The seminar meets on Wednesdays, 15:10-16:30, in Math -101

2017–2018–B meetings

Mar 7 Equations with singular moduli: effective aspects Yuri Bilu (University of Bordeaux)

A singular modulus is the j-invariant of an elliptic curve with complex multiplication. André (1998) proved that a polynomial equation F(x,y)=0 can have only finitely many solutions in singular moduli (x,y), unless the polynomial F(x,y) is “special” in a certain precisely defined sense. Pila (2011) extended this to equations in many variables, proving the André-Oort conjecture on C^n. The arguments of André and Pila were non-effective (used Siegel-Brauer).

I will report on a recent work by Allombert, Faye, Kühne, Luca, Masser, Pizarro, Riffaut, Zannier and myself about partial effectivization of these results.

Mar 14 Infinite loop spaces in motivic homotopy theory Maria Yakerson (University of Duisburg-Essen (Essen))

Classically in homotopy theory, infinite loop spaces are recognized as spaces with an additional structure: grouplike $E_{\infty}$-spaces. The category of such spaces is equivalent to the category of connective spectra. Replacing topological spaces with smooth schemes, we end up in the realm of motivic homotopy theory, where an analogous statement was sought for since the theory has appeared. In this talk we will discuss the motivic recognition principle, which provides an equivalence between the category of motivic connective spectra and the category of grouplike motivic spaces with so called framed transfers.

This is joint work with Elden Elmanto, Marc Hoyois, Adeel Khan and Vladimir Sosnilo.

Mar 21 The structure of flat modules and the module structure of flat algebras over commutative rings Leonid Positselski (University of Haifa)

The Govorov-Lazard theorem describes flat modules over any associative ring as filtered direct limits of finitely generated free modules, but a more informative description may be desirable. I will explain how to obtain the flat modules over a Noetherian commutative ring which has either Krull dimension 1, or an arbitrary Krull dimension but countable spectrum, as the direct summands of transfinitely iterated extensions of localizations of the ring with respect to countable multiplicative subsets. An even more precise description, with localizations of the ring at its elements (= multiplicative subsets generated by one element), is obtained for the underlying modules of flat finitely presented commutative algebras over arbitrary commutative rings. I will start with a discussion of complete cotorsion pairs in module categories and proceed to formulate the above mentioned results and explain some of the ideas behind their proofs. This talk is based on the speaker’s joint work with Alexander Slavik.

Apr 11 The polylog quotient and the Goncharov quotient in computational Chabauty-Kim theory Ishai Dan-Cohen (BGU)

Polylogarithms are those multiple polylogarithms which factor through a certain quotient of the de Rham fundamental group of the thrice punctured line known as the polylogarithmic quotient. In joint work with David Corwin, building on work that was partially joint with Stefan Wewers, we push the computational boundary of our explicit motivic version of Kim’s method in the case of the thrice punctured line over an open subscheme of Spec ZZ. To do so, we develop a greatly refined version of an algorithm I constructed a few years ago, tailored specifically to this case, and we focus attention on the polylogarithmic quotient with a vengeance. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth-one part of the mixed Tate Galois group studied extensively by Goncharov. We also discover an interesting consequence of the symmetry-breaking nature of the polylog quotient which forces us to symmetrize our polylogarithmic version of Kim’s conjecture. Finally, we apply our refined algorithm to verify Kim’s conjecture in an interesting new case.

Apr 25 The fibration method over real function fields Ambrus Pál (Imperial College London)

We prove that for a smooth, projective, geometrically irreducible variety X equipped with a dominant morphism f with a smooth generic fibre onto a smooth projective rational variety over the function field R(C) of a smooth, irreducible projective curve C over the reals R such that the cohomological obstruction introduced by Colliot-Thélène is the only one to the local-global principle for rational points for the smooth fibres of f over R(C)-valued points, then the same holds for X, too, by adopting the fibration method similarly to Harpaz–Wittenberg. We also show that the strong vanishing conjecture for n-fold Massey products holds for fields of virtual cohomological dimension at most 1 using a theorem of Haran. Joint work with Endre Szabó.

