I will present an algorithm that takes as input a d-regular bipartite graph G, runs in time exp(O(n/d log^3 d)), and outputs w.h.p., a (1 + o(1))-approximation to the number of independent sets in G. As a by-product of the intermediate steps to this algorithm, We also obtain, for fixed d, an FPTAS to approximate the number of independent sets in d-regular bipartite ``expanding'' graphs. More than the result itself, I will focus on the techniques used, which combine combinatorial methods (graph containers) with statistical physics methods (abstract polymer models and cluster expansion) and mention other recent applications. I will start from the basics, and no prior knowledge of any of the topics is assumed. Joint work with Matthew Jenssen and Will Perkins.

Building on breakthrough results of Andr\'e, Bhatt, Gabber and others, Ma and the speaker introduced a theory of mixed characteristic test ideals / multiplier ideals. There was a gap in this theory, it was defined only for complete local rings and the formation of these ideals did not seem to commute with localization. By utilizing ideas from Bhatt-Ma-Patakfalvi-Tucker-Waldron-Witsazek and the author (also see Takamatsu-Yoshikawa), we introduce a notion of multiplier / test ideals for normal schemes finite type over a complete local ring (in particular, our notion commutes with localization). We use our theory to study the non-nef locus and so obtain mixed characteristic versions of results on the non-nef locus for varieties over fields due to Ein-Lazarsfeld-Mustata-Nakamaye-Popa, Mustata, and Nakayama. This is joint work with Christopher Hacon and Alicia Lamarche.

Erdős and Turán proved a classical inequality on the distribution of roots for a complex polynomial in 1950, depicting the fundamental interplay between the size of the coefficients of a polynomial and the distribution of its roots on the complex plane. Various results have been dedicated to improving the constant in this inequality, while the optimal constant remains open. In this paper, we give the optimal constant, i.e., prove the sharp Erdős-Turán inequality. To achieve this goal, we reformulate the inequality into an optimization problem, whose equilibriums coincide with a class of energy minimizers with the logarithmic interaction and external potentials. This allows us to study their properties by taking advantage of the recent development of energy minimization and potential theory, and to give explicit constructions via complex analysis. Finally the sharp Erdős-Turán inequality is obtained based on a thorough understanding of these equilibrium distributions.

In geometric stability theory, the duality between locally modular and not plays a central structural role. For example, in many important cases, non-local modularity implies the interpretability of a field, a phenomenon known as Zilber's dichotomy. Hrushovski's classical counterexample to this behavior, while not locally modular, still has a relatively rudimentary forking geometry: it is CM-trivial, which prevents the interpretability of a field. More recently, a relative generalization of local modularity, inspired by the behavior of compact complex spaces, was defined: the Canonical Base Property (CBP). It was shown to hold in many key structures, for example DCF0, where it was used by Pillay and Ziegler to show Zilber's dichotomy. It was first conjectured that the CBP held for all superstable finite rank structures, until Hrushovski, Palacín and Pillay produced the first counterexample, as a reduct of an algebraically closed field of characteristic zero. Since then, all counterexamples produced involved a field, and it is natural to ask if this is necessary. In this talk, I will answer this question negatively by presenting a CM-trivial structure without the CBP. Joint with Thomas Blossier.

I’ll give an account of the Pila-Wilkie counting theorem and some of its extensions and generalizations. We exploit semialgebraic cell decomposition more thoroughly to simplify the deduction from the main ingredients of the original proof. Only very basic knowledge of o-minimality will be assumed; this is joint work with Prof. Lou van den Dries.

