The problem of algorithmic bias, where machine learning algorithms reflect biases that are prevalent in their training datasets, is widely recognized as a major concern. In this talk, I will discuss two of my projects related to algorithmic biases that are caused by underrepresentation of minority groups. In the first project, we demonstrate that when learning representations from standard contrastive learning methods, the representations of minority groups merge with the representations of certain similar majority groups. We refer to this phenomenon as representation harm and demonstrate that it leads to allocation harms in downstream classification tasks. In the second project, we investigate whether enforcing group fairness is aligned with improving model performance. In light of the long-held belief that enforcing fairness comes at the cost of reduced model performance, we present an alternative perspective on the problem. In cases where the machine bias is due to the underrepresentation of minority groups, we show that enforcing fairness is often in line with improving model performance on a balanced test dataset. Furthermore, we derive necessary and sufficient conditions for such an alignment. 

Algebraic geometry is the study of shapes defined by polynomial equations called algebraic varieties. One natural approach to study them is to construct a moduli space, which is a space parameterizing such shapes of a given type (e.g. algebraic curves). After surveying this topic, I will focus on the problem of constructing moduli spaces parametrizing Fano varieties, which are a class of positively curved complex manifolds that form one of the three main building blocks of varieties in algebraic geometry. While algebraic geometers once considered this problem intractable due to various pathologies that occur, it has recently been solved using K-stability, which is an algebraic definition introduced by differential geometers to characterize when a Fano variety admits a Kahler-Einstein metric.

The index for 1D quantum cellular automata (QCA) was introduced to measure the flow of the information by Gross, Nesme, Vogts, and Werner. Interpreting the index as the ratio of the Jones index for subfactors leads to a generalization of the index defined for QCA on more general abstract spin chains. These include fusion spin chains, which arise as the local operators invariant under a global (categorical/MPO) symmetry, and as the boundary operators on 2D topologically ordered spin systems. We introduce our generalization of index and show that it is a complete invariant for the group of QCA modulo finite depth circuits for the fusion spin chains built from the fusion category Fib. This talk is based on a joint work with Corey Jones.

We survey some of the newest results about ordered Ramsey numbers of graphs, that is, about a variant of Ramsey numbers for graphs with linearly ordered vertex sets. In particular, we will focus on one of the well-known problems in the area about estimating on off-diagonal ordered Ramsey numbers of ordered matchings versus a triangle. This is a joint work with Marian Poljak.

I will discuss joint work with Jacek Jendrej and Wilhelm Schlag about the two dimensional harmonic map heat flow for maps taking values in the sphere. It has been known since the 80s-90’s that solutions can exhibit bubbling along a well-chosen sequence of times — the solution decouples into a superposition of well-separated harmonic maps and a body map accounting for the rest of the energy. We prove that every sequence of times contains a subsequence along which such bubbling occurs. This is deduced as a corollary of our main theorem, which shows that the solution approaches the family of multi-bubbles in continuous time. The proof is partly motivated by the classical theory of dynamical systems and uses the notion of “minimal collision energy” developed in joint work with Jendrej on the soliton resolution conjecture for nonlinear waves. 

Classical Scaling is perhaps the main method for Multidimensional Scaling (MDS), which is an area of Statistics (although initiated in Psychometrics) where the central task is the embedding of a weighted graph as a configuration of points in a Euclidean space in such a way as to match, as much as possible, the edges weights with the Euclidean distance between the corresponding points. The presentation will introduce this old method (dating back to the 1930s) and then go over more recent advances (last 20 years) in terms of computation, perturbation bounds, and more. 

Bayesian estimation and uncertainty quantification depend on prior assumptions. These assumptions are often chosen to promote specific features in the recovered estimate like sparsity. The form of the chosen prior determines the shape of the posterior distribution, thus the behavior of the estimator, and the complexity of the associated optimization and sampling problems. Here, we consider a family of Gaussian hierarchical models with generalized gamma hyperpriors designed to promote sparsity in linear inverse problems. By varying the hyperparameters we can move continuously between priors that act as smoothed ℓp penalties with flexible p, smoothing, and scale. We introduce methods for efficiently tracking MAP solutions along paths through hyperparameter space. Path following allows a user to explore the space of possible estimators under varying assumptions and to test the robustness and sensitivity of solutions to changes in the prior assumptions. By tracing paths from a convex region to a non-convex region, the user can find local minimizers in non-convex, strongly sparsity-promoting regimes that are consistent with a convex relaxation drawn from the same family of posteriors. We show experimentally that these solutions are less error-prone than direct optimization of the non-convex problem. The same relaxation approach allows sampling from highly non-convex multi-modal posteriors in high dimension via a variational Bayesian formulation. We demonstrate predictor-corrector methods for estimator and sample continuation.

