# MSCS Seminar Calendar

Monday October 18, 2021

**Combinatorics and Probability Seminar**

Faster algorithms for counting independent sets in regular bipartite graphs

Aditya Potukuchi (UIC)

2:00 PM in 636 SEO

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.

**Algebraic Geometry Seminar**

Test ideals for quasi-projective schemes in mixed characteristic

Karl Schwede (University of Utah)

3:00 PM in Zoom

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.

**Analysis and Applied Mathematics Seminar**

The Sharp Erdős-Turán Inequality

Ruiwen Shu (University of Oxford)

4:00 PM in Zoom

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.

Tuesday October 19, 2021

**Logic Seminar**

The canonical base property and CM-triviality

Leo Jimenez (Fields)

3:00 PM in 636 SEO

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.

**Logic Seminar**

On the Pila-Wilkie theorem

Neer Bhardwaj (UIUC)

4:00 PM in 636 SEO

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.

Wednesday October 20, 2021

**Statistics and Data Science Seminar**

Information-based Optimal Subdata Selection for Clusterwise Linear Regression Model

Yanxi Liu (University of Illinois, Chicago)

4:00 PM in Zoom

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.

Thursday October 21, 2021

Monday October 25, 2021

**Combinatorics and Probability Seminar**

How often is a random symmetric matrix invertible?

Marcus Michelen (UIC)

2:00 PM in 636 SEO

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.

**Algebraic Geometry Seminar**

Vanishing theorems in equal characteristic zero

Takumi Murayama (Princeton University)

3:00 PM in Zoom

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.

**Analysis and Applied Mathematics Seminar**

Efficient distribution classification via optimal transport embeddings

Caroline Moosmueller (UCSD)

4:00 PM in 636 SEO

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.

Tuesday October 26, 2021

Wednesday October 27, 2021

**Statistics and Data Science Seminar**

Evidence factors from multiple, possibly invalid, instrumental variables

Youjin Lee (Brown University)

4:00 PM in Zoom

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.

Monday November 1, 2021

**Combinatorics and Probability Seminar**

Random polynomials near the unit circle

Oren Yakir (Tel Aviv)

2:00 PM in Zoom

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.

**Analysis and Applied Mathematics Seminar**

Applications of the Shear-flow Induced Enhanced Dissipation

Siming He (Duke University)

4:00 PM in Zoom

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.

Tuesday November 2, 2021

Wednesday November 3, 2021

**Statistics and Data Science Seminar**

Balancing Inferential Integrity and Disclosure Risk via Model Targeted Masking and Multiple Imputation

Bei Jiang (University of Alberta)

4:00 PM in Zoom

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).

Thursday November 4, 2021

Monday November 8, 2021

**Analysis and Applied Mathematics Seminar**

The reference map technique for simulating complex materials and multi-body interactions

Chris Rycroft (Harvard)

4:00 PM in 636 SEO

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.

Wednesday November 10, 2021

Friday November 12, 2021

**Departmental Colloquium**

Symmetry, invariance and the structure of matter

Richard D. James (University of Minnesota)

3:00 PM in 636 SEO

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.

Monday November 15, 2021

Wednesday November 17, 2021

Friday November 19, 2021

Monday November 22, 2021

Monday November 29, 2021

**Algebraic Geometry Seminar**

Quotient singularities in positive characteristic

Christian Liedtke (Technical University Munich)

2:00 PM in Zoom

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).

Wednesday December 1, 2021

Thursday December 2, 2021

Wednesday March 9, 2022