| Date | Speaker | Topic |
|---|---|---|
| M Sep. 8 | Neta Rabin (Tel Aviv University) | Title: Multi-Scale Kernel Methods: Applications to Grid Refinement and Data Augmentation Abstract: Multi-scale models provide a simple yet powerful framework for approximating and extending functions defined over grid-based or scattered datasets. In this talk, we focus on multi-scale kernel methods, where convolution with Gaussian kernels of progressively decreasing bandwidths yields a multi-scale representation. In statistics, this approach is closely related to the Nadaraya–Watson estimator. The resulting high-order approximation is constructed by iterating until the difference between the function and its approximation falls below a predefined error threshold. |
| M Sep. 15 | Katya Epshteyn (U. Utah) | Title: Structure-Preserving Algorithms for Hyperbolic Balance Laws and Related PDE-Based Models Abstract: In this talk, we will discuss progress in the design of structure-preserving numerical methods for hyperbolic and related nonlinear PDE-based models, including systems with uncertainty. As a primary example, shallow water equations will be considered, but the developed ideas can be extended to a wider class of models, including different models of conservation and balance laws. Shallow water systems are widely used in many scientific and engineering applications related to the modeling of water flows in rivers, lakes, and coastal areas. Thus, stable and accurate numerical methods for shallow water models are needed. Although some algorithms are well-studied for deterministic shallow water systems, more effort should be devoted to handling such equations with uncertainty. We will show that the structure-preserving numerical methods that we developed for these models deliver high resolution and satisfy important stability conditions. We will illustrate the performance of the designed algorithms on a number of challenging numerical tests. Current and future research will be discussed as well. Part of this talk is based on the recent work with Dihan Dai, Akil Narayan, Yinqian Yu, and is partially supported by the NSF-DMS Award 2207207 and Simons Foundation Fellowship Award SFI-MPS-SFM-00010667. |
| M Sep. 22 | Joseph Nasser (Brandeis) | Title: A Calculus for Transcription Abstract: What language should we use to describe the natural world: words, pictures, math, computer programs, something else? The discipline of physics has historically used mathematics with great success. The use of mathematics in biology has been more sporadic. I will begin by highlighting some historical uses of mathematics in biology. Then I will describe recent efforts to formulate a “calculus for transcription”. Transcription is the process by which RNA is synthesized from a DNA template. A “calculus for transcription” refers to a hypothetical mathematical framework that can be used to reason about transcription. This talk will be self contained and no previous knowledge of transcription will be assumed. |
| M Sep. 29 | Daniel McKenzie (Mines) | Title: Faster Decision-Focused Learning over Polytopes Abstract: Many real-world problems can be distilled into an optimization problem, for which many good algorithms exist. However, it is often the case that certain key parameters in the optimization problem are not observed directly. Instead, one can observe large amounts of data that is correlated with these parameters, but in ways that are not easy to describe. This raises the possibility of combining machine learning (to predict the unknown parameters) with optimization (to solve the problem of interest). This combination is sometimes called decision-focused learning. In this talk I’ll give an introduction to this field and describe some recent work done by myself and collaborators. |
| M Oct. 6 | Deepanshu Verma (Clemson) | Title: Neural Network Approaches for Optimal Control: Implicit Hamiltonians and Transferable Policies Abstract: This talk presents two neural network methodologies advancing optimal control beyond current limitations. First, we address implicit Hamiltonians in practical problems like space shuttle reentry, where existing methods fail without explicit feedback control formulas. Our end-to-end implicit deep learning approach directly parameterizes value functions to handle the underlying implicit structure while enforcing physical principles through the relationship between optimal control and value function gradients, bridging Pontryagin’s Maximum Principle and Dynamic Programming. Using Jacobian-Free Backpropagation, we efficiently train the implicit networks for high-dimensional feedback controllers in previously intractable scenarios. Second, we tackle the computational burden of re-solving problems when objectives change. Our function encoder framework learns reusable neural basis functions enabling zero-shot adaptation through offline-online decomposition: basis functions are learned once, while adaptation requires only lightweight coefficient estimation. Experiments demonstrate near-optimal performance across diverse dynamics with minimal overhead. These approaches expand neural HJB applicability by handling structural complexity through implicit Hamiltonians and enabling operational flexibility through transferable policies for real-time deployment. |
| M Oct. 20 | Sarah Cannon (Claremont McKenna) | Title: Learning About Political Districting Plans via Random Sampling Abstract: How can you tell if a political districting plan is gerrymandered? This is a hard question: compactness of districts or proportionality of election outcomes don’t tell the whole story. One method is to look at where a districting plan falls within the space of all possible districting plans – if it’s an outlier, it might be gerrymandered. However, there are far too many possible districting plans to look at all of them. Instead, we use random sampling algorithms: by picking a random subset of possible districting plans, we can still get a good idea of what this space of all possible districting plans looks like. How do we generate random political districting plans, and how do we know these plans are “random enough” for our purposes? This talk will provide an introduction to this area of work, including what’s been done, some recent results, and what we still don’t know, as well as some cautions about the limitations of this method. It will finish with a look at an application of these methods to proposals for reforming the Los Angeles City Council. |
