Oct 30, 15:10–16:25

Computation of padic multiple zeta values and motivic Galois theory

David Jarossay (BGU)

Multiple zeta values can be written as sums of series and as integrals. Their integral expression makes them into periods of the prounipotent fundamental groupoid of $\mathbb{P}^{1}  {0,1,\infty}$. pAdic multiple zeta values are defined as padic analogues of these integrals. We will show how to express them as sums of series, which allows in particular to compute them explicitly.
We will mention the role of finite multiple zeta values defined by Kaneko and Zagier, and of a question asked by Deligne and Goncharov on a relation between the computation of padic multiple zeta values and their algebraic properties. To express the results we will introduce new objects in relation with motivic Galois theory of periods.

Nov 6, 15:10–16:25

Factorization statistics for restricted polynomial specializations over large finite fields

Alexei Entin (TAU)

For a polynomial $F(t,A_1,...,A_n)$ in $\mathbb{F}_p[t,A_1,...,A_n]$ ($p$ being a prime number) we study the factorization statistics of its specializations $F(t,a_1,...,a_n)$ in $\mathbb{F}_p[t]$ with $(a_1,...,a_n) \in S$, where $S=I_1\times\dots\times I_n\subset\mathbb{F}_{p^n}$ is a box, in the limit $p\rightarrow\infty$ and $deg(F)$ fixed. We show that under certain fairly general assumptions on $F$, and assuming that the box dimensions grow to infinity with one of them growing faster than $p^{1/2}$, the degrees of the irreducible factors of $F(t,a_1, \dots,a_n)$ are distributed like the cycle lengths of a random permutation in $S_n$.
This improves and generalizes previous results of Shparlinski and more recent results of KurlbergRosenzweig, which in turn generalize the classical PolyaVinogradov estimate of the number of quadratic residues in an interval.

Nov 13, 15:10–16:25

Enumerating pseudoholomorphic curves with boundary

Sara Tukachinsky (IAS)

Open GromovWitten (OGW) invariants should count pseudoholomorphic maps from curves with boundary to a symplectic manifold, with Lagrangian boundary conditions and various constraints on boundary and interior marked points. The presence of boundary of real codimension 1 poses an obstacle to invariance. In a joint work with J. Solomon (20162017), we defined genus zero OGW invariants under cohomological conditions. The construction is rather abstract. Nonetheless, in a recent work, also joint with J. Solomon, we find that the generating function of OGW has many properties that enable explicit calculations. Most notably, it satisfies a system of PDE called open WDVV (WittenDijkgraafVerlindeVerlinde) equation. For the projective space, this PDE generates recursion relations that allow the computation of all invariants. Furthermore, the open WDVV can be reinterpreted as an associativity of a suitable version of a quantum product.
No prior knowledge of any of the above notions will be assumed.

Nov 20, 15:10–16:25

DGAs with polynomial homology

Haldun Özgür Bayindir (Haifa)

Differential graded algebras(DGAs) are one of the most important
objects of study in homological algebra. These are chain complexes
with an associative and unital multiplication. Examples of DGAs
include cochain complexes of topological spaces equipped with the cup
product.
In this talk, I present our recent classification results on DGAs with
polynomial homology. These results are obtained by exploiting
interesting interactions between DGAs and stable homotopy theory. I am
going to start my talk by stating these classification results. For
the rest of the talk, I am going to present how stable homotopy theory
comes into play for the classification of DGAs. This presentation is
going to be accessible to a general audience.

Nov 27, 15:10–16:25

The Prasad conjecture

Hengfei Lu (Weizmann)

Period Problem is one of the most popular interesting problems in recent years, such as the GanGrossPrasad conjectures. In this talk, we mainly focus on the local period problems, so called the relative Langlands programs. Given a quadratic local field extension E/F and a quasisplit reductive group G defined over F with associated quadratic character $\chi_G$, let $\pi$ be an irreducible admissible representation of G(E). Assume the LanglandsVogan conjecture. Dipendra Prasad uses the enhanced Lparameter of $\pi$ to give a precise description for the multiplicity $\dim Hom_{G(F)}(\pi,\chi_G)$ if the Lpacket $\Pi_\pi$ contains a generic representation. Then we can verify this conjecture if G=GSp(4).

Dec 4

The Loxton  van der Poorten conjecture, and an elliptic analogue

Ehud de Shalit (HUJI)

The conjecture of Loxton and var der Poorten is a criterion for a formal power series
to be the expansion at 0 of a rational function, and is related to a famous theorem of Cobham
in the theory of finite automata. It was proved by Adamczewski and Bell in 2013. Recently,
Schafke and Singer found a novel approach that lead also to a simple conceptual proof of
Cobham’s theorem. We shall explain these results and the cohomological machinery
behind them, and discuss what is missing from the picture to establish an elliptic analogue.

