The Intermingling of Symmetry and Parametrization in Matrix Product States
2026-04-24 12:00 - 13:00 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Daniel Spiegel
Title: The Intermingling of Symmetry and Parametrization in Matrix Product States
Abstract: In the study of quantum spin systems, it is by now well-known that interesting phases of quantum matter can arise from gapped ground states when the system is invariant under a symmetry group G or when the system varies continuously with a parameter in a topological space X. In these cases, phases are characterized by indices taking values in group cohomology of G or the cohomology of X, respectively. The situation where one has both a symmetry and a parametrization is much less studied but can lead to interesting phases even when both the aforementioned indices are trivial. In this talk, I will discuss work in progress on a simple construction for general index for symmetry protected parametrized systems of matrix product states and will show some illustrative examples.

Compression Is All You Need: Modeling Mathematics
2026-04-24 14:00 - 16:30 | Virtual
Freedman Seminar
Speaker: Mike Freedman, Harvard CMSA
Title: Compression Is All You Need: Modeling Mathematics
Abstract: The talk will exposit a recent eponymous arXiv posting with coauthors Vitaly Aksenov, Eve Bodnia, and Mike Mulligan. The approach is to think like a physicist and model a seemingly complex bit of reality: mathematics, by a simple toy model where exact computations can be carried out and then compared with observation. The models are finitely generated monoids and the data is derived from MathLib a large Lean-based repository. The hierarchical nature of definitions and lemmas in math is modeled by adding redundant generators to the monoids – think of the powers of 10 within the natural numbers which support place notation. Place notation confers an exponential compression of how we describe numbers; exploration of MathLib shows that this theme persists to (human) mathematics writ large. We hope that the observables we describe will help our agents navigate to interesting mathematical destinations.

Mathematics and Biology II: Mathematics and Science of Behavior
2026-04-27 09:00 - 2026-04-30 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Mathematics and Biology II: Mathematics and Science of Behavior
Dates: April 27–30, 2026
Location: Harvard CMSA, Room G10, 20 Garden Street, Cambridge MA
Youtube Playlist

This meeting will explore the emerging mathematics and science of embodied cognition—the idea that behavior arises not solely from the brain but through the dynamic interaction of brain, body, and environment. Understanding how animals sense, move, decide, and coordinate, from individual sensorimotor loops to collective dynamics, demands mathematical frameworks that integrate geometry, dynamics, stochastic processes, control theory, and multiscale physics. The meeting will bring together experimentalists studying behavior across species with theorists and engineers building mathematical models and bio-inspired machines, to identify shared principles of adaptive behavior.

Co-organizers: L. Mahadevan (Harvard), Francesco Mori (Harvard CMSA), Venkatesh Murthy (Harvard)
Speakers

Pulkit Agrawal, MIT
Kristin Branson, HHMI
Antonio C. Costa, Sorbonne University/Paris Brain Institute
Noah Cowan, Johns Hopkins University
Robert Datta, Harvard Medical School
Ben de Bivort, Harvard University
Ofer Feinerman, Weizmann Institute of Science
Deborah Gordon, Stanford University
Albert Kao, UMass Boston
Ann Kennedy, Scripps Research Institute
Hungtang Ko, Tufts University
George Lauder, Harvard University
Bence Ölveczky, Harvard University
Kirstin Petersen, Cornell University
Pavan Ramdya, EPFL
Elizabeth Tibbetts, University of Michigan
Robert Wood, Harvard University

