent-constructible correspondence for arbitrary smooth toric

Let G be a split, semisimple p-adic group. We construct a derived localization functor Loc : Dbmfg D Sh from the compactified category of [BK2 associated to G to the category of equivariant sheaves on the Bruhat-Tits building whose stalks have finite-multiplicity isotypic components as representations of the stabilizer. Our construction is motivated by the "coherent-constructible correspondence" functor in toric mirror symmetry and a construction of [CCC]. We show that Loc has a number of useful properties, including the fact that the sections LIP, Loc(V) = V when V is an object of Smfg compactifying the finitely-generated representation V.We also construct a depth-< e "truncated" analogue Loc(e) which has finite-dimensional stalks, and satisfies the property RIP, Loc(e) (V) = V for any V of depth < e. We deduce that every finitely-generated representation of G has a bounded resolution by representations induced from finite-dimensional representations of compact open subgroups, and use this to write down a set of generators for the K-theory of G in terms of K-theory of its parahoric subgroups. Thesis Supervisor: Roman Bezrukavnikov Title: Professor of Mathematics

AbstractGiven an abelian variety $A$ of dimension $g$ over a number field $K$, and a prime $\ell $, the ${\ell }^{n} $-torsion points of $A$ give rise to a representation ${\rho }_{A, {\ell }^{n} } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ( \mathbb{Z} / {\ell }^{n} \mathbb{Z} )$. In particular, we get a mod-$\ell $representation ${\rho }_{A, \ell } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{F} }_{\ell } )$ and an $\ell $-adic representation ${\rho }_{A, {\ell }^{\infty } } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{Z} }_{\ell } )$. In this paper, we describe the possible determinants of subquotients of these two representations. These two lists turn out to be remarkably similar.Applying our results in dimension $g= 1$, we recover a generalized version of a theorem of Momose on isogeny characters of elliptic curves over number fields, and obtain, conditionally on the Generalized Riemann Hypothesis, a generalization of Mazur’s bound on rational isogenies of prime degree to number fields.

Given an elliptic curve E over a number field K, the ℓ‐torsion points E[ℓ] of E define a Galois representation Gal(K¯/K)→GL2(픽ℓ) . A famous theorem of Serre (Invent. Math. 15 (1972) 259–331) states that as long as E has no complex multiplication (CM), the map Gal(K¯/K)→GL2(픽ℓ) is surjective for all but finitely many ℓ.

A spherical code is a finite set of points on the surface of a sphere in n dimensional space. The spherical codes problem asks for the maximum number of points in a spherical code where the angle between any two points with respect to the center is at least α. A traditional approach to this problem is to define an energy function for a spherical code as the sum of the inverse distance power of every pair of points and to optimize for minimum energy. However, a method to globally minimize the energy function for any given parameters is unknown, especially for spherical codes in higher dimensions. In our work, we improve the optimization by imposing certain symmetry groups on the spherical codes, as most configurations are invariant under reflection groups of certain Euclidean lattices. Furthermore, we develop a novel algorithm that individually separates pairs of points by treating them as unit vectors and applying gradient flow on their dot products. Using this approach, we were able to reproduce configurations for many of the best known spherical codes found in literature and find new configurations for dimension 6.

These are lecture notes that arose from a representation theory course given by the first author to the remaining six authors in March 2004 within the framework of the Clay Mathematics Institute Research Academy for high school students, and its extended version given by the first author to MIT undergraduate math students in the Fall of 2008. The notes cover a number of standard topics in representation theory of groups, Lie algebras, and quivers, and contain many problems and exercises. They should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra, and may be used for an undergraduate or introductory graduate course in representation theory.