Distal structures and their theories are characterized by a variety of model-theoretic and combinatorial properties.

In this talk, we will make the case that distality is the right context for studying incidence combinatorics, and construct explicit distal cell decompositions for useful structures.

We will examine properties of distal-definable graphs such as the distal regularity lemma and the strong Erdős-Hajnal property, and see how new examples from continuous logic can help shed light on these graph criteria.

As a step towards outer Lipschitz classification of the real surface germs definable in a polynomially bounded o-minimal structure, which remains a major open problem in Lipschitz geometry since the inner Lipschitz classification was established by Lev Birbrair in 1999, we consider a special case of a surface germ: the union of two Lipschitz Normally Embedded (LNE) Hölder triangles. Although a single LNE Hölder triangle is the simplest building block in the Lipschitz geometry of surface germs, characterized by a single exponent, a pair of LNE Hölder triangles exhibits unexpected metric and combinatorial complexity. In our 2022 paper, an invariant of pairs of LNE Hölder triangles, called $\sigma\tau$-pizza, was defined, conjectured to be a complete invariant of the outer Lipschitz equivalence class of such special surface germs. This conjecture was proved in our 2023 paper, where even more combinatorial complexity of pairs of LNE Hölder triangles was discovered. Moreover, we formulated conditions on the $\sigma\tau$-pizza invariant necessary and sufficient for existence and uniqueness, up to outer Lipschitz equivalence, of a pair of LNE Hölder triangles.

Using a new algebraic definition of the winding number we prove in [PR2] a fully general complex root counting result, improving a former result of [E]. This result is closely related to our quantitative approach of the Fundamental Theorem of Algebra [PR1]. While complex analysis methods compute the winding number through an integral, the algebraic method uses only real algebraic geometry methods on univariate polynomials, counts properly zeroes at vertices or edges and is valid for any real closed field.

Joint work with Daniel Perrucci, University of Buenos Aires, Argentina.

• [E] M. Eisermann, The fundamental theorem of algebra made effective: an elementary real-algebraic proof via Sturm chains. Amer. Math. Monthly 119 (2012), no. 9, 715–752.
• [PR1] D. Perrucci, M.-F. Roy, Quantitative fundamental theorem of algebra. Q. J. Math. 70 (2019), no. 3, 1009–1037.
• [PR2] D. Perrucci, M.-F. Roy, Algebraic winding number. arXiv:2305.08638v1 [math.AG]

In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation $T$, one takes averages of a given integrable function over the intervals $(x, Tx, T^2 x,...,T^n x)$ in front of the point x. We prove a “backward” ergodic theorem for a countable-to-one pmp $T$, where the averages are taken over subtrees of the graph of $T$ that are rooted at x and lie behind x (in the direction of $T^{-1}$). Surprisingly, this theorem yields forward ergodic theorems for countable groups, in particular, one for pmp actions of free groups of finite rank, and can be extended to yield ergodic theorems for pmp actions of free semigroups as well. In each case, the averages are taken along subtrees of the standard Cayley graph rooted at the identity. This is joint work with Anush Tserunyan.

(joint with Rideau-Kikuchi)

One of the most striking results of the model theory of henselian valued fields is the Ax-Kochen/Ershov principle, which roughly states that the first order theory of a henselian valued field that is unramified is completely determined by the first order theory of its residue field and the first order theory of its value group.

Our leading question is: Can one obtain an Imaginary Ax-Kochen/Ershov principle?

In previous work, I showed that the complexity of the value group requires adding the stabilizer sorts. In previous work, Hils and Rideau-Kikuchi showed that the complexity of the residue field reflects by adding the interpretable sets of the linear sorts. In this talk we present recent results on weak elimination of imaginaries that combine both strategies for a large class of (almost all) henselian valued fields of equicharacteristic zero.