Schedule (pdf)
Monday, Apr. 27, 2026
9:00–9:30 am: Breakfast
9:30–10:15 am: Deborah Gordon, Stanford University
Title: The dynamics of collective behavior in changing environments
Abstract: Collective behavior operates without central control, using interactions among participants adjust to changing conditions. There is enormous diversity in the dynamics of collective behavior, including in the rate of response to conditions, in feedback regimes that set whether interactions stimulate or inhibit activity, and the extent of centralization or modularity of information flow. An ecological perspective suggests how this diversity of collective behavior reflects the dynamics of the environment, including its stability, the ratio of resources spent to resources gained, and the distribution of resources in time and space.
As examples, I will discuss field studies and modelling of the regulation of foraging behavior in two species of ants, Harvester ant colonies in the desert regulate foraging to manage high costs, in water loss, to obtain scattered and stable resources. They use a centralized system, with the default to remain inactive unless stimulated, that is slow to adjust foraging activity. In contrast, the turtle ant colonies form trail networks in the canopy of the tropical forest, in unstable conditions where activity costs are low, to find and collect ephemeral and patchy resources. They use a highly modular system, with the default to sustain activity unless inhibited, that can rapidly adjust trail networks to changing resources and conditions.
10:15–10:30 am: Discussion
10:30–11:00 am: Tea Break
11:00–11:45 am: Hungtang Ko, Tufts University
Title: Collective mechanical intelligence: how fluid environments mediate self-organization of swarms
Abstract: Biological collectives across scales self-organize within fluid environments. The mechanical coupling between swarming agents and fluid fields provides opportunities for both passive self-assembly and active, fluid-mediated communication. However, while sporadic evidence of collective mechanical intelligence exists, its underlying mechanisms remain elusive. In this talk, I will focus on two key systems: fire ant rafts and fish schools. Using a combination of experiments and mathematical models, I will show that fire ant rafts leverage passive interfacial forces for self-assembly and self-stabilization. Furthermore, I will demonstrate how schools of giant danio utilize mechanically intelligent formations in 3D, and discuss how swarm robotics may provide the key to future research in collective mechanical intelligence.
11:45 am–12:00 pm: Discussion
12:00–1:30 pm: Catered Lunch
1:30–2:15 pm: Albert Kao, UMass Boston
Title: The limits and potential of collective wisdom
Abstract: Over the past several years, many studies have demonstrated, both in theory and in experiments, the ability of groups to make better decisions than individuals — a phenomenon known as collective wisdom. However, the task types and experimental paradigms used vary considerably, making comparisons across studies, and consequently a unified theory of collective wisdom, difficult. Here, I derive a measure called the effective group size which allows for such comparisons. I use this measure to demonstrate several limitations to collective wisdom, including when groups are large, when there are correlations in opinions in the group, and when information is passed down in chains. In addition, collective wisdom is fragile in the face of an individual who has a disproportionate amount of power, even if far from being a dictator.
2:15–2:30 pm: Discussion
2:30–3:15 pm: Ann Kennedy, Scripps Research Institute
Title: Neural mechanisms that gate the expression of motivated behaviors
Abstract: In order to survive and reproduce, animals must set and weigh off goals in a way that is adaptive and responsive to the context of the environment. Intense evolutionary pressure has wired these algorithms of survival into the connectivity and gene expression patterns of the brain. In this talk, I will present our lab’s recent work on the structure of animal behavior and its neural correlates, showing how theory and modeling can uncover the computational mechanisms by which the brain sets survival goals and weighs off competing survival needs. I will first show how at the macroscopic level, models of behavior as a feedback control system can help uncover control principles by which pressing survival needs can override less urgent drives. Next, I will present new work exploring how macroscopic drives are translated into moment-to-moment behavioral choices.

3:15–4:30 pm: Discussion
4:30–5:30 pm: CMSA Colloquium: Ofer Feinerman, Weizmann Institute of Science
Title: Enacted collective cognition: Brainless problem-solving in weaver ants
Abstract: Unlike most ants, weaver ants construct their nests by pulling together leaves. Because individual ants are small relative to the leaves, they assemble their bodies into temporary tools that bend the leaves into a hollow structure, later stabilized with larval silk. Remarkably, they achieve functional nests across a wide range of leaf shapes and configurations, suggesting that this distributed system is capable of solving complex, open-ended problems.
To understand how this is possible, we performed laboratory experiments using controlled leaf configurations. In simple cases, we show that ants can rely on a zipping heuristic that produces closed nests, and we use differential geometry to demonstrate how flexible leaves are transformed into rigid structures. Crucially, this zipping behavior forms a feedback loop in which ants continuously read and modify the evolving structure. In this sense, the nest itself functions as a shared physical information system.
This suggests that cognition in this system is not located within individual ants, but is enacted through the co-dynamics of the colony and the structure it builds. We present preliminary experiments with more complex leaf configurations, showing that this process can solve increasingly challenging construction problems. Together, these results point to a distributed, brainless, and enactive form of cognition.

