PNNL

Join Pacific Northwest National Laboratory at the 2026 Joint Mathematics Meetings (JMM)! JMM is the world’s largest gathering of mathematics experts and professionals. The American Mathematical Society, in collaboration with many partnering organizations, will host this exciting annual event in Washington, D.C.

PNNL Organized Special Sessions

AMS Special Session on Mathematics for AI Robustness, Explainability, and Safety, I 

Date: January 4, 2026

Organizer: Scott Mahan 

Co-Organizers: Eric Yeats, Henry Kvinge, and Tim Doster

Answering questions around the safety, robustness, and explainability of AI models is becoming increasingly critical. Mathematics helps us understand AI failure modes and make AI more transparent and reliable. This special session features mathematics research that analyzes and addresses AI assurance concerns, showcasing areas such as algebraic geometry, probability theory, and computational topology, which provide the insights required for AI systems to meet the needs of real-world applications.

AMS Special Session on Augmenting, not Automating: Machine Learning Tools for Mathematical Discovery, I and II

Date: January 5, 2026 

Organizer: Helen Jenne

Co-Organizers: Henry Kvinge and Max Vargas 

The last several years have seen an explosion of interest in the application of machine learning for mathematical discovery. This special session will consist of a morning session highlighting recent developments in this area, followed by an interactive afternoon problem session. The afternoon session will begin with tool demonstrations, leading into small group exploration of datasets that the organizers will provide using tools discussed earlier in the session.

AMS Special Session on Mathematical Advances in Mission-Aligned Research, I and II

Date: January 7, 2026

Organizer: Emilie Purvine

Co-Organizers: Brett Jefferson and Audun Myers

The U.S. Government has a wide range of mission priority areas, including energy, security, health, AI, cybersecurity, and more. Moreover, these priorities come with unique challenges, including scale, temporal data, and the need for transparent solutions. Mathematics, both theoretical and applied, plays a role in all these areas. This session will highlight advances in mathematics that have enabled progress in these areas to demonstrate the wide-ranging uses of mathematics.

PNNL Presentations 

Retraining Emulation: A General Framework for Machine Unlearning

Date: January 4, 2026 

Presenter: Yiran Jia 

Authors: Eric Yeats and Scott Mahan

As large-scale models are increasingly deployed in sensitive domains, there is a critical need for machine unlearning: the task of selectively removing the influence of a specific data subset from a trained model without the prohibitive cost of full retraining. The central challenge is to ensure complete removal of target information while preserving the model’s overall utility on remaining data.

Detecting Collateral Damage in Unlearning for Diffusion-Based Image Generation Models

Date: January 4, 2026 

Presenter: Aaron Jacobson 

Author: Scott Mahan 

Recent generative AI models have demonstrated remarkable growth in capabilities, size, and data requirements. As this technology continues to develop, privacy and security risks associated with training these models on sensitive data become more common and harder to prevent. Retraining large generative models from scratch without sensitive data is cost-prohibitive; the field of machine unlearning seeks to provide cost- and time-efficient update methods to remove the effects of sensitive data without causing collateral damage to related, non-sensitive data. To understand the effects of these methods, we investigate the internal representations of data in large neural networks and diffusion-based generative models. We observe that the unlearning process induces changes in latent-space representations of data; importantly, the local intrinsic dimension of data manifolds is increased when the corresponding classes of data are unlearned. Using this insight, we propose a method to identify data that may be subject to collateral damage from unlearning and measure the degree to which they were affected. In the context of diffusion-based image generation, this method works by sampling non-target images from the normal space of the target data manifold and projecting them to the space of natural images. The result is a collection of natural images that are close to the target data in latent space, and these results can be captioned to produce a list of classes that may be collaterally damaged by unlearning.

Investigating Bijection Discovery with LLMs: A Case Study Using Catalan Objects

Date: January 5, 2026

Presenting Author: Helen Jenne

Recent breakthroughs have highlighted the potential of large language models (LLMs) to advance mathematics by combining program synthesis with evolutionary search. Systems such as FunSearch seem to be particularly effective for combinatorial optimization problems, such as the cap set problem, where it is straightforward to verify a proposed solution.

We ask whether similar approaches can address the more creative challenge of bijection discovery. Finding bijections requires mathematical intuition and deep familiarity with the combinatorial objects of interest, suggesting an opportunity for LLM-based systems that combine broad prior knowledge, code-writing proficiency, and the ability to do computational exploration at a scale far beyond what is possible for humans. We investigate this question using objects counted by the Catalan numbers, with a pipeline based on OpenEvolve (an open-source analog of AlphaEvolve). In this talk, we present our framework and progress, and share key lessons learned. We will also give a brief overview of the AI-for-combinatorics efforts at Pacific Northwest National Laboratory, highlighting the Algebraic Combinatorics Dataset Repository—a collection of datasets representing foundational results and open problems in algebraic combinatorics. This talk represents joint work with Davis Brown, Herman Chau, Jesse He, Max Vargas, Sara Billey, Mark Raugas, and Henry Kvinge.