Translation surfaces are geometric objects that can be defined as a collection of polygons with sides identified in parallel opposite pairs by translation. If we generalize slightly and allow for polygons with sides identified by both translation and dilation, we get a new family of objects called dilation surfaces. While translation surfaces are well-studied, much less is known about dynamics on dilation surfaces and their moduli spaces. In this talk, we will survey recent progress in understanding the topology of moduli spaces of dilation surfaces. We will do this by understanding the action of the mapping class group on the moduli space of dilation surfaces. This talk represents joint work with Paul Apisa and Matt Bainbridge.

I will present an analog of the Chebotarev Density Theorem which works over local fields. As an application, I will use it to prove a conjecture of Bhargava, Cremona, Fisher, and Gajović. This is joint work with Asvin G and Yifan Wei.

[pre-talk at 1:20PM]

Ligand-receptor binding and unbinding are fundamental molecular processes, whereas water fluctuations impact strongly their thermodynamics and kinetics. We develop a variational implicit-solvent model (VISM) and a fast binary level-set method to calculate the potential of mean force and the molecule-water interfacial structures for dry and wet states. Monte Carlo simulations with our model and method provide initial configurations for efficient molecular dynamics simulations. Moreover, combined with the string method and stochastic simulations of ligand molecules, our hybrid approach enables the prediction of the transition paths and rates for the dry-wet transitions and the mean first-passage times for the ligand-pocket binding and unbinding. Without any explicit description of individual water molecules, our predictions are in a very good, qualitative and semi-quantitative, agreement with existing explicit-water molecular dynamics simulations. 

This talk reviews a series of works done in collaboration with L.-T. Cheng, S. Zhou, Z. Zhang, S. Liu, H.-B. Cheng, J. Dzubiella, C. Ricci, and J. A. McCammon. 

A translation surface is a collection of polygons with edge identifications given by translations. In spite of the simplicity of the definition, the space of translation surfaces has connections to different areas of math such as the moduli space of hyperbolic surfaces. A  guiding principle centers on turning questions on a fixed translation surface into a dynamical one on the space of all translation surfaces. We consider an instance of this philosophy related to the slope gap distribution of holonomy vectors of a translation surface. We use this as a jumping off point to consider expanding translates in different spaces such as non-arithmetic hyperbolic manifolds. Aspects of this talk represent different works with L. Kumandari and J. Wang, and with K. Ohm.

Groups of matrices with integer entries, aka arithmetic groups, are prominent objects of mathematics.From a geometric point of view, they appear as the fundamental groups of locally symmetric space. Topological invariants of such spaces could be seen as group invariants and vice versa. 

In my talk I will relate this useful link between topology and arithmetics with the theory of unitary representations. More precisely, I will focus on the cohomology of arithmetic groups with unitary coefficients, presenting a recent joint work with Roman Sauer which completely clarifies the theory in small degrees.

By the end of the talk I will discuss the relation of the above with the phenomenon of spectral gap and state various related conjectures.

I will make an effort to present the subject to a general audience. 

Constrained estimation problems are prevalent in statistics, machine learning, and engineering. These problems encompass constrained generalized linear models, constrained deep neural networks, physics-inspired machine learning, algorithmic fairness, and optimal control. However, existing estimation methods under hard constraints rely on either projection or regularization, which may theoretically exhibit optimal efficiency but are impractical or unreasonably fail in reality. This talk aims to bridge the significant gap between practice and theory for constrained estimation problems.

I will begin by introducing the critical methodology used to bridge the gap, called Stochastic Sequential Quadratic Programming. We will see that SQP methods serve as the workhorse for modern scientific machine learning problems and can resolve the failure modes of prevalent regularization-based methods. I will demonstrate how to make SQP adaptive and scalable using various modern techniques, such as stochastic line search, trust region, and dimensionality reduction. Additionally, I will show how to further enhance SQP to handle inequality constraints online.

Following the methodology, I will present some selective theories, emphasizing the consistency and efficiency of the SQP methods. Specifically, I will show that online SQP iterates asymptotically exhibit normal behavior with a mean of zero and optimal covariance in the Hajek and Le Cam sense. Significantly, the covariance does not deteriorate even when we apply modern techniques driven by practical concerns. The talk concludes with experiments on both synthetic and real datasets.