May 2 A Nullstellensatz for noncommutative polynomials: advances in determinantal representations Jurij Volcic (BGU)
May 9 Disjointness of models Mahendra Verma (BGU)
May 16 Cacti groups, real locus of Deligne-Mumford compactification $\bar{M_{0,n+1}}$. Anton Khoroshkin (Higher School of Economics (Moscow))

The real locus of Deligne-Mumford compactification $\bar{M_{0,n+1}}(R)$ is known to be the Eilenberg-Maclane space of the so called pure cacti group. This group has a lot of common properties with pure braid group. In particular, (pure) cacti group acts naturally on tensor products of representations of quantum groups $U_q(g)$. This action has a well-defined simple combinatorial limit for $q \to 0$. I will report on old and new results on the (pure) cacti group. In particular, I will explain how the language of operads provides the description of the rational homotopy type of $\bar{M_{0,n+1}}(R)$.

The talk is based on the joint work with Thomas Willwacher.

May 23 Topological H^1 of algebraic groups Ariel Davis (HUJI)
Jun 6 Homogeneous spherical varieties over large fields Stephan Snigerov (TAU)

Let $k_0 \subset k$ be a field extension such that $k$ is an algebraic closure of $k_0$ and both are of characteristic 0. We assume that $k_0$ is a large field (for example it can be $\mathbb{R}$, or a $p$-adic field). Let $G$ be a connected reductive group defined over $k$, $H \subset G$ a spherical $k$-subgroup. Let $G_0$ be a form of $G$ over $k_0$. Two natural questions one can ask are: (i) Is $H$ conjugate to a subgroup defined over $k_0$? (ii) Is there a $G_0$ equivariant $k_0$-model of $G/H$? An obvious necessary condition for (ii) (and hence for (i)) is that $\Gamma=Gal(k/k_0)$ ``preserves the combinatorial invariants of $G/H$”, so we assume this throughout. We show that if $G_0$ is a quasi-split form and $H$ has self-normalizing normalizer in $G$ then the above condition is sufficient for (i). When $H$ has self-normalizing normalizer, but $G_0$ is not assumed to be quasi-split, we give a necessary and sufficient Galois-cohomological condition for (ii) (assuming $\Gamma$ preserves the combinatorial invariants of $G/H$).

We will explain what all these words mean, and if time permits discuss the proofs, which rely on the existence of rational points in wonderful varieties over large fields.

Jun 13 Genuine equivariant factorization homology Asaf Horev (HUJI)

Factorization homology is a method for constructing quantum field theories from $\mathbb{E}_n$-algebras. We describe a genuine $G$-equivariant version of factorization homology for a finite group $G$. A $G$-factorization homology theory assigns to each smooth manifold with an action of a subgroup H of G a genuine $H$-spectrum. Following Ayala and Francis we give an axiomatic characterization of such theories as satisfying a monoidal version of excision and intertwining topological induction of manifolds with multiplicative transfer of spectra. As a future application we present real THH as genuine $\mathbb{Z}/2$ factorization homology.

Jun 20 The abelianization of inverse limits of groups Ilan Barnea (HUJI)

The abelianization is a functor from groups to abelian groups, which is left adjoint to the inclusion functor. Being a left adjoint, the abelianization functor commutes with all direct limits. It is thus natural to wonder about the behavior of the abelianization functor under inverse limits. There is always a natural map from the abelianization of an inverse limit of groups to the inverse limit of their abelianizations. In this lecture I will present results giving restrictions on the kernel and cokernel of this natural map, in certain cases. These cases include countable directed inverse limits of finite groups, and can thus help in the calculation of the abelianization of certain profinite groups. If time permits I will also consider other families of functors into abelian groups.

Seminar run by Dr. Ishai Dan-Cohen