The seminar features talks in all areas of applied and computational mathematics, including applications in science and engineering. Please contact one of the organizers if you are interested in giving a talk.
Unless noted otherwise, the seminar takes place on Tuesdays 1.30–2.30pm in Kemeny 307.
Past seminar talks can be found here.
Organizers: Chris Vales, David C. Freeman
Tue Sep 22, 2026 | 1.30pm | Kemeny 343
Microfold transformers
From a mathematical perspective, a transformer model can be viewed as a many-body particle system in which the laws of interaction are parameterized and efficiently learned. We generalize the transformer model to the setting in which the embedding space is a Riemannian manifold. The key-query-value linear maps in transformers are replaced by more general, possibly nonlinear, maps in the manifold setting. In this setting, there are two essential choices: (i) the degree of nonlinearity of the interaction laws and (ii) the dimension of the embedding manifold. A microfold transformer is a transformer on a relatively low-dimensional manifold with highly nonlinear interaction laws. We study microfold transformers in an effort to understand the mathematics of transformers at a foundational level.
Tue Sep 29, 2026 | 1.30pm | Kemeny 307
Dynestyx: a probabilistic programming library for dynamical systems
State-space models (SSMs) are the standard formalism for Bayesian treatment of dynamical systems, with natural applications in statistics, signal processing, and machine learning. Despite their importance in both theory and application, dynamical systems have proven difficult to incorporate in modern probabilistic programming languages (PPLs), making state-of-the-art methods less accessible to practitioners and introducing friction in following the "Bayesian workflow." We introduce dynestyx, a probabilistic programming library with first-class support for SSMs, including state-of-the-art methods in the estimation of both states and parameters. Through a single, unified interface, users may specify arbitrary priors for discrete-time or continuous-time dynamical systems, perform inference over mixed-effect data, and make state and parameter estimates with principled uncertainty quantification. We discuss problems, applications, and opportunities that dynestyx introduces.
Tue Oct 6, 2026 | 1.30pm | Kemeny 307
The residual prior transform: from signal recovery to data assimilation
Recovering signals, images, and physical states
from noisy or incomplete data is typically framed as a
regularization problem: enforce that some transform
$\mathcal{L}\mathbf{x}$ of the unknown $\mathbf{x}$ is sparse.
Classical choices such as total variation and its higher-order
variants fix a single assumed order of smoothness, which forces
a trade-off between staircasing and ringing whenever the true
variability changes across the domain—exactly the situation
for piecewise smooth signals and for physical states governed by
hyperbolic conservation laws, whose solutions develop shocks
even from smooth initial data.
We introduce the residual transform operator $R=T-S$,
built from two distinct operators, $T\neq S$, that nevertheless
satisfy $T\mathbf{x}\approx S\mathbf{x}$ to the same order of
approximation in the transform domain.
Because $T$ and $S$ are matched in this way regardless of the
underlying signal's local smoothness, $R$ acts as an annihilating
operator: it is small wherever the underlying signal is smooth
and large only at genuine discontinuities, without ever needing
to know the signal's variability in advance.
Placed inside a generalized sparse Bayesian learning framework,
$R$ yields both a robust point estimate as well as uncertainty
quantification, with the regularization strength learned
automatically from the data rather than hand-tuned.
We then carry $R$ into the 3D-Var variational data assimilation
setting for hyperbolic conservation laws, where the usual
Gaussian background term is ill-suited to a shock's genuinely
multimodal position uncertainty.
Replacing this with residual-based regularization yields analyses
that remain accurate even when the background-error covariance is
misspecified, with the advantage over standard baselines growing
precisely as observations become sparser—the regime where
this structure-preserving approach earns its keep.
This research is done in collaboration with Yao Xiao and
Adityvikram Viswanathan, and is partially supported by
DOD (ONR MURI) #N00014-20-1-2595 and DOE ASCR #DE-SC0025555.
