Algebraic Geometry and Number Theory Ical Atom

Injective modules are fundamental in homological algebra over rings. In this talk, we explain how to generalize this notion to higher algebra. The Bass-Papp theorem states that a ring is left noetherian if and only if an arbitrary direct sum of left injective modules is injective. We will explain a version of this result in higher algebra, which will lead us to the notion of a left noetherian derived ring. In the final part of the talk, we will specialize to commutative noetherian rings in higher algebra, show that the Matlis structure theorem of injective modules holds in this setting, and explain how to deduce from it a generalization of Grothendieck’s local duality theorem over commutative noetherian local DG rings.

Let E be a quadratic imaginary field and p a prime which is inert in E. Let S be the special fiber (at p) of a unitary Shimura variety of signature (n,m) and hyperspecial level subgroup at p, associated with E/Q.

We study a natural foliation in the tangent bundle of S, which is originally defined on the \mu-ordinary stratum only, but is extended to a certain non-singular blow-up of S. We identify the quotient of S by the foliation with a certain irreducible component of a Shimura variety with parahoric level structure at p. As a result we get new results on the singularities of the latter.

We study integral submanifolds of the foliation and end the talk with a new conjecture of Andre-Oort type.

Planar algebras are a type of diagrammatic graded algebra, introduced to axiomatize the standard invariant of subfactors. The fundamental example is the Temperley-Lieb algebra which can be constructed as End(X^n), where X is a quantum sl_2 module. Recently, there has been interest in a finite dimensional version of quantum sl_2, known as restricted quantum sl_2, and it has been conjectured that its representation theory is equivalent to a logarithmic conformal field theory. We aim to generalize the Temperley-Lieb construction to the restricted case, giving generators and relations of the planar algebra, and describing morphisms between indecomposable modules diagrammatically.

I will discuss the KdV integrable hierarchy, and its tau functions and wave functions.

Witten conjectured that the tau functions are the generating functions of intersection numbers over the moduli of curves (now Kontsevich’s theorem). Recently the following was conjectured: The KdV wave function is a generating function of intersection numbers on moduli of “Riemann surfaces with boundary” (Pandharipande-Solomon-T,Solomon-T,Buryak).

I will describe the conjecture, its generalization to all genera (Solomon-Tessler), and sketch its proof (Pandharipande-Solomon-T in genus 0, T,Buryak-T for the general case). If there will be time, I’ll describe a conjectural generalization by Alexandrov-Buryak-T.

A finite set has an interesting numerical invariant - its cardinality. There are two natural generalizations of “cardinality” to an (homotopy) invariant for (suitably finite) spaces. One, is the classical Euler characteristic. The other is the Baez-Dolan “homotopy cardianlity”. These two invariants, both natural from a certain perspective, seem to be very different from each other yet mysteriously connected. The question of the precise relation between them was popularized by John Baez as one of the “mysteries of counting”. Inspired by this, we show that (p-locally) there is a unique common generalization of these two invariants satisfying some desirable properties. The construction of this invariant relies on a certain l-adic continuity property of the sequence of Morava-Euler characteristics of a given space, which seems to be an interesting “trans-chromatic” phenomenon by itself.

The Birch and Swinnerton-Dyer conjecture predicts that the group of rational points on an elliptic curve E over Q has rank equal to the order of vanishing of the L-function of E. A generalization of this conjecture to all geometric Galois representations V was formulated by Bloch and Kato. I will explain a proof of a version of the Bloch-Kato conjecture in p-adic coefficients, when V is attached to a p-ordinary Hilbert modular form of any weight and the order of vanishing is 1. The case of elliptic curves corresponds to classical modular forms of weight two, and was treated by Perrin-Riou in 1987 using the modular points on E(Q) constructed by Heegner. The proof in the general case relies on the universal p-adic deformation of Heegner points and a formula for its height.

Singular spaces appear everywhere. And the singularity is often non-isolated, i.e. the singular locus is of positive dimension. These non-isolated singularities are more complicated and less studied.

Let X be a variety with singular locus Z, the simplest example being the surface {y^2=x^2z}. Generically along Z the singularity “factorizes”, i.e. X is locally at each point the product: (the germ of Z)\times (the germ of space with an isolated singularity).

