February 3

Hamilton Sawczuk

• Title: Line Graphs, Graham's Conjecture, and the WL-1 Algorithm

• Abstract: The line graph construction replaces edges of a graph by vertices. While a single application is well understood, the behavior of graphs under repeated iteration is less clear. This talk explores the dynamics of iterated line graphs through examples with a focus on the Graham sequence, which records vertex counts of successive iterates. We discuss Graham equivalence and Graham’s Tree Reconstruction Conjecture. Finally, we discuss connections with the Weisfeiler–Leman (WL-1) and "forgetful walks" which lead us to a conjecture of our own.

February 10

Aaron Zoll

• Title: What is… Interpolation Theory?

• Abstract: The design of optimal methods for a class of optimization problems (e.g. optimized gradient span methods for smooth, convex minimization) has been of recent interest. While many methods have been optimized up to their big-O complexity for decades, only recently have we begun to close the gap between upper and lower bounds in terms of their constants. However, this problem of optimizing optimization methods inherently possesses infinite constraints despite the iterative methods we hope to optimize containing only a finite amount of data. Interpolation theory aims to bridge the domains between discrete observations—the points, function values, and gradient seen by the algorithm—and continuous function which build the allowable problems over which we measure an algorithms performance. This talk will outline the necessary and sufficient conditions for an interpolation to exist for a wide variety of examples. With enough time, the talk will further discuss applications and recent extensions. 

February 17

Ian McPherson

• Title: Neural Dynamic Portfolio Control with Provable Learning Guarantees

• Abstract: high-dimensional dynamic portfolio optimization, but existing methods typically rely on parametric return models or prespecified Markovian state representations and are largely supported by empirical evidence rather than comprehensive performance guarantees. We develop a provable learning framework for discrete-time, finite-horizon portfolio control that takes a step toward closing these gaps. We parameterize the investor’s decision at each rebalancing time as a monetary allocation that depends directly on the realized return path, rather than as portfolio weights modeled as functions of Markovian states. This formulation is operationally natural, avoids explicit dynamics modeling, and accommodates return processes with general temporal dependence. Under self-financing dynamics and concave terminal utility, the resulting objective is convex with respect to the policy function. Exploiting this geometry, we model each period’s policy using a wide two-layer neural network and, via a mean-field analysis, establish global convergence guarantees for noisy gradient descent with weight decay. In particular, the geometric convergence rate is independent of the planning horizon. We further derive finite-sample bounds that control both generalization error and regularization bias. Numerical experiments show that these theoretical guarantees come with empirical competitive performance relative to standard baselines across representative return environments. Joint Work: Yizhe Huang, Rui Gao, Shuang Li, Luhao Zhang.

February 24

Pedro Izquierdo Lehmann

• Title: Active Set Identification and Rapid Local Convergence

•  for Primal-Dual Degenerate Problems

• Abstract: Primal-dual methods for solving convex optimization problems with functional constraints often exhibit a distinct two-stage behavior. Initially, they converge towards a solution at a sublinear rate. Then, after a certain point, all iterates accurately identify the set of active constraints, and their convergence accelerates to linear. Theory characterizing this phenomenon spans over three decades. However, most existing work only guarantees eventual active set identification and relies heavily on nondegeneracy conditions, such as strict complementarity, which often fail to hold in practice. We characterize mild conditions on the problem geometry and algorithm under which this phenomenon provably occurs. Our guarantees are entirely nonasymptotic and, importantly, do not rely on strict complementarity. Our framework encompasses several widely-used algorithms, including the proximal point method, primal-dual hybrid gradient method, alternating direction method of multipliers, and extragradient method.

March 3

Wilson Gregory

• Title: Tensor learning with orthogonal, Lorentz, and symplectic symmetries

• Abstract: Tensors are a fundamental data structure for many scientific contexts, such as time series analysis, materials science, and physics, among many others. Improving our ability to produce and handle tensors is essential to efficiently address problems in these domains. In this paper, we show how to exploit the underlying symmetries of functions that map tensors to tensors. More concretely, we develop universally expressive equivariant machine learning architectures on tensors that exploit that, in many cases, these tensor functions are equivariant with respect to the diagonal action of the orthogonal, Lorentz, and/or symplectic groups. We showcase our results on three problems coming from material science, theoretical computer science, and time series analysis. For time series, we combine our method with the increasingly popular path signatures approach, which is also invariant with respect to reparameterizations. Our numerical experiments show that our equivariant models perform better than corresponding non-equivariant baselines.

March 10

Hongyu Cheng

• Title: Linear Threshold for Oertel's Conjecture on the Mixed-Integer Volume

• Abstract: Grünbaum's inequality guarantees that the centroid of a convex body has halfspace depth at least 1/e: every halfspace containing the centroid captures at least a 1/e fraction of the body's volume. For mixed-integer convex sets S = C ∩ (Z^n × R^d), where C is a convex body, Oertel conjectured that there exists y ∈ S such that every closed halfspace H containing y satisfies H_d(S ∩ H) ≥ (1/(2^n e)) H_d(S), where H_d denotes the d-dimensional Hausdorff measure. This conjecture is closely connected to complexity bounds for cutting plane methods and information complexity in mixed-integer convex optimization. Basu and Oertel proved the conjecture for sets of sufficiently large lattice width, where the required lower bound on lattice width grows exponentially with the dimension. More recently, Cristi and Salas reduced this threshold to a polynomial one under the assumption that the projection of C onto R^n contains a Euclidean ball of radius at least 1178 d^2 n^(3/2). In this paper we show that if the projection of C onto R^n contains an l_infinity ball of radius k ≥ (3e/2)(n + d), which scales linearly with the dimensions n and d, then there exists y* ∈ S such that every closed halfspace H containing y* satisfies H_d(S ∩ H) ≥ (1/e − 3(n + d)/(2k)) H_d(S). In particular, when k ≥ 3e(n + d), we obtain H_d(S ∩ H) ≥ (1/(2e)) H_d(S) ≥ (1/(2^n e)) H_d(S), thus verifying Oertel's conjecture for a significantly larger class of sets than previous results. We also show that this linear scaling is necessary: if the radius is sublinear in the total dimension, the maximum achievable halfspace depth can be arbitrarily small relative to the total mixed-integer volume, so no dimension-independent constant fraction lower bound is possible under such an assumption alone. The conjecture remains open in full generality.

