In 2014 Zilber asked: If M is a nonstandard model of full Arithmetic and n is a nonstandard element of M congruent to 1 modulo all standard integers m, does the ring M/nM interpret Arithmetic? While the answer is clearly no, the question motivated the model theoretic analysis of the rings M/nM in the more general context of M model of Peano Arithmetic. I will present what we know of such rings.
This is a joint work with A. Macintyre.
Zilber's Trichotomy Conjecture is one of the most thoroughly refuted conjectures in model theory. So much so, that more than 35 years after its refutation, no alternative reformulation has been suggested. This is all the more frustrating, since there is growing evidence that the conjecture tends to be correct in many natural settings, and when that was the case, it also had some striking applications. After a quick survey of the Trichotomy's history, I will describe the new (and developing) axiomatic framework of (Loric) Hausdorff Geometric Structures (introduced by B. Castle, J. Ye and myself) for studying a wide class of instances of the Trichotomy. As time permits, I will describe how these techniques can be applied to show (this is joint with B. Castle) that non-locally modular strongly minimal sets definable in RCVF are 2-dimensional.
This talk combines two phenomena motivated by physics: Rigged Hilbert spaces and finite dimensional approximations. Rigged Hilbert spaces are a way of accounting for objects that don't fit into a Hilbert space, such as Dirac delta functions, modelled by suitable distributions. In this talk a structure resembling a rigged Hilbert space is constructed by modifying the metric ultraproduct construction. The original space is embedded into the ultraproduct by building its spectral measure via an ultraproduct of scaled counting measures in finite dimensional spaces. Working with different norms in the ultraproduct, one can then find vectors corresponding to distributions. If these are chosen carefully, one can use them to calculate time evolution in a fashion encountered in beginning physics textbooks.
This is joint work with Tapani Hyttinen.
Zilber's trichotomy conjecture has had an extraordinary influence on the course taken by model theory in the last half-century. I will recall the conjecture and its background, the totally categorical case, counterexamples, Zariski geometries, the trichotomy for differential algebra, for difference equations, for o-minimal structures; and speculate about a possible new chapter with globally valued fields.
Zilber's exponential-algebraic closedness conjecture is a form of existential closedness, and it predicts that the graph of exp intersects algebraic varieties as often as feasible, that is to say, compatibly with Ax-Schanuel. It implies that complex exponentiation is quasiminimal, but also that the associated quasiminimal pregeometry coincides with the pregeometry given by transversal intersections between exp and algebraic varieties. The latter pregeometry is meaningful for other functions satisfying Ax-Schanuel style properties, and so we can ask about existential closedness even when quasiminimality is not in the cards, such as for the j-function.
After an introduction to the above topics, and a summary of the state of the art, I'll discuss joint work with Vahagn Aslanyan and Sebastian Eterović about j and its derivatives, where even the simplest questions seem non-trivial: we prove, for instance, that the second derivative of j has infinitely many 'non-trivial' zeroes. Our argument combine previous techniques for exp and j (without derivatives) with new tricks to deal with the derivatives of j, which are less well behaved (they are not invariant under the action of the modular group).
The abelian exponential-algebraic closedness conjecture, stated by Bays and Kirby building on Zilber's work on the model theory of the complex exponential function, predicts sufficient conditions for solvability of systems of equations involving algebraic operations and the exponential map of a complex abelian variety. It is phrased geometrically, interpreting existence of solutions to these systems as existence of points in the intersection of the graph of the exponential and an algebraic subvariety of the tangent bundle of the abelian variety. In this talk I will report on work in progress concerning the case of the conjecture in which the subvariety of the tangent bundle splits as a product of subvarieties, discussing how a method developed by Peterzil and Starchenko to approximate complex algebraic varieties and a previous result of mine on a simpler case of the conjecture can be used to tackle this problem.
I will report on joint work with Moshe Kamensky establishing a binding group theorem, in characteristic zero difference-closed fields, for certain quantifier-free types that are quantifier-free internal to the fixed field. This is used to obtain the expected bound on the Morley power of a qf-type that one has to take to witness nonorthogonality to the fixed field. The results can be articulated in terms of, and has applications to the birational geometry of, rational dynamical systems.
Jana Marikova in 2007, was first to study automorphism-invariant groups in o-minimal structures and showed that they can be endowed with a group topology, similarly to the case of definable groups. In this talk I describe results on invariant rings and fields. While 0-dimensional invariant fields may include the fields of rational or algebraic numbers (clearly not definable), we show that a positive dimensional type-definable or Ind-definable field must be definable, as is any invariant positive dimensional sub-field of a definable ring.