| M Oct. 27 | Linh Huynh (Dartmouth) | Title: Spin-Glass-Based Active Inference for Transformers Abstract: Spin glasses is a subfield of high-dimensional probability and statistical physics that studies large networks with random interactions and conflicting combinatorial energy optimization constraints. Originally developed to understand disordered magnets, spin glass models have since found powerful applications in Artificial Intelligence/Machine Learning, and are increasingly being connected to problems in eco-evolutionary biology. Active inference is a partially observable Markov Decision Process framework where an agent uses Bayesian inference to perceive its surrounding environment and then makes decisions by minimizing the associated free energy. In this talk, I will discuss my integration of these two methods to design a mathematical framework for transformers in Large Language Models, which contributes to a broader goal of studying adaptive dynamics on high-dimensional optimization landscapes. |
| M Nov. 3 | Miriam Gordin (Brown) | Title: Vector-Valued Concentration on the Biased Discrete Cube Abstract: While scalar-valued concentration inequalities, such as Poincare or log-Sobolev inequalities, are classical, few are known for functions that take values in a general Banach space. We present a vector-valued concentration inequality for the biased measure on the discrete cube with an optimal dependence on the dimension, bias parameter, and Rademacher type of the target Banach space. This inequality yields novel lower bounds on the average distortion when embedding the biased discrete cube into Banach spaces of nontrivial type and allows us to characterize completely when nonembeddability in high-dimension occurs. Furthermore, we derive scaling limits of the inequality, leading to a vector-valued concentration inequality for a product of Poisson distributions. |
| M Nov. 10 | Daniel Pickard (MIT) | Title: Revealing Material and Structural Failure Mechanisms with Hypersonic Multiphysics Simulation Abstract: Materials and structures subjected to the extreme conditions of hypersonic flight undergo complex thermochemical degradation and fracture. Understanding these effects is critical for low-cost, sustainable space access and various scientific, industrial and national security objectives. We elucidate the fundamental mechanisms governing both ceramic and ablative thermal protection systems using a theoretical formulation and large-scale simulation framework for fracturing solids with complex post-fracture thermochemical response. First, the salient features of our methodology are highlighted by computational analyses of thermal shock-induced fragmentation. Then, a rigorous constitutive theory is shown to capture molecular diffusion through passively oxidizing ultra-high temperature ceramics. Three-dimensional coating simulations expose the channeling mechanisms and a transition from decussating to circumferential cracking that may explain the rich variety of surface cracks found in experiments. The distinct fracture morphology regimes are corroborated by a simple structural theory. Finally, we analyze recently-observed ablative material spallation and discuss exciting avenues for future research. Keywords: Multiphysics; Hypersonic; Fracture; |
| M Nov. 17 | Varun Kharuna (Brown) | Title: Training Guarantees of Neural Network Classification Two-Sample Tests by Kernel Analysis Abstract: We construct and analyze a neural network two-sample test to determine whether two datasets came from the same distribution (null hypothesis) or not (alternative hypothesis). We perform time-analysis on a neural tangent kernel (NTK) two-sample test. In particular, we derive the theoretical minimum training time needed to ensure the NTK two-sample test detects a deviation-level between the datasets. Similarly, we derive the theoretical maximum training time before the NTK two-sample test detects a deviation-level. By approximating the neural network dynamics with the NTK dynamics, we extend this time-analysis to the realistic neural network two-sample test generated from time-varying training dynamics and finite training samples. A similar extension is done for the neural network two-sample test generated from time-varying training dynamics but trained on the population. To give statistical guarantees, we show that the statistical power associated with the neural network two-sample test goes to 1 as the neural network training samples and test evaluation samples go to infinity. Additionally, we prove that the training times needed to detect the same deviation-level in the null and alternative hypothesis scenarios are well-separated. Finally, we run some experiments showcasing a two-layer neural network two-sample test on a hard two-sample test problem and plot a heatmap of the statistical power of the two-sample test in relation to training time and network complexity. |
| M Nov. 24 | Boris Landa (Yale) | Title: Reliable Detection and Recovery of Low-Rank Structures under Heterogeneous Noise Abstract: Detecting and recovering low-rank structure from noisy data matrices is a fundamental task in many data analysis pipelines. Classical spectral methods—such as singular value thresholding—are well understood in the homoskedastic setting, where the noise exhibits uniform variance across all entries. However, in many practical scenarios, the noise is heteroskedastic, with variances that vary substantially across rows and columns, thereby distorting the spectral behavior of the data and posing significant challenges for reliable signal detection and recovery. In this talk, I will present a principled framework for stabilizing heteroskedastic noise by appropriately rescaling the rows and columns of the data matrix. This approach aims to restore the standard spectral behavior of homoskedastic noise—the celebrated Marchenko–Pastur law—enabling simple and effective signal detection and recovery. I will describe two complementary methods for determining the required scaling factors: one designed for count-valued data (such as Poisson or negative binomial observations) accommodating general variance structures, and another applicable to more general data types under a more restricted variance model. I will demonstrate the effectiveness of these methods through simulations and real data examples, highlighting their advantages for high-dimensional low-rank recovery under heterogeneous noise conditions. |
| M Dec. 1 | Lucas Janson (Harvard) | |
| M Dec. 8 | Chris Criscitiello (UPenn) |