The model-theoretic construction of the binding group in the presence of internality has proven to be one of the most important group existence theorems. It saw numerous applications, including those in differential Galois theory, envisaged by Poizat, and pursued by Pillay in his generalisation of the strongly normal theory of Kolchin.
We show how to view the binding group construction as an instance of Janelidze's categorical Galois theory. This allows us to apply it in areas beyond model theory, for example to the scheme-theoretic treatment of Pillay's differential Galois theory.
We will discuss connections with Hrushovski’s work on groupoids.
I will discuss an application to probability theory (more precisely, to the theory of exchangeable random variables) of recent developments in affine logic, the fragment of continuous logic in which only affine connectives are allowed.
An affine theory is simplicial if its type spaces, which are always compact convex sets, form Choquet simplices. A remarkable dichotomy holds: for a complete simplicial theory, either all type spaces are Bauer simplices (i.e., the set of extreme points is closed) or they are all Poulsen simplices (i.e., the extreme points are dense).
On the other hand, simplices of invariant measures arising in ergodic theory and probability theory can be represented as *quantifier-free* type spaces of appropriate universal affine theories. I will discuss a quantifier-free adaptation of the dichotomy result mentioned above and its application to the geometry of certain simplices of invariant measures arising in exchangeability theory. Relative property (T) for Polish groups will play an important role.
In this talk, we study the numerical approximation of the stochastic heat equation on the sphere driven by an additive Lévy process. For this, we first prove new regularity results for the solution of the stochastic heat equation under different regularity assumptions on the initial value and driving Lévy process. In these settings, we perform a spectral approximation based on the truncation of the series expansion with respect to the real-valued spherical harmonic functions. Further, we apply a forward resp. backward Euler-Maruyama scheme for the temporal approximation. We prove strong and weak convergence rates for the introduced approximation scheme and present numerical simulations that confirm our theoretical results.
This is joint work with Annika Lang and Andrea Papini.
As AI agents become capable of autonomously conducting empirical research, a basic question arises: when can we trust the discoveries they produce? Autonomous systems can formulate hypotheses, implement and evaluate them, learn from intermediate results, and repeatedly redirect their search. The resulting hypotheses, their number, and their dependence structure are themselves outcomes of a random and highly adaptive research process, while multiplicity and information leakage can make spurious findings appear convincing. We develop a reusable statistical methodology for rigorous autonomous asset-pricing research that separates essentially unrestricted exploration from formal post-registration confirmation. A time-stamped boundary ledger records and freezes the realized research output and its confirmation designs; fresh post-boundary data are then reserved for statistical certification.
The key insight is that adaptive exploration need not compromise statistical validity once the realized search output is frozen and subsequent evidence is valid conditional on that output. We formalize this principle through conditional anytime e-closed testing for random, dependent hypothesis families. Our central result controls conditional expected loss at level $\alpha$, even when the reporting time is data dependent and the registered family and boundary-specified design choices are random. Different loss functions recover FDR, FWER, FDX, and other error criteria within a common framework; for indicator losses, the guarantees hold simultaneously across confirmation checkpoints.
We instantiate the exploration layer with SEADS, an autonomous system that proposes, implements, tests, and refines predictors of daily U.S. equity returns. Using two fresh years of post-registration data, we subject its discoveries to FDR- and FWER-controlled confirmation using the same evidence paths. Without being supplied known anomaly formulas or labels, SEADS reconstructs much of the established short-horizon structure of the equity cross section from primitive building blocks, including reversal, low-risk and volatility effects, turnover, and profitability. The results demonstrate how open-ended autonomous exploration can be coupled with rigorous fresh-data confirmation to produce statistically credible asset-pricing discoveries.
This is joint work with Xiaowei Ding (Nanjing), Jason Pan (Stanford), and Markus Pelger (Stanford).