Joint work with Wolfgang Hackbusch and Daniel Kressner

(joint work with Sergei Starchenko)

Let p:C^n ->A be the covering map of a complex abelian variety and let X be an algebraic variety of C^n, or more generally a definable set in an o-minimal expansion of the real field. Ullmo and Yafaev investigated the topological closure of p(X) in A in the above two  settings and conjectured that the frontier of p(X) can be described, when X is algebraic as finitely many cosets of real sub tori of A, They proved the conjecture when dim X=1. They make a similar conjecture for X definable in an o-minimal structure.

In recent work we show that the above conjecture fails as stated, and prove a modified version,  describing the frontier of p(X) as finitely many families of cosets of subtori. We prove a similar result when X is a definable set in an o-minimal structure and p:R^n-> T is the covering map of a real torus.  The proofs use model theory of o-minimal structures as well as algebraically closed valued fields.

In various research fields such as machine learning, compressed sensing and operations research, optimization problems which seek sparsity of solutions by the cardinality constraint or rank constraint are studied. We formulate such problems as DC (Difference of two Convex functions) optimization problems and apply DC Algorithm (DCA) to them. While a subproblem needs to be solved in each DCA iteration, its closed-form solution can be easily obtained by soft-thresholding operation. Numerical experiments demonstrate the efficiency of the proposed DCA in comparison with existing methods.
This is a joint work with J. Gotoh (Chuo Univ.) and K. Tono (U. Tokyo). 

There are several conjectures in the literature suggesting that absolute Galois groups of fields tend to be "as free as possible," given their "almost-abelian" data.
This can be made precise in various ways, one of which is via the notion of "1-formality" arising in analogy with the concept in rational homotopy theory.
In this talk, I will discuss several examples which illustrate this phenomenon, as well as some surprising diophantine consequences.
This discussion will also include some recent joint work with Guillot, Mináč, Tân and Wittenberg, concerning the vanishing of mod-2 4-fold Massey products in the Galois cohomology of number fields.

We discuss how one can define transseries in several variables. The idea is
to combine the construction of the univariate transseries with a blow up procedure. The
latter allows to normalize transseries in an arbitrary number of variables which makes
them manageable as usual transseries.
 

In joint work with Sylvy Anscombe we had found an abstract
valuation theoretic condition characterizing those fields F for which
the power series ring F[[t]] is existentially 0-definable in its
quotient field F((t)). In this talk I will report on recent joint work
with Sylvy Anscombe and Philip Dittmann in which the study of this
condition leads us to some beautiful results on the border of number
theory and model theory. In particular, I will suggest and apply a
p-adic analogue of Lagrange's Four Squares Theorem.