Oct 25

Injective modules in higher algebra 
Liran Shaul (Ben Gurion University ) 
Injective modules are fundamental in homological algebra over rings. In this talk, we explain how to generalize this notion to higher algebra. The BassPapp 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.

Nov 1

Foliations on unitary Shimura varieties in positive characteristic 
Ehud de Shalit (Hebrew University ) 
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 \muordinary stratum only, but is extended to a certain nonsingular blowup 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 AndreOort type.

Nov 15

NonSemisimple Planar Algebras from Restricted Quantum sl_2 
Stephen Moore (BGU) 
Planar algebras are a type of diagrammatic graded algebra, introduced to axiomatize the standard invariant of subfactors. The fundamental example is the TemperleyLieb 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 TemperleyLieb construction to the restricted case, giving generators and relations of the planar algebra, and describing morphisms between indecomposable modules diagrammatically.

Nov 22

Integrable hierarchies, wave functions and open intersection theories 
Ran Tessler (ETH) 
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” (PandharipandeSolomonT,SolomonT,Buryak).
I will describe the conjecture, its generalization to all genera (SolomonTessler), and sketch its proof (PandharipandeSolomonT in genus 0, T,BuryakT for the general case). If there will be time, I’ll describe a conjectural generalization by AlexandrovBuryakT.

Nov 29

Homotopy cardinality and the ladic continuity of MoravaEuler characteristic (Joint with Tomer Schlank) 
Lior Yanovski (Hebrew University ) 
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 BaezDolan “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 (plocally) there is a unique common generalization of these two invariants satisfying some desirable properties. The construction of this invariant relies on a certain ladic continuity property of the sequence of MoravaEuler characteristics of a given space, which seems to be an interesting “transchromatic” phenomenon by itself.

Dec 6

On the padic BlochKato conjecture for Hilbert modular forms 
Daniel Disegni (Université ParisSud ) 
The Birch and SwinnertonDyer 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 Lfunction 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 BlochKato conjecture in padic coefficients, when V is attached to a pordinary 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 PerrinRiou in 1987 using the modular points on E(Q) constructed by Heegner. The proof in the general case relies on the universal padic deformation of Heegner points and a formula for its height.

Dec 13

Discriminant of the ordinary transversal singularity type 
Dmitry Kerner (BGU) 
Singular spaces appear everywhere. And the singularity is often nonisolated, i.e. the singular locus is of positive dimension. These nonisolated 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 equisingular (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.