Tuesday, April 28, 2026
9:00–9:30 am: Breakfast
9:30–10:15 am: Ben de Bivort, Harvard University
Title: Bayesian Inference on biophysical models of connectomes
Abstract: Recent progress in connectomics has opened new frontiers for understanding the underlying principles of neural circuits. By leveraging high-resolution maps of synaptic connections, computational models can simulate neural dynamics with unprecedented detail. However, it remains challenging to parsimoniously integrate circuit activity data with connectomic information to make biological in- sights. We propose a Bayesian framework as a principled method for bringing to bear existing data, enabling uncertainty quantification for inferring parameters of interest, as well as for predicted circuit outputs. To demonstrate this approach, we implement a simple spiking neuron model using leaky- integrate-and-fire dynamics in the Drosophila olfactory circuit, incorporating available firing rate data. We evaluate how models with varying levels of biological detail fit experimental data and examine how training on different subsets of data influences model predictions.
10:15–10:30 am: Discussion
10:30–11:00 am: Trainee talk: Yasuko Isoe, Harvard University
Title: Divergent spatiotemporal integration of whole-field visual motion in medaka and zebrafish larvae
Abstract: Cross-species comparisons offer powerful leverage for identifying conserved and divergent neural computations underlying innate behavior. Visual motion integration is a fundamental operation that stabilizes an animal’s position relative to its environment, yet how its underlying algorithms vary across closely related vertebrate brains remains poorly understood. We investigated how zebrafish (Danio rerio) and medaka (Oryzias latipes) larvae implement visual motion integration using both free-swimming behavioral assays and head-fixed, tail-free preparations, the latter allowing us to confirm and extend our findings under precise stimulus control. Using whole-field motion stimuli, we found that the two species employ distinct spatiotemporal filtering strategies. Medaka pool motion signals over larger visual fields and weight peripheral inputs more strongly, whereas zebrafish rely more on motion signals directly beneath the body. Temporally, zebrafish respond robustly to brief stimuli, while medaka require longer stimulus durations and sustain motion-driven activity well after stimulus offset. Decomposition of turning behavior revealed separable control modules for large and small corrective maneuvers, with species differences arising primarily from prolonged temporal integration in medaka. Together, our results demonstrate how alterations in basic computational motifs — spatiotemporal pooling, gain, and persistence — can generate divergent visuomotor strategies across closely related vertebrate brains, offering a window into the evolutionary diversification of sensorimotor computation.
11:00–11:30 am: Trainee Talk: Siddharth Jayakumar, Harvard University
Title: Mice follow scent trails using predictive policies
Abstract: Animals must extract reliable information from noisy sensory signals. In olfaction, this is especially challenging, since cues are sparse and must be actively sampled. We asked how mice navigate odor trails under these conditions. Using an “infinite” paper treadmill, we find that mice rapidly learn to track trails with high precision. Disrupting bilateral sampling introduces systematic, lateralized errors, consistent with a comparison of signals across the two sides. Individual inhalations near the trail trigger rapid corrective movements.
Interestingly, we find that mice do not follow trails purely reactively: deviations in tracking at unexpected trail bends reflected recent history, indicating the use of short-term memory. We have begun to investigate the neural substrates of this behavior, focusing on how sensory signals and predictive information are represented in the brain. Broadly, our results suggest that odor-guided navigation depends on combining immediate sensory input with a short-term internal estimate, enabling reliable tracking despite sparse cues.
11:30 am–12:00 pm: Discussion
12:00–1:30 pm: Catered Lunch
1:30–2:15 pm: Noah Cowan, Johns Hopkins University
Title: Toward a Control Theory for Active Sensing
Abstract: Active sensing is often defined as “movement for the purpose of sensing.” Here, I take a different perspective—that active sensing in biological systems is not a distinct class of behaviors, but rather a set of movement phenomena that arise from a control objective. Biological sensors adapt to persistent stimuli, acting like high-pass filters that tend to block “DC.” Such “change-detecting” sensors can support efficient coding with a high dynamic range, and in engineering, bio-inspired event cameras are similar: they transmit information only when a pixel changes and, as such, are extremely fast and make efficient use of bandwidth for the right applications. However, such “AC” sensors pose technical challenges for control. Specifically, event-like biological sensors can cause a nonlinear system (1) to lose local linear observability, and (2) to become impossible to stabilize about an equilibrium point (Biswas, Sontag, Cowan, Eur J Control, 2025). Active sensing behaviors must emerge for stable control, even in the somewhat paradoxical setting where the task-level goal is to remain stationary. Here, I will discuss my lab’s progress in analyzing how animals use active sensing behaviors to format sensory information, enhancing observability and control. I will also present our efforts to formalize controller synthesis with event-like sensors.
2:15–2:30 pm: Discussion
2:30–3:15 pm: Robert Datta, Harvard Medical School
Title: Unveiling structure in natural behavior
Abstract: Ethologists describing animals in the wild have long appreciated that naturalistic, self-motivated behavior is built from modules that are linked together over time into predictable sequences. Many such sequences are built to extract information from the environment.
And yet, it remains unclear how the brain regulates the selection of individual behavioral modules for expression at any given moment, or how it dynamically composes these modules into the fluid behaviors observed when animals act of their own volition, and in the absence of experimental restraint, task structure or explicit reward. Here we use novel methods for characterizing spontaneous mouse behavior to reveal mechanisms used by the brain to create the architecture of self-guided behavior.
3:15–4:30 pm: Discussion

