Research Seminar

Abstract: We introduce the notion of sp-homogeneous and weakly sp-homogeneous linear orderings --- linear orderings which become homogeneous or weakly homogeneous when expanded by partial functions for successor and predecessor.

We demonstrate that these orderings are always relatively $\Delta_4$ categorical and determine exactly which ones are (uniformly) relatively $\Delta_3$ categorical.

We also provide a classification for sp-homogeneity and weak sp-homogeneity. This classification is optimal in that the set of sp-homogeneous linear orderings is $\Pi_5^0$-complete and that the set of weakly sp-homogeneous linear orderings is $\Sigma_6^0$-complete. This result is obtained in the setting of computability theory and also in the setting of descriptive set theory.

This work is joint with Wesley Calvert, David Gonzalez, Valentina Harizanov, Keng Meng Ng.

Abstract: It is natural to study the algebraic properties of interesting transcendental holomorphic functions. For example, the complex exponential function is a group homomorphism. Moreover, one can characterize precisely which algebraic varieties are mapped coordinatewise by exp to other algebraic varieties. Analogous characterizations are known for other functions that, like exp, satisfy differential equations. These results are proved using techniques that exploit the differential equations and do not adapt cleanly to differentially transcendental functions. The Gamma function, which extends the factorial function to complex numbers, is an important differentially transcendental function. I will characterize the algebraic varieties mapped by Gamma to other algebraic varieties of the same dimension. This is joint work with Sebastian Eterović and Roy Zhao.

Abstract: Thematically following Blaise's seminar from last week, I will describe what is known about existential definability of henselian valuation rings within their fields of fractions, working in the language of rings, sometimes allowing parameters. There is a story---somewhat parallel to the one Blaise recounted---in which one may classify such valuation rings (at least in equal characteristic) by a first-order property of their residue field. I will try to explain the key ingredients and examples, together with some open questions. This is based on non-recent work, some joint with Arno Fehm, and some also with Philip Dittmann.

Abstract: Mirror symmetry is an interesting phenomenon in both mathematics and physics. Emerging in physics more than 30 years ago, as a kind of duality between symplectic and complex geometry, has called the attention of mathematicians who have proposed several programs to understand it. One of the central objects in mirror symmetry is the so-called mirror map which is still a bit mysterious. On the other hand, from the model theory side, Boris Zilber has proposed a program for both mathematics and physics based on the idea of logically perfect structures, a notion close to the idea of categoricity and a kind of extension of his program in abstract algebraic geometry based on the idea of Zariski geometries. 

In this talk, I will propose a set of ideas looking for understanding mirror maps from the point of view of logically perfect structures. The main idea behind is to explore how model theoretic tools could help to understand the way different geometries are related.

Abstract: In this talk, we will discuss bi-colored structures of the form $(\mathfrak{M},p^{\mathfrak{M}})$ in the style of the Hrushovski construction.  There are two aspects that we will address here.

Firstly, we will see how the expanded structure inherits the model-theoretic tameness, such as NIP, NTP2, and simplicity, from the original structure. Secondly, we will explore the nature of definable sets and, in particular, in the case of a (black) bi-colored field $(\mathbb{F},p^{\mathbb{F})$, we show that if a valuation is definable in $(\mathbb{F},p^{\mathbb{F})$ then it should be already definable in $(\mathbb{F}$.

Abstract: Cellular automata were introduced as models of computation by Ulam and von Neumann. We are concerned with one such class of automata, that have vector states and are additive (hence do not simulate universal Turing machines). We will show how the temporal dynamics of these can be much more complicated than the scalar state case, but can be understood in terms of dynamics of endomorphisms of algebraic vector groups over the algebraic closure of a finite field. For example, studying the asymptotics of periodic orbit lenghts leads to a generalization of the prime polynomial theorem, that Gauss proved in the upublished eight chapter of the Disquisitiones. (Joint work with Jakub Byszewski.)

Abstract: The distribution of ranks of elliptic curves is a problem of importance to number theorists and logicians alike. Many random matrix heuristics predict the distribution of ranks of reasonable families of elliptic curves, as well as the distribution of p-Selmer groups. However, these heuristics can fail for families of elliptic curves with extra structure — for example, in families with isogenies. In this talk, I will report on a work in progress with H. Helfgott, Z. Klagsbrun, and A. Weiss, showing that the distribution of 2-Selmer groups in a quadratic twist family of elliptic curves with 2-isogenies coincides with the distribution of the kernels of random alternating matrices.