Registration & Funding
If you are interested in attending this event, please register here. There is some funding available for PhD students to cover travel expenses. Please contact Lorna Gregory for more information.
Schedule
12:15 – 13:00: Arrival and Lunch (in the Common Room on the fifth floor of Huxley Building)
13:00 – 13:50: Ivan Tomašić
13:55 – 14:45: Melissa Özsahakyan
14:45 – 15:15: Tea & Coffee Break
15:15 – 16:05: Abhiram Natarajan
16:10 – 17:00: Pietro Frenni
After the talks, we will go to the pub and have dinner at The Bugis Singapore Restaurant by Gloucester Road Station at 18:30.
Titles & Abstracts
Ivan Tomašić. Categorical logic and difference algebra
Model theory has been extremely successful at studying difference fields from the point of view of stability theory. In this talk, we will show how categorical logic can be used to study homological algebra of difference rings and modules, to define difference schemes and develop difference algebraic geometry and cohomology theory.
Melissa Özsahakyan. VC-Density in Divisible Oriented Abelian Groups and their Pairs
In this talk, we introduce oriented abelian groups and present some tameness properties of these structures and their pairs. We show that the VC-density of formulas in divisible oriented abelian groups is bounded by the size of the parameter variable. We further show that this bound becomes twice the size of parameter variables in pairs of divisible oriented abelian groups, and that these bounds are optimal. Finally, we establish that such pairs are not dp-minimal. This is joint work with Ebru Nayir.
Abhiram Natarajan: Semi-Pfaffian geometry – tools, and applications
We generalize the seminal polynomial partitioning theorems of Guth and Katz [1, 2] to a set of semi-Pfaffian sets. Specifically, given a set $\Gamma \subseteq \mathbb{R}^n$ of $k$-dimensional semi-Pfaffian sets, where each $\gamma \in \Gamma$ is defined by a fixed number of Pfaffian functions, and each Pfaffian function is in turn defined with respect to a Pfaffian chain $\vec{q}$ of length $r$, for any $D \ge 1$, we prove the existence of a polynomial $P \in \mathbb{R}[X_1, \ldots, X_n]$ of degree at most $D$ such that each connected component of $\mathbb{R}^n \setminus Z(P)$ intersects at most $\sim \frac{|\Gamma|}{D^{n – k – r}}$ elements of $\Gamma$. Also, under some mild conditions on $\vec{q}$, for any $D \ge 1$, we prove the existence of a Pfaffian function $P’$ of degree at most $D$ defined with respect to $\vec{q}$, such that each connected component of $\mathbb{R}^n \setminus Z(P’)$ intersects at most $\sim \frac{|\Gamma|}{D^{n-k}}$ elements of $\Gamma$. To do so, given a $k$-dimensional semi-Pfaffian set $\gamma \subseteq \mathbb{R}^n$, and a polynomial $P \in \mathbb{R}[X_1, \ldots, X_n]$ of degree at most $D$, we establish a uniform bound on the number of connected components of $\mathbb{R}^n \setminus Z(P)$ that $\gamma$ intersects; that is, we prove that the number of connected components of $(\mathbb{R}^n \setminus Z(P)) \cap \gamma$ is at most $\sim D^{k+r}$. Finally, as applications, we derive Pfaffian versions of Szemeredi-Trotter-type theorems and also prove bounds on the number of joints between Pfaffian curves.
These results, together with some of my other recent work (e.g., bounding the number of distinct distances on plane Pfaffian curves), are steps in a larger program – pushing discrete geometry into settings where the underlying sets need not be algebraic. I will also discuss this broader viewpoint in the talk.
This talk is based on multiple works, some of which are joint works, with subsets of Saugata Basu, Antonio Lerario, Martin Lotz, Adam Sheffer, and Nicolai Vorobjov.
[1] Larry Guth, Polynomial partitioning for a set of varieties, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 159, Cambridge University Press, 2015, pp. 459–469.
[2] Larry Guth and Nets Hawk Katz, On the Erdős distinct distances problem in the plane, Annals of Mathematics (2015), 155–190.
Pietro Frenni: Cuts and derivations in o-minimal expansions of fields
After a brief introduction to Hardy fields, I will describe how some axioms of the theory of H-fields can be generalized so as to cover the case of differential fields of germs at a non-principal cut in an o-minimal ordered field. I will then show how this can be used to analyse types in o-minimal structures, answering two natural questions. First I will sketch the proof that in polynomially bounded structures expanded by exponentiation, a symmetric cut is weakly orthogonal to the cut above its additive invariance group, answering a question of Tress; then I will characterize exponential o-minimal theories where the definable image of a pseudolimit (wim-cut) over a (T-convexly valued) model is always an image by a composition of exponentials, translations and sign-changes of another pseudolimit. Finally, I will give some motivation for the questions answered.