Transseries emerged in connection with Ecalle's work on Dulac's problem and Dahn and Goering's work on nonstandard models of real exponentiation, and they can often be viewed as asymptotic expansions of solutions to differential equations. A few years ago, decisive results on the model theory and algebra of the differential field of logarithmic-exponential transseries were achieved by Aschenbrenner, van den Dries, and van der Hoeven. Since then, this differential field has been extended to much larger structures encompassing far more rates of growth, for example including solutions to more functional equations. In this talk, I will consider pairs of transseries-like fields, more precisely, two models of the theory of transseries with one a proper extension of the other. My aim is to describe work in progress on the model theory of such pairs, including a model completeness result for them. This follows a long line of model-theoretic work on pairs in different contexts, going back to pairs of real closed fields, which our context generalizes.

This talk concerns the topology of Vandermonde varieties in the setting of Type B symmetry. Recent results of Basu and Riener leverage symmetry relative to the action of the symmetric group $S_n$ on $R^n$ in the study of the cohomology of semi-algebraic sets. Vandermonde varieties, which are classically defined by the first several generators of the ring of $S_n$-symmetric polynomials, play a key role in these arguments. With an eye towards extending the results of Basu and Riener to broader classes of symmetry, we examine Vandermonde varieties defined relative to the next major type of reflection symmetry. This talk will describe Vandermonde varieties within this setting and present new results (joint with Dr Saugata Basu), including the topological regularity of the intersection of a Type B Vandermonde variety with a fundamental region of the group's action.

Eventual polynomial growth is a common theme in combinatorics and commutative algebra. The quintessential example of this phenomenon is the Hilbert polynomial, which eventually coincides with the linear dimension of the graded pieces of a finitely generated module over a polynomial ring. A later result of Kolchin shows that the transcendence degree of certain field extensions of a differential field is eventually polynomial. More recently, Khovanskii showed that for finite subsets A and B of a commutative semigroup, the size of the sumset A+tB is eventually polynomial in t. I will present a common generalization of these three results in terms of finitary matroids (also called pregeometries). I’ll discuss other instances of eventual polynomial growth (like the Betti numbers of a simplicial complex) as well as some applications to bounding model-theoretic ranks. This is joint work with Antongiulio Fornasiero.

O-minimality has proved a useful tool in proving certain number theoretic special point conjectures, such as the Mordell-Lang conjecture. In 1998, Pillay showed that these special point conjectures can be used to construct model theoretically tame structures by expanding algebraically closed fields by a predicate identifying special points. Pillay called structures of this form 'structures of Lang type'. In more recent years G\"unaydin, Gorman and Hieronymi have constructed an abstract framework for proving a weak form of quantifier elimination based on based on the Modell-Lang conjecture. This talk will aim to provide a brief overview of special point conjectures and their model theoretic consequences.

We construct several algebras of functions definable in $R_{an,\exp}$ which are stable under parametric integration.

Given one such algebra $A$, we will study the parametric Mellin and Fourier transforms of the functions in $A$. These are complex-valued oscillatory functions, thus we leave the realm of o-minimality. However, we will show that some of the geometric tameness of the functions in $A$ passes on to their integral transforms.

Büchi automata are the natural extension of finite automata, also called finite-state machines, to a model of computation that accepts infinite-length inputs. We say a subset X of the reals is r-regular if there is a Büchi automaton that accepts (one of) the base-r representations of every element of X, and rejects the base-r representations of each element in its complement. We can analogously define r-regular subsets of higher arities, and these sets often exhibit fractal-like behavior -- e.g., the Cantor set is 3-regular. There are compelling connections between fractal geometry and Büchi automata, and we will consider them from the perspective of model theory. In this talk, we will give a characterization of when different notions of fractal dimension do or do not agree for definable sets in an expansion of the real ordered additive group by a ternary predicate with a remarkable connection to Büchi automata. This is joint work with Christian Schulz.

Fomin and Shapiro conjectured that the link of the stratified space formed by the totally nonnegative part of the unipotent radical of a split semi-simple algebraic group over $\mathbb{R}$ is a regular cell complex. Though resolved by Hersch, interest remains in alternate proof routes. The concept of monotonicity may provide a means to a simpler proof, as graphs of monotone maps are regular cells. In type $A$, examples suggest that the strata in question may be graphs of monotone maps, though the symbolic manipulation grows difficult for $n$ even as small as 4.

Meanwhile, Davis, Hersh, and Miller have shown that the contractibility of the fibers of maps associated to a given totally nonnegative space would imply the regularity of the decomposition of the original space. The fibers are somewhat simpler to describe, and in type A we had hoped to prove this conjecture by demonstrating that the fibers stratify into monotone cells. This is indeed true for $n=4$, but counterexamples to monotonicity (though not regularity) begin to arise in the $n=5$ case. We present the successes, counterexamples, and current state of our investigation of the monotonicity of the strata of the fibers of maps to totally nonnegative spaces.