Information Theory in a Variety of Contexts

Date: January 5, 2026 

Presenting Author: William Kay 

I am a research mathematician who went from academia to a Federally Funded Research and Development Center. In this talk, I will discuss how my background in information theory found a variety of applications across domain sciences. No background on information theory or applied science is necessary for the audience.

Complex Mathematical Models of Data

Date: January 5, 2026

Presenting Author: Emilie Purvine

Real world systems are complex and messy, not straightforward and simple. Take academic collaborations as an example. We work together in teams to innovate and move the needle on scientific knowledge, and then we publish papers on our work. When you zoom out from an individual collaboration among a group of colleagues to the full research landscape you see interconnected groups of people from different organizations with different funding sources interacting on different topics in a collaboration web that is full of complexities and special cases. As another example, consider computer networks. Processes are constantly starting and completing, communications are being sent, authentications are being made within individual computers and across a network. All these entities and systems are interacting together and being orchestrated by both humans and automated processes. Biological networks, social networks, the power grid, etc., there is no shortage of complex interconnected systems in the world. These systems all have common traits: they are large, have objects of different types, and these objects are related in some ways. In the age of data science, we collect so much information from these systems for real-time and future investigation. At Pacific Northwest National Laboratory, I work with a team focused on representing these kinds of systems using complex mathematical models like graphs, hypergraphs, geometric or topological systems, and even category theory! In this talk I will give a broad (and shallow) overview of this work with examples of how we represent data from these complex systems mathematically, how we analyze those mathematical models to learn something new about the real systems, and how we use what we learn to make an impact in these domains. I will also touch on how we interface these mathematical structures with machine learning models.

Model Editing and Machine Unlearning for Mission Priorities

Date: January 7, 2026

Presenting Author: Scott Mahan

Authors: Henry Kvinge, Tim Doster, Eric Yeats, Darryl Hannan, Yiran Jia, Aaron Jacobson, and Wilson Fearn 

Generative AI models are advancing rapidly and showcasing increasingly sophisticated capabilities, making them useful in many U.S. government mission priorities. However, the massive size of training datasets increases the likelihood of exposure to undesirable or flawed data, potentially resulting in unwanted downstream model behaviors. Model editing and machine unlearning offer effective mechanisms for AI alignment, enabling the modification of factual associations or the removal of problematic information from generative models. In this work, we present mathematical innovations that enhance the effectiveness of model alignment techniques. Moreover, we introduce methods to predict unexpected failures in aligned generative AI systems and propose strategies for mitigating these risks. To demonstrate the real-world impact of these advances, we explore applications in areas critical to mission priorities, such as cybersecurity, where enhanced AI capabilities can strengthen defense mechanisms, detect threats, and safeguard sensitive data.

Evaluation of AI Systems Beyond Accuracy and Leaderboards

Date: January 7, 2026

Presenting Author: Helen Jenne 

Authors: Robert Jasper, Henry Kvinge, Sarah McGuire, Grace O’Brien, and Andrew Aguilar 

Recent years have seen dramatic advances in the capabilities of AI systems, but our methods for ensuring these systems work correctly haven’t kept pace. Even before the explosion in use of generative AI, we had limited understanding of failure modes, performance nuances, and unexpected behaviors. With generative models, we face an even more fundamental challenge: it is difficult to specify what it means for a model to be correct. In this talk, we will give an overview of current evaluation challenges and approaches and present frameworks we’ve developed to address these complex challenges. This represents joint work with Andrew Aguilar, Robert Jasper, Henry Kvinge, Grace O’Brien, and Sarah McGuire Scullen.

Comparison of Binaries Using Sequence Alignment

Date: January 7, 2026

Presenting Author: Brett Jefferson

Author: Stephen Young 

Comparing binary files of software (or firmware) has inherent challenges: register allocations can change, reordering of code components due to optimizers and different compilers can change the binaries. This talk will present one method of comparison that borrows techniques from biology to measure the change between versioned binaries.

A Categorical Framework for Compliance Architecture

Date: January 7, 2026

Presenting Author: Alyson Gauthier 

In large information systems found in national intelligence agencies, banking, and health care, every resource in a database is tagged with "control markings" related to data acquisition and accessibility. Every user or agent in the system has some "entitlement" reflecting their authority, need-to-know, and training. One way to model thie system is a directed acyclic graph whose vertex set is the union of the control markings and entitlements, to express (1) the inheritance structures present in each of these sets, and (2) the access rules relating the two. Such a graph is sufficient to describe "read access", but not to describe the variety of different kinds of access – write, purge, re-label, time limitation, etc – which occur in practice.

We extend the above system to more accurately model real-life use cases: cast vertices as objects in a bounded preorder, and attach to each arc an element of a quantale (a closed monoidal preorder with all joins) to represent some grade of privilege. Any path between two vertices is enriched by the monoidal product of the markings over all the arcs in the path. The privilege that some entitlement has with respect to data with some control marking is the join in of such monoidal products over all paths from to . A compliance architecture is associated with the resulting enriched category, computed using a transitive closure operation. This architecture has a minimal matrix representation analogous to the transitive reduction of an acyclic digraph.

Careers at PNNL

As a national laboratory that conducts an abundance of research using advanced mathematics, we are always searching for talented individuals looking to be a part of our mission.