Dec 11

Irreducibility of Galois representations associated to low weight Siegel modular forms

Ariel Weiss (HUJI)

If f is a cuspidal modular eigenform of weight k>1, Ribet proved that its associated padic Galois representation is irreducible for all primes. More generally, it is conjectured that the padic Galois representations associated to cuspidal automorphic representations of GL(n) should always be irreducible.
In this talk, I will prove a version of this conjecture for low weight, genus 2 Siegel modular forms. These twodimensional analogues of weight 1 modular forms are, conjecturally, the automorphic objects that correspond to abelian surfaces.

Dec 18

Flatness and Completion Revisited

Amnon Yekutieli (BGU)

https://www.math.bgu.ac.il/~amyekut/lectures/flatcomprevis/abstract.html

Dec 25

Fourier transforms on the basic affine space

Nadya Gurevich (BGU)

For a quasisplit group $G$ over a local field $F$, with Borel subgroup $B=TU$ and Weyl group $W$,
there is a natural geometric action of $G\times T$ on $L^2(X),$ where $X=G/U$ is the basic affine space of $G$.
For split groups, Gelfand and Graev have extended this action to an action of
$G\times (T\rtimes W)$ by generalized Fourier transforms $\Phi_w$. We shall extend this result for quasisplit groups, using a new interpretation
of Fourier transforms for quasisplit groups
of rank one.
This is joint work with David Kazhdan.

Jan 1

Monogenic cubic fields and local obstructions

Ari Shnidman (HUJI)

A number field is monogenic if its ring of integers is generated by a single element. It is conjectured that 0% of degree d number fields are monogenic (for any d > 2). There are local obstructions that force this proportion to be < 100%, but beyond this very little is known. I’ll discuss work with Alpoge and Bhargava showing that a positive proportion of cubic fields (d = 3) have no local obstructions and yet are still not monogenic. This uses new results on integral points and ranks of Selmer groups of elliptic curves in twist families.

Jan 8

Irreducibility problem for Severi varieties

Ilya Tyomkin (BGU)

Severi varieties parameterize reduced irreducible curves of given geometric genus in a given linear system on an algebraic surface. The first irreducibility result for Severi varieties was established in 1986, and it is due to Harris, who considered the classical case of planar curves in characteristic zero. Few more irreducibility results have been obtained since then, but none of the known approaches is applicable in positive characteristic. In my talk, I will discuss the history and the state of the art in the irreducibility problem, and will also announce new results obtained in a joint work with Karl Christ and Xiang He.

Jan 15

Degenerating plane curves via tropicalization

Karl Christ (BGU)

In my talk, I will describe how simultaneous stable reduction and tropical geometry can be used to construct degenerations of plane curves. This is the main ingredient in a new proof for irreducibility of Severi varieties of the projective plane. The crucial feature of this construction is that it works in positive characteristic, where the other known methods fail. The talk will be a follow up on last week’s talk and is based on joint work with Xiang He and Ilya Tyomkin.

Jan 22

Commutative DG Rings and their Derived Categories

Amnon Yekutieli (BGU)

The commutative DG rings in the title are more commonly known as “nonpositive strongly commutative unital differential graded cochain Kalgebras”, where K is a commutative base ring. In the literature the standard assumption is that K is a field of characteristic zero  but one of our themes in this talk is that this assumption is superfluous (K = Z works just as well).
There are two kinds of derived categories ralated to commutative DG rings. First, given a DG ring A, we can consider D(A), the derived category of DG Amodules, which is a Klinear triangulated category. This story is well understood by now, and I will only mention it briefly.
In this talk we shall consider another kind of derived category. Let DGRng denote the category whose objects are the commutative DG rings (the base K is implicit), and whose morphisms are the DG ring homomorphisms. The derived category of commutative DG rings is the category D(DGRng) gotten by inverting all the quasiisomorphisms in DGRng. (In homotopy theory the convention is to call it the “homotopy category”, but this is an unfortunate historical accident.)
I will define semifree DG rings, and prove their existence and lifting properties. Then I will introduce the quasihomotopy relation on DGRng, giving rise to the quotient category K(DGRng), the “genuine” homotopy category. One of the main results is that the canonical functor from K(DGRng) to D(DGRng) is a faithful right Ore localization.
I will conclude with a theorem on the existence of the left derived tensor product inside D(DGRng), and with the pseudofunctor from D(DGRng) to the TrCat, sending a DG ring A to the triangulated category D(A).
Next semester I will talk about the geometrization of these ideas: “The Derived Category of Sheaves of Commutative DG Rings”.
