Location: MALL
Title: Residue field domination in some theories of valued fields
A paraphrase of the Ax-Kochen-Ersov theorem for some theories of valued fields is that the elementary theory is determined by the theory of the value group and the residue field. At the level of types, the intuition is that a type should be controlled by its trace in each of the residue field and value group. In this talk, I will explore some ways in which this intuition can be made precise, and also some limitations to that preliminary intuition. I will try to give lots of examples to keep the discussion concrete.
Location: MALL 1
Title: Two New Inequalities for Cardinal Characteristics of the Continuum
NOTE date and time change: Thu 1pm
Over the last decades the theory of cardinal characteristics of the continuum has emerged as one among several important subfields of set theory. Some of the classical results in it precede the invention of forcing and arguably the aforementioned emergence. Open problems in this field have inspired the invention over ever more versatile constructions of forcing notions and much of the progress has consisted of proving the values of cardinal characteristics not to be ZFC-provably related. A recent outlier has been the celebrated result by Malliaris and Shelah that $\mathfrak{p}$ is equal to $\mathfrak{t}$. I had guessed that there might be more ZFC-provable relations between the hitherto defined characteristics and I am going to talk about what I found up to now. This is to say that I am going to present some ZFC-provable inequalities. In particular I am going to show that the evasion number is at most the subseries number.
These cardinal characteristics have been introduced in work by Blass, Brendle, Brian, and Hamkins and originate from Algebra and Analysis, respectively. The proof interpolates via the pair-splitting number which is due to Minami.
Location: MALL 2
Title: Around definable types in valued fields
NOTE time change: 11am
Haskell, Hrushovski, and Macpherson showed that the theory ACVF of algebraically closed valued fields has elimination of imaginaries after adding the so-called "geometric sorts" to the language. The same result holds in $p$-adically closed fields ($p$CF) by work of Hrushovski, Martin, and Rideau. In the case of ACVF, one way to prove this is to encode imaginaries using definable types, and then encode definable types in the geometric sorts. While $p$CF does not have "enough" definable types to encode imaginaries, the encoding of definable types carries over. Surprisingly, the geometric sorts are unnecessary: any definable type in $p$CF has a code in the home sort (the field sort). This fact and its proof have some unexpected applications to definable groups and definable topological spaces in $p$CF. For example, certain quotient groups are definable rather than interpretable, and there is a unified notion of "definable compactness" for definable topological spaces. Parts of this talk are joint work with Pablo Andújar Guerrero.
Sometimes more abstract concepts in general topology are considered as having little relation with areas outside of topology. In this talk we will explore a beautiful construction that unexpectedly links the notion of n-Hausdorffness and a special topology in the dynamical systems setting.
Location: MALL 1
Title: Elimination of imaginaries in ordered abelian groups of bounded regular rank
NOTE time change: 4pm
In this talk I will present some results about elimination of imaginaries in pure ordered abelian groups. For the class of ordered abelian groups with bounded regular rank (equivalently with finite spines) we obtain weak elimination of imaginaries once we add sorts for the quotient groups $\Gamma/ \Delta$ for each definable convex subgroup $\Delta$, and sorts for the quotient groups $\Gamma/(\Delta+ \ell\Gamma)$ where $\Delta$ is a definable convex subgroup and $\ell \in \mathbb{N}_{\geq 2}$. We refer to these sorts as the quotient sorts. For the dp-minimal case we obtain a complete elimination of imaginaries if we also add constants to distinguish the cosets of $\Delta+\ell\Gamma$ in $\Gamma$, where $\Delta$ is a definable convex subgroup and $\ell \in \mathbb{N}_{\geq 2}$.
Location: MALL
Title: Adelic Geometry via Geometric Logic
NOTE time change: 3pm
On a syntactic level, the peculiarity of geometric logic can be seen from the choice of logical connectives used (in particular, we allow for infinitary disjunctions but do not allow negation), but there are deep ramifications of this seemingly innocuous move. One, geometric logic is incomplete if we restrict ourselves to set-based models, but is complete if we also consider models in all toposes (i.e. not just $\mathrm{Set}$) — as such, geometric logic can be viewed as an attempt to pull our mathematics away from a fixed set theory. Two, there is an intrinsic continuity to geometric logic, which is furnished by the definition of the classifying topos. Indeed, since every Grothendieck topos is a classifying topos of some geometric theory, this provides yet another way of viewing Grothendieck toposes as generalised spaces.
Both insights will inform the content of this talk. We shall start by giving a leisurely introduction to the theory of geometric logic and classifying toposes, before introducing a new research programme (joint with Steven Vickers) of developing a version of adelic geometry via topos theory.
The first step of this programme is to define the geometric theory of absolute values of $\mathbb{Q}$ and provide a point-free account of exponentiation. The next step is to construct the classifying topos of places of $\mathbb{Q}$, which incidentally provides a topos-theoretic analogue of the Arakelov compactification of $\operatorname{Spec}(\mathbb{Z})$. Interestingly, whereas the classical picture views the Archimedean place as a single point "at infinity", our picture reveals that the Archimedean place resembles a blurred interval living below $\operatorname{Spec}(\mathbb{Z})$. This raises challenging questions to our current understanding of the number theory, particularly in regards to reconciling the Archimedean vs. the non-Archimedean aspects.
Location: MALL 1
Title: Is o-minimality of the open-core an elementary property?
Let $M$ be a structure endowed with a dense linear order without endpoints. Its open-core is its reduced generated by the collection of all of its open definable sets (in any Cartesian power of $M$). We prove that the property of having an o-minimal open core isn't elementary. This is a joint work with Alexi Block-Gorman.
Location: MALL
Title: Functional interpretations and the contraction rule therein
Functional interpretations are generalisations of Goedel's Dialectica interpretation of Heyting arithmetic into a primitive-recursive functional system, which arose from a disbelief in universal quantifiers and integrated the ideas of functionals and proofs; the Dialectica interpretation shows the consistency of Heyting arithmetic (hence that of Peano arithmetic) with respect to that primitive-recursive functional system. We give an introduction to the idea of functional interpretations, and in the course, we will see how a seemingly innocuous inference rule called contraction (from A\vee A, deduce A) interferes with the interpretations.
Grothendieck rings were introduced in model theory in the early 2000s. They appear especially in motivic integration, where they are used to express formulas for certain counting functions in a uniform manner. There also is a dictionary of the combinatorial properties of a structure and of the algebraic properties of its Grothendieck ring. It wasn't known until recently wether there exist finite Grothendieck ring. In this talk, we will show that for any integer $N$, we can construct a structure whose Grothendieck ring is $\mathbb{Z}/N\mathbb{Z}$.
The strong existential closedness problem was introduced in Zilber's work on pseudoexponentiation. Since then, it has been naturally adapted to many situations in arithmetic geometry. In this talk I will introduce the problem, review some important Diophantine questions that are connected to it, and discuss some of the known results.
