Given an ideal \(I\) in a regular local ring \(A\), the cohomological dimension of \(I\) in \(A\) is the index of the highest non-vanishing local cohomology of \(A\) supported at \(I\). Determining effective upper bounds on the cohomological dimension in terms of topological invariants of \(\textrm{Spec}(A/I)\) is a central problem in commutative algebra: foundational results include the Hartshorne--Lichtenbaum Vanishing Theorem and the Second Vanishing Theorem.

In equal characteristic, Faltings established in 1980 a general bound on the cohomological dimension of an ideal in terms of its “big height”. In this talk, we extend Faltings’ results to the unramified mixed characteristic setting and show that the resulting bound is sharp.

To a triple of a ring \(R\), an \(R\)-module \(M\), and an ideal \(I\) we can associate the local cohomology modules \(H^n_I(M)\). One classical problem dating back to a question of Grothendieck is to identify when these vanish for all \(n > i\) and all modules \(M\), the smallest such \(i\) we denote the cohomological dimension of \(R\) with respect to \(I\). Grothendieck showed that this is bounded by \(d = \dim R\), while a very simple condition on \(\widehat{R}\) and \(I\widehat{R}\) controls the vanishing at \(d\). A more subtle topological condition controls the vanishing at \(d-1\) for regular local rings as posed by Hartshorne in the late 60s, and was proven in equicharacteristic in the early 70s. After 50 years it was shown to be true in unramified mixed characteristic, and the main result of this talk is the ramified mixed characteristic case.

Discriminant is a basic invariant associated with an algebraic number field. Its notion was
first introduced by Dedekind in 1871. The problem of effective computation of discriminant as
well as an integral basis of an infinite family of algebraic number fields which are defined over
the field $\mathbb{Q}$ of rational numbers by certain types of irreducible polynomials has been tackled
by several mathematicians. In this lecture we shall review the progress made regarding this problem in the case of pure fields. We shall present a formula for the exact power of any prime $p$ dividing the discriminant of the field  $K_n=\mathbb{Q}(\alpha_n),$ where $\alpha_n\in\mathbb{C}$ is a root of the $n$th exponential Taylor polynomial
$\frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}+\cdots+\frac{x^2}{2!}+\frac{x}{1!}+1,
$ in terms of the $p $-adic expansion of positive integer $n$. We also describe an explicit $p$-integral basis of $K_n$ for each prime $p$. These local bases lead naturally to the construction of an integral basis of $K_n$.

Modern generative diffusion models are distinguished by their ability to generalize, consistently and robustly, in very high dimensional spaces. They produce a combinatorial explosion of novel images from a relatively small training set, subverting normal concerns about the curse of dimensionality. Yet, their generations also sometimes fall short, exhibiting distinctive flaws such as spatial inconsistency (e.g. excessive limbs). I will discuss an analytical theory that, making only the assumptions of A) locality and B) (broken) equivariance, explains 1) how models are able to generalize combinatorially, mixing and matching from different images in their training data, 2) why models are able to generalize consistently and robustly in high dimensional spaces, and 3) mechanistically explains the origins of spatial consistency issues such as the “excess limbs” phenomenon. This theory is totally solvable in terms of the training dataset, and we show that it is able to predict on a case-by-case basis the behavior of certain classes of weak diffusion models: 1) small convolutional neural networks, and 2) diffusion models early in their training process. I will then comment on what is still needed to further explain the mysteries of generalization in more powerful models.

When a diffusion model is not memorizing the training samples, what does it generate, and why? In this talk, I will describe a quantitative framework for understanding the distribution produced by a learned diffusion model through a data-driven geometric object: a log-density ridge manifold of the smoothed training distribution. This manifold acts as a backbone for generation and reveals a three-stage inference behavior: trajectories first reach the ridge, then align in normal directions, and finally slide along tangent directions. 

This perspective allows us to quantify how training error influences generation in different directions, and to explain when inter-mode generations arise. I will also present a random feature example in which the model’s inductive bias can be decomposed explicitly into architectural bias and optimization error, and tracked along the inference dynamics. Experiments on synthetic multimodal distributions and MNIST latent diffusion support the theory in both low- and high-dimensional settings.

The goal of this talk is to consider two instances of a class of reconstruction problems that aim to recover an unknown signal from indirect measurements that are algebraic in nature. Such problems are paramount in mathematics, enjoying applications in a wide array of fields like molecular imaging, machine learning, and geo-positioning. In this talk, we will motivate and study the generic crystallographic phase retrieval problem and the orthogonal beltway problem and deduce conditions in each setting that guarantee signal recovery. In the latter problem, we will also develop a polynomial-time algorithm that recovers the signal while remaining robust to small amounts of noise. We resolve both of the central problems of this talk by recasting them as special instances of the problem of recovering a signal from its second moment under the multi-reference alignment (MRA) model, for which a rich theory has been developed in prior literature.

Abstract: Given a simplicial projective toric variety with at most one singular point, we study the asymptotic structure of the sumsets arising from its parametrization. We introduce a notion of sumsets regularity and relate it to the Castelnuovo–Mumford regularity of the variety, translating an algebraic problem into a combinatorial one. We then obtain new upper bounds on the sumset regularity, and hence on the Castelnuovo–Mumford regularity of these varieties. This talk is based on joint work with Ignacio García-Marco and Philippe Gimenez. 

Density functional theory is a standard tool for computing energies (eigenvalues) throughout chemistry and in many parts of physics and materials science. I will present the basics of the subject and then show how semiclassical analysis can improve the accuracy of its results. Specifically, we compensate errors in approximate density functionals using a normalization correction derived from semiclassical spectral asymptotics.