But at some special points of Z the picture degenerates and the family of sections of X, transversal to Z, becomes not equi-singular (in whichever sense). These points form the discriminant of transversal singularity type. We study this discriminant, assuming X,Z are locally complete intersections and X is of “ordinary type” generically along Z.

First I will define the discriminant, as a subscheme of Z, and formulate its properties. This discriminant is a (effective) Cartier divisor in Z, nef but not necessarily ample, with nice pullback/pushforward properties under some maps. The discriminant deforms flatly under some deformations of X.

Then I will give the formula for the class of this discriminant in the cohomology/Chow group/Picard group of Z. This class “counts the number of points” where the transversal type degenerates as one travels along the singular locus. In most cases this class is not zero (when Z is complete or projective). This places a “topological” obstruction to the naive expectation (from differential geometry): “In the generic case the transversal type does not degenerate”.

The talk is based on arXiv:1705.11013 and arXiv:1308.6045.

The functoriality conjecture is a key ingredient in the theory of automorphic forms and the Langlands program. Given two reductive groups G and H, the principle of functoriality asserts that a map r:G^->H^ between their dual complex groups should naturally give rise to a map r*:Rep(G)->Rep(H) between their automorphic representations. In this talk, I will describe the idea of functoriality, its connection to L-functions and recent work on weak functorial lifts to the exceptional group of type G_2.

I shall review the framework of algebraic families of Harish-Chandra modules, introduced recently, by Bernstein, Higson, and the speaker. Then, I shall describe three of their applications. The first is contraction of representations of Lie groups. Contractions are certain deformations of representations with applications in mathematical physics. The second is the Mackey bijection, this is a (partially conjectural) bijection between the admissible dual of a real reductive group and the admissible dual of its Cartan motion group. The third is the hidden symmetry of the hydrogen atom as an algebraic family of Harish-Chandra modules.

This is about joint work with Shuji Saito. We will begin the talk with a quick introduction to Voevodsky’s theory of motives, and how Kahn-Saito-Yamazaki generalise this theory to allow one to treat cohomology theories that are not necessarily A^1-invariant, and have a notion of ramification. We finish by discussing a blowup formula in Kahn-Saito-Yamazaki’s category of motives with modulus, and how this produces a new proof of this blowup formula for cyclic homology.

In this talk, I present an analogue of the Hardy-Littlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields. I will discuss the definition of a “short interval” on a curve as an additive translation of the space of global sections of a sufficiently positive divisor E by a suitable rational function f, and show how this definition generalizes the definition of a short interval in the polynomial setting. I will give a sketch of the proof which includes a computation of a certain Galois group, and a counting argument, namely, Chebotarev density type theorem.

This is a joint work with Tyler Foster.

De Rham homology and cohomology of algebraic varieties over a field of characteristic 0 were studied by R. Hartshorne in a 1975 paper. In the same paper Hartshorne gave an analogous definition for complete local rings of equicharacterisitc 0 and proved that in this complete local case the properties of de Rham homology and cohomology were similar to the global case. In particular, both in the local and in the global case there exist Hodge-to-deRham spectral sequences for homology and cohomology. In the local case one gets those spectral sequences from surjecting a regular local ring onto the local ring in question (and in the global case by embedding the algebraic variety in question into a regular algebric variety)..

Recently my student Nick Switala proved the following in the complete local case: beginning with the E_2 page the Hodge-to-deRham spectral sequences both for homology and cohomology are finite-dimensional and the isomorphism classes of those spectral sequences depend only the local ring in question, not on the surjection from a regular local ring. I am going to explain Switala’s results in my talk.

In 1980s, Hida constructed p-adic families of ordinary cusp forms. He showed that the Fourier expansions of the ordinary normalized Hecke eigen cusp forms can be p-adically interpolated. Moreover, their associated Galois representations can also be interpolated via a big Galois representation. The Galois representations associated to cusp forms are known to be pure. This suggests a notion of purity for big Galois representations. In this talk, we will discuss this notion and explain its role in the study of variation in p-adic families.