As the data size increases rapidly, the relationship between input and output variables may not be homogeneous anymore. Conventional statistical models such as generalized linear models (GLMs) may not be well-suited to heterogeneous relationships. Using a Mixture of Expert models is a good solution. The Mixture of Expert models can combine different statistical models to detect heterogeneous patterns while maintaining the benefits of conventional statistical modeling techniques. However, it needs a considerable amount of computer resources, particularly when working with big data. To address this issue, an attractive idea is to analyze a subsample of the data retaining the rich information of the full data. Information-Based Optimal Subdata Strategy (IBOSS), proposed by Wang et al. (2019), is such a strategy. The IBOSS strategy captures most of the relevant information in the full data through a judicious selection of the subdata by "maximizing" the Fisher information matrix. This project aims to develop an algorithm for the Clusterwise Linear Regression model, a type of Mixture of Experts, to select subdata based on IBOSS strategy. However, the Fisher information matrix of the model has no explicit form, which is a major challenge of the work. To overcome this challenge, we propose a surrogate matrix which is proved to be asymptotically equivalent to the Fisher information matrix, and it is used to construct the IBOSS subdata. Further, the proposed subdata selection is proved to be asymptotically optimal, i.e., no other method is statistically more efficient than the proposed one when the full data size is large.

You've heard of Topology without Tears, now get ready for...

Let A_n be chosen uniformly at random from the set of n x n symmetric matrices with entries -1 or 1. How often is A_n invertible? While this is similar to the problem discussed in the previous week, many of the tools used for the asymmetric case (provably) only go so far for the symmetric case. I'll discuss a recent work of mine (joint with Marcelo Campos, Matthew Jenssen, and Julian Sahasrabudhe) that shows the first exponential upper bound on the probability that A_n is singular.

In 1953, Kodaira proved what is now called the Kodaira vanishing theorem, which states that if L is an ample divisor on a complex projective manifold X, then H^i(X,-L) = 0 for all i < dim(X). Since then, Kodaira's theorem and its generalizations due to Grauert–Riemenschneider, Kawamata–Viehweg, Kollár, and others have become indispensable tools in algebraic geometry over fields of characteristic zero, in particular in birational geometry and the minimal model program. Even in this context, however, it is often necessary to work with schemes that are not of finite type over fields, and a fundamental problem in this more general context has been the lack of Kodaira-type vanishing theorems. We prove generalizations of Kodaira's vanishing theorem for proper morphisms of schemes of equal characteristic zero in arbitrary dimension, answering questions of Boutot, Kollár, and Kawakita. These results are optimal given known counterexamples to these vanishing theorems in positive and mixed characteristic.

Detecting differences and building classifiers between distributions, given only finite samples, are important tasks in a number of scientific fields. Optimal transport (OT) has evolved as the most natural concept to measure the distance between distributions, and has gained significant importance in machine learning in recent years. There are some drawbacks to OT: Computing OT is usually slow, and it often fails to exploit reduced complexity in case the family of distributions is generated by simple group actions. In this talk, we discuss how optimal transport embeddings can be used to deal with these issues, both on a theoretical and a computational level. In particular, we'll show how to embed the space of distributions into an L^2-space via OT, and how linear techniques can be used to classify families of distributions generated by simple group actions in any dimension. The proposed framework significantly reduces both the computational effort and the required training data in supervised settings. We demonstrate the benefits in pattern recognition tasks in imaging and provide some medical applications.

Instrumental variables have been widely used to estimate the causal effect of a treatment on an outcome in the presence of unmeasured confounders. When several instrumental variables are available and the instruments are subject to possible biases that do not completely overlap, a careful analysis based on these several instruments can produce orthogonal pieces of evidence (i.e., evidence factors) that would strengthen causal conclusions when combined. We develop several strategies, including stratification, to construct evidence factors from multiple candidate instrumental variables when invalid instruments may be present. Our proposed methods deliver nearly independent inferential results each from candidate instruments under the more liberally defined exclusion restriction than the previously proposed reinforced design. We apply our stratification method to evaluate the causal effect of malaria on stunting among children in Western Kenya using three nested instruments that are converted from a single ordinal variable. Our proposed stratification method is particularly useful when we have an ordinal instrument of which validity depends on different values of the instrument. This is based on joint work with Anqi Zhao, Dylan Small, and Bikram Karmarkar.