March 24

Beatrix Wen

• Title: Proximal Identification and Estimation in Front-Door Causal Structures with Unobserved Confounding of the Mediator

• Abstract: Unobserved confounding is a fundamental obstacle in causal inference problems. In the graphical modeling literature, a general theory has been developed that allows identification in the presence of hidden variables, with some limitations. In particular, Pearl's celebrated front-door criterion allows nonparametric identification in the presence of unobserved common causes of the treatment and the outcome, however it requires the presence of an unconfounded variable that mediates all causal influence from the treatment to the outcome. This stringent requirement limited the applicability of the front-door criterion in applied problems.
  
  We propose proximal generalizations of the front-door criterion, allowing both arbitrary treatment/outcome confounding, and unobserved confounders of the mediator, provided informative proxies for the latter type of confounders are observed. In addition to deriving three new identification strategies in this setting, we provide plug-in and influence function-based estimation strategies for the resulting functionals, and evaluate their performance through simulations.

March 31

Matthew Hudes

• Title: Spontaneous Stochasticity and the Kelvin Helmholtz Instability

• Abstract: Spontaneous stochasticity has emerged as a key concept for understanding turbulence and its statistical structure. A flow exhibits spontaneous stochasticity if, in the infinite-Reynolds-number limit, the probability distribution over admissible weak solutions is nontrivial and universal—independent of specific regularizations and perturbations.
  
  I will present a numerical renormalization group (RG) scheme to investigate whether the Kelvin-Helmholtz instability exhibits spontaneous stochasticity, building on existing numerical evidence for this phenomenon. The method combines ideas from theoretical physics with pseudo-spectral simulations of 2D Navier-Stokes to probe the universal statistics that may emerge in the zero viscosity and zero noise limit. Time permitting, I will also discuss complementary projects: extending rigorous existence results for vortex sheets to the statistical setting, and applying the Functional RG to a 1D model of spontaneous stochasticity.

April 7

Kailee Lin

• Title: What is... a matroid?

• Abstract: Matrix analysis taught us what linear independence means. But what other things can be independent in a similar way? Matroids generalize the idea of linear algebraic independence to a broader notion of “independence”, which can be used in other areas of math, especially graph theory, which will be the focus of the talk. 

April 14

Zekun (Bill) Wang

• Title: Representing a Collection of Large Language Models as a Gaussian Mixture

• Abstract: Motivated by the prevalence of black-box large language models (LLMs), we aim to understand the statistical properties of LLMs via response embeddings. We consider prompt augmentation, with each augmentation corresponding to one augmented LLM. Statistical consistency of the response embeddings is established. We define a measure of dissimilarity between response embeddings for a collection of augmented LLMs, which leads to a matrix of comparative dissimilarities. We consider, via multidimensional scaling (MDS), representing this dissimilarity matrix in low-dimensional Euclidean space. Under regularity conditions, we prove a row-wise central limit theorem for the MDS representation associated with the collection of augmented LLMs. That is, for a given query set, the MDS embedding of each augmented LLM asymptotically follows a Gaussian mixture model distribution when the augmentations are drawn from a mixture distribution.

April 21

George Kevrekidis

• Title: How to apply to post-docs (and jobs, and other jobs and post-docs)

• Abstract: This talk is actually going to be a panel about what to expect when applying to post-docs and more generally jobs, whether in academia or industry. On the post-doc route, I will give a quick overview of the material you may need to prepare and the general timeline for the application process. On the industry side, we will be joined by our recent alum, Vittorio Loprinzo, currently at Pfizer, to talk about what the recruitment process looks like from the non-academic perspective. Finally our own Wilson and Alan who also have different experiences from the most recent cycle will join! The point of the panel is to give background information on what preparing applications towards the end of the PhD program looks like - it should be useful even if you are not planning to graduate imminently within the next couple of years.

April 28

Barbara Fiedorowicz

• Title: Adaptive and Nonadaptive Strategies for Fault Detection in Trees

• Abstract: Network analysis for fault detection is widely studied in application, especially within electrical networks or circuits. In practice, fault detection must be done in the most economical way. Therefore, the problem setting we have is the following: given a network modeled as a graph, any edge can be the site of a fault, and one is allowed to make probes on the network to detect if a fault is present, and if so, where it occurred within the network. The problem objective is to minimize the number of probes needed to do so.
  
  In this talk, we discuss networks which are modeled as trees, applicable for applications such as telecommunication networks and chemical structures. We provide strategies for both the adaptive (value from probe used to decide the next one) and nonadaptive (all probes selected a priori) versions of the problem. This brings together techniques from optimization, combinatorial and spectral graph theory, and information theory into one applied problem that studies natural network phenomena.