Wednesday, April 29, 2026
9:00–9:30 am: Breakfast
9:30–10:15 am: Kristin Branson, HHMI
Title: How can generative AI help us understand animal behavior?
Abstract: Understanding animal behavior at an algorithmic level — what animals attend to, how they form internal world models, goals, and plans, and how state maps to action — remains a central challenge in neuroethology. Large-scale behavioral experiments now produce trajectory datasets of extraordinary scale and complexity, but existing approaches necessarily compress this complexity to just a few dimensions. We argue that generative AI offers a path toward rich, query-able models of the data. We adapt transformer-based sequence modeling to multi-agent animal keypoint trajectories, treating behavior forecasting as analogous to next-token prediction. Our agent-based network inputs biologically-motivated sensory representations and outputs the distribution of future pose velocities. We show that the model captures statistical properties of the behavioral distribution. We have built a Python library that encapsulates the complexity of transforms relating raw keypoints and model inputs and outputs to make these tools extensible by the NeuroAI community and accessible to theorists and experimentalists. Finally, we argue that mechanistic interpretability methods allow us to query trained models through the natural framework of artificial neuroethology experiments.
10:15–10:30 am: Discussion
10:30–11:00 am: Tea Break
11:00–11:30 am: Trainee talk: Golnar Gharooni Fard, Harvard University
Title: The Geometry and Dynamics of Embodied Cognition: From Collective Architecture to Interspecies Navigation
Abstract: Biological behavior is fundamentally an emergent property of the coupling between an agent’s physical form, its environment, and local interaction rules. In this talk, I explore the mathematical principles of this “embodied cognition” across two distinct scales: the stigmergic spatial memory of honeybee collectives and the real-time dynamic coordination of human-bird mutualism. I’ll start by discussing static embodied intelligence through the lens of honeycomb construction. Using 3D-printed foundations to introduce controlled geometric frustration (including misalignment angles and lattice shifts) I demonstrate how honeybee collectives resolve structural mismatches through the adaptive placement of topological defects. I will show how these complex behavioral responses can be modeled as a physics-based potential minimization problem, proving that the hive’s “intelligence” is a distributed response to local geometric cues. In the second part, I transition to “dynamic” coordination by examining the mutualistic search for honeybee nests between humans and honeyguide birds in Africa. Unlike the persistent memory of the wax comb, this interspecies cooperation requires real-time processing of noisy, stochastic signals. I present a data-driven model of this interaction as a coupled tracking problem. By analyzing the interplay between human engagement and a leaky integrator memory constant, I identify the sweet spots of temporal integration required to successfully filter bird behavior and maintain goal-oriented navigation. Together, these two projects demonstrate that a data-driven physics-inspired modeling framework, can uncover the fundamental rules of agent-environment coupling that drive adaptive behavior across biological scales.
11:30 am–12:00 pm: Trainee talk: Wenyi Zhang, Harvard University
Title: Mechanisms of Setpoint Control in Drosophila Navigation System
Abstract: Navigation provides a powerful system for studying how animals balance behavioral persistence with flexibility. During navigation, fruit flies often default to fast straight walking (or “menotaxis”) in a barren environment, maintaining a stable heading setpoint over a long period of time. Conversely, when the local environment is enriched with sensory stimuli, flies often explore the environment with more frequent heading changes, either through directed steering driven by a sequence of updating setpoints, or through undirected turning driven by temporarily lifting the setpoint control. Although this framework suggests a central role for the setpoint in guiding navigation, the neural mechanisms for flexible setpoint control remain unclear.
