Let X be a complex manifold and let M be a meromorphic connection on X with
poles along a normal crossing divisor D. Levelt-Turrittin theorem asserts that the pull-back of M to the formal neighbourhood of a codimension 1 point in D decom poses (after ramification) into elementary factors easy to work with.
This decomposition may not hold at some other points of D. When it does, we say
that M has good formal decomposition along D. A conjecture of Sabbah, recently
proved by Kedlaya and Mochizuki independently, asserts roughly the
Title: Towards Chabauty-Kim loci for the polylogarithmic quotient over an arbitrary number field
Abstract: Let K be a number field and let S be an open
subscheme of Spec O_K.
Minhyong Kim has developed a method for
bounding the set of S-valued points on a
hyperbolic curve X over S; his method opens
a new avenue in the quest for an "effective
Mordell conjecture".
But although Kim's approach has lead to the
construction of explicit bounds in special
cases, the problem of realizing the potential
Higher Etale obstructions are cohomological obstructions introduced by Yonatan Harpaz and Tomer Schlank for solutions of algebraic equations over a field. Their definition is based on the theory of relative etale homotopy type. In my talk I will explain the construction of relative etale homotopy type and the resulting obstruction theory.
I will also present the calculation of these obstructions for quadratic equations of the form a_1x_1^2 + ... + a_nx_n^2 = 1. This is a joint work with Edo Arad and Tomer Schlank.
A well-known result of Borcherds yields the modularity of Heegner divisors on complex orthogonal Shimura varieties (i.e. Grassmannian quotients). These varieties are typically non-compact, and one way of completing them to compact varieties is via toroidal compactifications. However, the boundary components there also contain divisors. We show how to extend the Heegner divisors to such compactifications in such a manner that the modularity result of Borcherds still holds. This is joint work with J. Bruinier.
I will talk about a new Abelian category associated to an open variety with normal-crossings (or more generally, logarithmic) choice of compactification, which behaves in remarkable (and remarkably nice) ways with respect to changes of compactification and duality, and which first appeared in work on mirror symmetry.
The talk is based on the joint work with Yanki Lekili. The associative Yang-Baxter equation
is a quadratic equation related to both classical and quantum Yang-Baxter equations. It appears naturally in connection with triple Massey products in the derived category of
coherent sheaves on elliptic curve and its degenerations. We show that all of its nondegenerate trigonometric solutions are obtained from Fukaya categories of some noncompact surfaces. We use this to prove that any two simple vector bundles on a cycle of projective lines are related by a sequence of spherical twists.
Abstract: my talk will be devoted to a basic theory of extensions of
complete real-valued fields L/K. Naturally, one says that L is
topologically-algebraically generated over K by a subset S if L lies
in the completion of the algebraic closure of K(S). One can then define
topological analogues of algebraic independence, transcendence degree, etc.
These notions behave much more wierd than their algebraic analogues. For example,
there exist non-invertible continuous K-endomorphisms of the completed
Abstract: Inseparable morphisms proved to be
an important tool for the study of algebraic
varieties in characteristic p. In particular,
Rudakov-Shafarevitch, Miyaoka and Ekedahl
have constructed a dictionary between
"height 1" foliations in the tangent bundle
and "height 1" purely inseparable quotients
of a non-singular variety in characteristic p.
In a joint work with Eyal Goren we use this
dictionary to study the special fiber S of a
unitary Shimura variety of signature (n,m),
I discuss some class of function of several elliptic variables,
this functions generalize multiple polylogarithms of D. Zagier.
I show some applications of developed formalism.
This is a joint work with F. Brown.
Abstract: I will start with a motivation of what algebraic (and model-theoretic) properties
an algebraically closed field of characteristic 1 is expected to have. Then I will explain
how these properties can be obtained by the well-known in model theory Hrushovski's
construction and then formulate very precise axioms that such a field must satisfy.
The axioms have a form of statements about existence of solutions to systems
of equations in terms of a 'multi-dimansional' valuation theory and the validity
Abstract: In the last five years Bridgeland stability has revolutionized
our understanding of the geometry of moduli spaces of sheaves on surfaces,
allowing us to compute ample and effective cones and describe different
birational models. In this talk, I will survey some of my joint work with
Daniele Arcara, Aaron Bertram, Jack Huizenga and Matthew Woolf on the
birational geometry of moduli spaces of sheaves on the plane. I will
describe the ample and effective cones of these moduli spaces,
Many of the main conjectures in Iwasawa theory can be phrased as saying
that the first Chern class of an Iwasawa module is generated by a p-adic
L-series.
In this talk I will describe how higher Chern classes pertain to the higher
codimension behavior of Iwasawa modules. I'll then describe a template
for conjectures which would link such higher Chern classes to elements
in the K-theory of Iwasawa algebras which are constructed from tuples of
Katz p-adic L-series. I will finally describe an instance in which a result of