In addition, while an arbitrary invariant field of positive dimension might not be definable we conjecture that it is isomorphic, via an invariant map, to a definable one.
This is joint work with Mirvat Mhameed and incorporates part of her MSc thesis.
We show that for $n\geq 3$ the theory of free generalized n-gons is complete, strictly stable and strictly 1-ample, yielding a new and easily accessible class of examples in the zoo of stable theories. The construction proceeds via Hrushovski amalgamation. (Joint with A.-M. Ammer)
Zilber's Quasiminimality Conjecture states that the complex field equipped with the exponential function is quasiminimal, i.e. every definable subset is countable or co-countable. Despite still being out of reach, this conjecture led to multiple new concepts and results. One of the directions inspired by the conjecture is the investigation of analogous conjectures where the exponential map is replaced with another function or a function-like object. In most cases the obtained conjecture seems to stay as difficult as the exponential one; as pointed out by Koiran and Wilkie, it even remains open whether adding all entire functions to the complex field would make it quasiminimal or non-quasiminimal. In this talk we provide two quasiminimal examples of this sort: first one involves a correspondence between two elliptic curves, while the second one considers the theory of generic functions, as introduced by Zilber in 2002.
In [2], Ben Yaacov et. al. extended the basic ideas of Scott analysis to metric structures in infinitary continuous logic. These include back-and-forth relations, Scott sentences, and the Lopez-Escobar theorem to name a few. In this talk, I will talk my work connecting the ideas of Scott analysis to the definability of automorphism orbits and a notion of isolation for types within separable metric structures.
Our results are a continuous analogue of the robuster Scott rank developed by Montalban in [1] for countable structures in discrete infinitary logic. However, there are some differences arising from the subtleties behind the notion of definability in continuous logic.
[1] Antonio Montalban. A robuster Scott rank. Proceedings of the American Mathematical Society, 143(12):5427--5436, April 2015.
[2] Itaı Ben Yaacov, Michal Doucha, Andre Nies, and Todor Tsankov. Metric Scott analysis. 318:46--87, 2017.
Categorical logic is
I will try to illustrate how techniques coming from the syntactic side of categorical logic (number 1) can solve problems coming from the semantic side (number 2):
In the book [2] the authors ask if it is true that in a λ-accessible category A, the λ-pure maps are strict monomorphisms. In [1] a counterexample is given.
I will sketch the proof of two positive results: 1) if A is axiomatizable in finitary coherent logic, then the answer is affirmative and 2) if there is a proper class of strongly compact cardinals then for every λ-accessible category A there is some μ > λ, such that when we see A as a μ-accessible category then the answer is affirmative (that is: μ-pure maps are strict monomorphisms).
The proof sketch will allow me to mention some powerful methods coming from category theory (motivated by geometry): A is sitting inside the λ-pro-completion of some small category, and to analyze this inclusion one has to understand Boolean-valued models from a sheaf theoretic perspective, following [4].
The talk is based on [3].
[1] J. Adamek, H. Hu, and W. Tholen. "On pure morphisms in accessible categories". In: Journal of Pure and Applied Algebra 107.1 (1996), pp. 1--8.
[2] J. Adamek and J. Rosicky. Locally Presentable and Accessible Categories. Vol. 189. London Mathematical Society Lecture Note Series. London: Cambridge University Press, 1994.
[3] Kristof Kanalas. Pure maps are strict monomorphisms. https://arxiv.org/abs/2407.13448. 2024.
[4] Jacob Lurie. Lecture notes in Categorical Logic. 2018. url: https://www.math.ias.edu/~lurie/278x.html.
This talk aims at developing a suitable learning framework within a model-theoretic context, thereby building a bridge between Model Theory and Statistical Learning Theory. Specifically, relations between the notions NIP, VC dimension and PAC learning are studied, with a particular emphasis on measurability aspects.
In the context of Statistical Learning Theory, I introduce the concepts of VC dimension and (agnostic) PAC learning. Further, I present the Fundamental Theorem of Statistical Learning, which states that a class of indicator functions is PAC learnable if and only if its VC dimension is finite. I scrutinize the concepts as well as the theorem from a measure-theoretic perspective, in order to extract the minimal measurability requirements needed for rigorous reasoning.
I then explain how the model-theoretic notion NIP can be applied in Statistical Learning Theory. In particular, we discuss in which cases the above-mentioned measurability requirements are partly or fully satisfied. Most prominently, we examine classes of indicator functions defined over o-minimal expansions of the reals.
This work is part of my doctoral research project, which is supervised by Professor Salma Kuhlmann and Dr. Lothar Sebastian Krapp at Universität Konstanz.