Mathematics seminars

A high-dimensional nonlinear recurrent neural network is driven by periodic, incoherent inputs while its synapses undergo Hebbian-like plasticity. Depending on drive and plasticity strengths, the network can retain autonomous limit-cycle activity after both the inputs and learning are removed. Dynamical mean-field theory and finite-size simulations characterize the resulting rhythmic memories and transitions between chaotic, fixed-point and oscillatory regimes. Memory robustness depends on the interval between removing the input and stopping plasticity. These results suggest a mechanism by which neural circuits learn and maintain biological rhythms.

Uri Cohen examines how stable memories can emerge in recurrent networks without imposing specially engineered dynamics such as a Lyapunov gradient. The analysis concerns discrete fixed points in networks with graded activity and noise. Random matrix methods characterize both the bulk and outlier eigenvalues of the local Jacobians. This yields a stability threshold: either all fixed points are stable or all are unstable, depending on their number. That threshold differs from the usual storage capacity, which counts achievable fixed points irrespective of stability. Relating the stability threshold to measurable pattern statistics and activation functions predicts advantages for sparse-like memories and threshold-linear responses. The talk connects these predictions to hippocampal CA3, a proposed auto-associative memory circuit. An assembled dataset of in-vivo patch-clamp recordings from CA3 pyramidal cells in behaving mice provides preliminary support for the predicted threshold-linear activation.

Amber Smith examines immune mechanisms underlying respiratory infections using mechanistic mathematical models combined with multimodal data. Using influenza, the seminar shows how models can expose biological processes that measurements alone do not reveal, explain why similar biomarker profiles can reflect different disease states, and improve predictions of outcomes. The approach links otherwise inaccessible mechanisms to a foundation for predictive, mechanism-based precision medicine.

Andrew Hodges examines the relationship between Turing’s 1936 computability foundations, his digital-computer plans of 1945, the first working electronic computers of 1948–51, and the emergence of computer science. He revisits whether von Neumann’s 1945 EDVAC report drew on Turing’s work, then focuses on Turing’s striking reticence after 1945 about his earlier foundations despite their relevance to his predictions about intelligent machines. Drawing on lesser-known documents, the lecture explores how this reticence influenced computing science and technology and why Turing can be considered a founding figure who avoided claiming either praise or responsibility.

Martin Raum describes how AI entered his mathematical workflow, focusing on manuscript review and research infrastructure while retaining human mathematical judgment. Joint work with Olav Richter and Matt Ortiz provides a case study of human-guided, AI-generated research. The project uses ordinary subscription tools rather than specialized opaque models, offering a workflow adaptable to other mathematical projects. It also raises questions about standards for solid intermediate research and the possible future of research at the highest level.

Alice Mark examines faculty participation when mathematics departments gradually coordinate their courses. The session considers who influences the process, whether instructors can opt out, how coordination affects teaching assignments, and what it means to ask colleagues to support a shared approach. It focuses on maintaining constructive discussion when faculty agree to coordinate but differ over implementation. Participants will explore these questions and develop strategies relevant to their own course-coordination efforts, rather than receive a prescribed set of answers.

Mammalian brains increasingly reuse the same neurons and regions for different tasks, while maintaining robust performance when the environment or stimulus context changes. Udo Ernst combines theoretical modelling with quantitative physiological observations to investigate the mechanisms that support both flexibility and resilience. Selective processing in the primate visual system provides the central example. One analytically tractable framework routes attended external signals through avalanches of spikes in recurrent neuronal networks; a second examines control circuits that preserve reliable routing across stimulus contexts. Mathematical analysis clarifies both mechanisms and how their operating ranges depend on physiological parameters.

Shadi Sharif Azadeh examines how artificial intelligence is turning operations research into a learning-enabled approach to decisions in dynamic, uncertain and data-rich systems. The talk develops two directions: integrating prediction, optimization and learning sequentially, and embedding AI within optimization algorithms. Applications address demand and supply in multimodal mobility with uncertain resources, train scheduling and modular autonomous vehicle operations. Rather than replacing optimization, AI can predict changing states, learn from operational feedback, improve computational efficiency and support adaptation where static models are inadequate.

Tito Homem-de-Mello presents decision-focused learning for two-stage stochastic optimization conditioned on contextual data. Recent theory shows that some two-stage stochastic linear programs can be solved using a single, initially unknown scenario. The talk establishes when learned scenario generators induce optimal policies and why one scenario component outside the observed support can suffice. A log-barrier smoothing operator makes that learning possible. The resulting method draws on interior-point optimization and is evaluated on standard contextual stochastic-programming benchmarks.

Nathan Alexander describes an introductory statistics curriculum that combines logic, theory and computation to develop critical quantitative analysis. Open-source tools support an interdisciplinary process in which students define concepts and propositions, then apply statistics to social questions. The approach aims to strengthen technical skills alongside critical reasoning. The talk also introduces the structure of the critstats package and its role in a community framework for building informational data systems that help users investigate particular histories, issues and challenges.

Large language models alternate multilayer perceptrons and self-attention to process token sequences. Gabriel Peyré compares the proof methods and approximation capabilities of these components. Whereas perceptron approximation is well understood, attention presents less explored mathematical questions. The talk connects their expressive power to approximation theory in infinite-dimensional spaces representing distributions of input tokens and identifies open problems at that interface.

Aditya Dwarkesh explains how geometric quantization extends from ordinary symplectic geometry to higher stacks represented by simplicial manifolds. Whereas ordinary prequantization produces line bundles, the shifted setting leads to bundle gerbes, higher-categorical analogues of principal bundles. For a compact simple Lie group G, the string group provides an example of a multiplicative bundle gerbe over G and can be understood through prequantization of BG with a 2-shifted symplectic structure. The talk develops this relationship and may include a sketch of a new model for String(G).