In this talk, we will explore several invariants for curves on (very) general hypersurfaces and complete intersections, which will have applications to measures of irrationality. This is joint work with Ben Church and Junyan Zhao, and separately with David Yang.

We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories. Graduate students are particularly encouraged to attend. We will prove the simple Kim-forking conjecture for isolated types with the definable Morley property, and then discuss examples of the definable Morley property and finite F_Mb(p).

Leonhard Euler stands among the most influential mathematicians in history. One of his earliest and most remarkable achievements, which helped establish his reputation, was the evaluation of the infinite series of the reciprocals of the squares: 1 + 1/4 + 1/9 + 1/16 + ... now recognized as a p-series with p=2. In this presentation, we will explore Euler’s original proof of this celebrated result and examine how he extended his methods to determine the values of other p-series.

We introduce the dynamical commensurator group for a generalized odometer action, that is for minimal equicontinuous group actions on Cantor sets. We show there is a map from the pointed mapping class group of a solenoidal manifold (ie a weak solenoid) to a dynamical commensurator group, and give conditions for when this map is either surjective or an isomorphism. Odden proved that this map is an isomorphism for the mapping class of the universal hyperbolic solenoid; Bering and Studenmund proved that the mapping class group of a universal solenoid over a compact K(G,1) manifold maps onto the commensurator group of G. We extend the results of both of these papers to arbitrary solenoidal manifolds. This work is joint with Olga Lukina.

Newelski introduced tools from topological dynamics into model theory to generalize the notion of a generic type from stable group theory to arbitrary first-order theories. He adapted Ellis’s theory to the definable setting and introduced what is now known as the Ellis Group Conjecture, which relates the Ellis group of the space of types concentrating on a definable group G, viewed as a G-flow, to the maximal compact quotient of G, a model-theoretic invariant, under suitable tameness assumptions. The conjecture has been established for definably amenable groups in NIP theories by Chernikov and Simon. In this talk, we will outline the conjecture’s motivation, introduce the key concepts surrounding it, and discuss possible directions for further research.

Precision medicine is the future of drug development, and subgroup identification plays a critical role in achieving the goal. In this presentation, we propose a powerful end-to-end solution squant (available on CRAN) that explores a sequence of quantitative objectives. The method converts the original study to an artificial 1:1 randomized trial, and features a flexible objective function, a stable signature with good interpretability, and an embedded false discovery rate (FDR) control. We demonstrate its performance through simulation and provide a real data example.

The study of classical scissors congruence addresses the following problem: if we cut a polytope up into finitely many pieces, and we are allowed to rearrange those pieces by isometries, which polytopes can we build? Higher scissors congruence, or scissors congruence K-theory, expands on the study of scissors congruence by not only searching for equivalent polytopes, but also equivalences between the symmetries themselves. Computations in higher scissors congruence are computations in K-theory, which are notoriously difficult. As such, higher scissors congruence groups have only been computed in dimension 1 for euclidean, spherical, and hyperbolic geometries — dimension 2 and above have been open. My results prove that for 2-dimensional euclidean space, higher scissors congruence groups are uncountable in almost every degree, with the potential exception of degree 1, where the group was conjectured to vanish prior to this work. I will walk the viewer through the strategy of computing approximations of all of the higher scissors congruence groups, and then push uncountably many nontrivial classes forward from these approximations of the groups into the higher scissors congruence groups themselves.

We begin by presenting work of the author and Julia Wolf from 2021 showing that any subset of an elementary abelian $p$-group of bounded VC_2-dimension is well approximated by a union of atoms of a quadratic factor of bounded complexity. This result relies on a general quadratic arithmetic regularity lemma of Green and Tao, and consequently, yields bounds on the linear and quadratic complexities of the factor which are of tower-type in $\varepsilon^{-1}$, where $\varepsilon$ is the approximation parameter. We then present more recent work, also joint with Julia Wolf, which shows the bound on the quadratic complexity of the factor appearing in the structure theorem for sets of bounded VC_2-dimension can improved drastically, specifically to a logarithm in a power of $\varepsilon^{-1}$.

In a series of results dating back to Morin, it is shown that smooth hypersurfaces in a large number of variables are unirational. The basic technique shows an important relationship between the spaces of k-planes in these hypersurfaces and their unirationality. We investigate these questions using the notion of strength coming from commutative algebra. In particular, we prove that hypersurfaces having high secondary strength are also unirational, providing a new source of examples of (singular) unirational hypersurfaces. Along the way, we see that notions of strength allow for a very short proof of a weak form of the de Jong-Debarre conjecture. This is joint with Daniel Erman.

In many applications of scientific and engineering interest, the accurate modeling of linear waves scattered by quasiperiodic media plays a crucial role. The ability to numerically simulate such configurations robustly and rapidly is of overwhelming importance in photonics applications. In this talk we will discuss the specific problem of electromagnetic radiation interacting with a two-dimensional multiply layered diffraction grating with quasiperiodic interfaces. We describe how the classical boundary perturbation method of Field Expansions can be extended to this two-dimensional problem, and with specific numerical experiments we will show the remarkable efficiency, fidelity, and high-order accuracy one can achieve with an implementation of this algorithm.

In a recent joint work with Michael Bersudsky and Nimish Shah, we study the homogeneous equidistribution phenomenon of polynomially bounded o-minimal curves in homogeneous spaces --- in particular, the limiting distribution of $\{ \phi(t) \mathbb Z^n \}$ in the space of unimodular lattices in $\mathbb R^n$, where $\phi(t)$ is an $n \times n$ matrix curve of determinant 1 whose coordinate functions are definable in a polynomially bounded o-minimal structure (which is a large family of functions that includes rational functions and more), and discuss an important condition for this equidistribution to hold. This extends the earlier work of Shah for polynomial trajectories and the work of Peterzil and Starchenko on trajectories on nilmanifolds that are definable in a polynomially bounded o-minimal structure. The talk will be made accessible to a general audience without a background in model theory or homogeneous dynamics.

In network settings, interference between units makes causal inference more challenging as outcomes may depend on the treatments received by others in the network. Typical estimands in network settings focus on treatment effects aggregated across individuals in the population. We propose a framework for estimating node-wise counterfactual means, allowing for more granular insights into the impact of network structure on treatment effect heterogeneity. We develop a doubly robust and non-parametric estimation procedure, KECENI (Kernel Estimator of Causal Effect under Network Interference), which offers consistency and asymptotic normality under network dependence. The utility of this method is demonstrated through an application to microfinance data, revealing the node-wise impact of network characteristics on treatment effects.

Many geometric spaces come equipped with a natural collection of special submanifolds that reflect their internal symmetries. Examples include abelian varieties with their sub-abelian varieties, locally symmetric spaces with totally geodesic subspaces, period domains with sub–period domains, and strata of abelian differentials with affine invariant submanifolds. In recent years, significant progress has been made in understanding such structures through the lens of unlikely intersections and functional transcendence. I will outline the general framework of variations of Hodge structures and period domains, and explain how the so-called completed Zilber–Pink philosophy provides a unifying way to describe the qualitative behaviour of these special loci. This perspective reveals deep connections between arithmetic geometry, Hodge theory, and dynamical systems.

We generalize an intersection-theoretic local inequality of Fulton–Lazarsfeld to weighted blowups. Using this together with the classification of 3-dimensional divisorial contractions, we prove nonrationality of many families of terminal Fano 3-folds. This is a joint work with Igor Krylov and Takuzo Okada.

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