It is well known that a random polynomial with iid coefficients has most of its roots close to the unit circle. Recently, Michelen and Sahasrabudhe found the limiting distribution for the closest root to the unit circle, in the case of Gaussian coefficients. We give a different proof of their result, which shows that the limit distribution is in fact universal (i.e. remains true for general coefficient distribution). Our new proof is inspired by earlier works of Konyagin and Schlag on the minimum modulus of the polynomial on the unit circle itself. Joint works with Nick Cook, Hoi Nguyen and Ofer Zeitouni.

In this talk, we consider the enhanced dissipation phenomena induced by shear flows. In the first part of the talk, I will introduce the idea of shear flow-induced enhanced dissipation and the recent developments on this topic. Then I will exhibit the applications of this phenomenon in various settings, ranging from suppression of chemotactic blow-ups to enhancement of chemical reactions.

There is a growing expectation that data collected by government-funded studies should be openly available to ensure research reproducibility, which also increases concerns about data privacy. A strategy to protect individuals' identity is to release multiply imputed (MI) synthetic datasets with masked sensitivity values (Rubin, 1993). However, information loss or incorrectly specified imputation models can weaken or invalidate the inferences obtained from the MI-datasets. We propose a new masking framework with a data-augmentation (DA) component and a tuning mechanism that balances protecting identity disclosure against preserving data utility. Applying it to a restricted-use Canadian Scleroderma Research Group (CSRG) dataset, we found that this DA-MI strategy achieved a 0% identity disclosure risk and preserved all inferential conclusions. It yielded 95% confidence intervals (CIs) that had overlaps of 98.5% (95.5%) on average with the CIs constructed using the full, unmasked CSRG dataset in a work-disability (interstitial lung disease) study. The CI-overlaps were lower for several other methods considered, ranging from 73.9% to 91.9% on average with the lowest value being 28.1%; such low CI-overlaps further led to some incorrect inferential conclusions. These findings indicate that the DA-MI masking framework facilitates sharing of useful research data while protecting participants' identities. This a joint work with Adrian Raftery (University of Washington), Russel Steele (McGill University) and Naisyin Wang (University of Michigan).

Conventional computational methods often create a dilemma for fluid–structure interaction problems. Typically, solids are simulated using a Lagrangian approach with grid that moves with the material, whereas fluids are simulated using an Eulerian approach with a fixed spatial grid, requiring some type of interfacial coupling between the two different perspectives. Here, a fully Eulerian method for simulating structures immersed in a fluid will be presented. By introducing a reference map variable to model finite-deformation constitutive relations in the structures on the same grid as the fluid, the interfacial coupling problem is highly simplified. The method is particularly well suited for simulating soft, highly-deformable materials and many-body contact problems, and several examples in two and three dimensions will be presented.

We begin with a general introduction to our work on “objective structures” with a focus on the relation between symmetry, invariance and structure, and applications to the periodic table, origami design and nanostructures. We then concentrate on Maxwell’s equations. We find solutions of Maxwell’s equations that are the precise analog of plane waves, but in the case that the translation group is replaced by the (largest) Abelian helical group. These waves display constructive/destructive interference with helical atomic structures, in the same way that plane waves interact with crystals. We show how the resulting far-field pattern can be used for structure determination, and we test the idea theoretically on the Pf1 virus from the Protein Data Bank. The underlying mathematical idea of this and our related work is: the structure of interest is the orbit of a group, and this group is an invariance group of the differential equations.

We study isolated quotient singularities by finite group schemes in positive characteristic. We compute invariants, study the uniqueness of the quotient presentation, and compute some deformation spaces. A special emphasis is laid on the dichotomy between quotient singularities by linearly reductive group schemes and by group schemes that are not linearly reductive. We essentially classify the linearly reductive ones, give applications, and make some conjectures. This is joint work with Gebhard Martin (Bonn) and Yuya Matsumoto (Tokyo).