Tue Oct 13, 2026 | 1.30pm | Kemeny 307
Modeling, interpreting, and optimizing functions in biological sequence space
A fundamental goal of genetics is to understand how variation in biological sequences gives rise to differences in measurable characteristics called phenotypes. The mapping from genotype (DNA, RNA or protein sequence) to phenotype can be difficult to model and interpret because the space of possible sequences is enormous and combinations of mutations interact in complex ways. We describe how to use Gaussian process regression to learn genotype-phenotype maps, which to a mathematician are real-valued functions over a discrete space of sequences with a fixed length and alphabet. We will discuss some mathematically nice properties of this modeling framework and their downstream applications for analyzing genotype-phenotype datasets.
Tue Oct 20, 2026 | 1.30pm | Kemeny 307
Physics-informed Gaussian processes for ODEs, PDEs, and SDEs
Differential equations describe scientific systems
through interpretable mechanisms, but learning their unknown
parameters from sparse and noisy observations remains challenging.
Numerical solvers can make inference computationally expensive,
particularly when some system components are unobserved.
In this talk, I will present Gaussian process methods that
incorporate differential equation structure into Bayesian inference
for parameters and latent states.
I will begin with manifold-constrained Gaussian process inference
(MAGI) for ordinary differential equations.
By conditioning a Gaussian process on the constraint that its
derivatives satisfy the governing equations, MAGI enables inference
without numerical integration, including for systems with
unobserved components.
I will then discuss the extension to nonlinear partial differential
equations, where an augmentation strategy yields an equivalent
system that is linear in its derivatives and allows Gaussian
process constraints to be constructed.
This approach provides uncertainty quantification for both the
unknown parameters and the solution while bypassing numerical
PDE solvers.
Finally, I will describe ongoing work on stochastic differential
equations, where nondifferentiable sample paths require a different
formulation.
The proposed SDE-informed Gaussian process model uses a
nonstationary Gaussian process with a Matérn 1/2 kernel and a
discrepancy measure based on Kullback–Leibler divergence to connect
the surrogate process to the SDE.
Tue Oct 27, 2026 | 1.30pm | Kemeny 307
Bridging scales in cell biology: transport, stochasticity, and particle methods
At the scale of a single cell, chemical processes are driven by a complex interplay of spatial transport and stochasticity. Capturing these dynamics requires mathematical models that bridge the microscopic and macroscopic worlds. In this talk, I will introduce the mesoscopic particle methods we use to investigate cellular processes, which we have applied to problems in cellular signaling and antibody-antigen interactions. I will explore several aspects of our recent research, which has included the development of accurate and efficient numerical simulation methods, the derivation and analysis of rigorous coarse-grained (PDE) limits, and applications to immune signaling. By surveying these different areas, this talk will offer a broad introduction to the mathematical and computational challenges of modeling cellular biology at the single cell scale.
Tue Nov 3, 2026 | 1.30pm | Kemeny 307
Inaccuracy of ensemble-based covariance propagation, beyond sampling error
The propagation, or evolution, of the estimation error covariance is an important aspect of the statistical estimation of dynamical systems, particularly for data assimilation and numerical weather forecasting. Ensemble-based methods, such as the ensemble Kalman filter, approximate the evolution of the covariance through the propagation of the individual ensemble members. Thus, it is typically assumed that if the discrete state propagation and resulting mean state estimates are accurate, then the ensemble-based discrete covariance propagation will be accurate as well, apart from sampling errors due to limited ensemble size. Through a series of numerical experiments supported by analytical results, I demonstrate that this assumption is false when correlation length scales approach grid resolution. I show that for states satisfying advective dynamics, although the discrete state propagation and ensemble mean state estimates are accurate, the corresponding ensemble covariances can be remarkably inaccurate, well beyond that expected from sampling errors or typical numerical discretization errors. The underlying problem is fundamental discrepancy between discrete covariance propagation and the continuum covariance dynamics, which I can identify because the exact continuum covariance dynamics are known. This work clarifies longstanding issues in stratospheric ozone forecasting and brings to light a fundamental problem with data assimilation schemes that propagate covariances using the same discrete dynamical model used to propagate the state.
Tue Nov 10, 2026 | 1.30pm | Kemeny 307
PDE dynamics on metric graphs and applications
Abstract TBA.
Tue Nov 17, 2026 | 1.30pm | Virtual
Title TBA
Abstract TBA.
Tue Nov 24, 2026 | 1.30pm | Kemeny 307
Dual Koopman embeddings for quantum simulation of plasma dynamics
Abstract TBA.