Here we identified h∆A, a central complex cell type involved in setpoint control. In an aversive heat paradigm, hΔA played an important role in the fly’s sensory-driven deviation from the menotactic goal direction. We characterized hΔA population activity and found that it carries two separable activity components: a bump-like signal that encodes a slowly varying travel-direction-related setpoint, and a spatially uniform signal associated with turning. We further identified modulatory inputs to hΔA that shape h∆A activity. Together, these results support a model in which short- and long-timescale setpoints compete for steering control, and suggest a circuit mechanism by which flies balance directional persistence with flexible reorientation under changing sensory conditions.
12:00–1:30 pm: Catered Lunch
1:30–2:15 pm: Bence Ölveczky, Harvard University
Title: Using neuro-biomechanical simulations to probe neural control of learned skills
Abstract: The goal of my lab is to decipher the circuit logic by which the brain learns and controls motor skills. The standard mechanistic approach is to dissect the underlying circuits brain area-by-brain area, inferring function by relating recordings and perturbations within each to behavior. This runs into fundamental problems in highly recurrent systems, where activity in any one node is shaped by the dynamics of the whole, a problem compounded by the fact that the circuits we probe control a complex biomechanical body and not measurable features of behavior. I will discuss these challenges and present results suggesting that neuro-biomechanical simulation, leveraging advances in physics simulation and AI, can offer a powerful alternative window into the neural circuits underlying learned skills.
2:15–2:30 pm: Discussion
2:30–3:15 pm: Pavan Ramdya, EPFL
Title: Object manipulation and affordance learning in Drosophila
Abstract: Many animals must manipulate objects to perform tasks like pushing away debris when navigating over complex, natural terrain. For previously unseen objects, efficient manipulation requires that their affordances–the possible actions one can perform upon them–first be learned through experience. However, the behavioral and neural mechanisms underlying the learning of object affordances remain largely unknown. To address this gap, we show that adult Drosophila melanogaster flies can learn to push novel spherical objects without being given any explicit reward. To do this, flies appear to learn the ball’s pushability affordance: pushing is delayed when animals are first exposed to an immobile ball, and manipulating one ball accelerates pushing of a second one in a new context. Behavioral quantification of a large-scale neural silencing screen reveals that specific visual projection neurons and olfactory sensory neurons regulate initial reactions to the object while dopaminergic neurons and the mushroom bodies, a center for learning and memory in insects, are critical for generalizing object affordances. These findings open the door to a mechanistic understanding of object manipulation and affordance learning.
3:15–4:30 pm: Discussion

Thursday, April 30, 2026
9:00–9:30 am: Breakfast
9:30–10:15 am: Pulkit Agrawal, MIT
Title: What Robots Are Missing: Force Intelligence and Lifelong Learning
Abstract: Modern robots can plan sophisticated motions, yet they remain slow, brittle, and unreliable on tasks humans find effortless. The missing piece is not better planning, but better force reasoning: knowing when, where, and how much force to apply under uncertainty and across diverse tasks. Force intelligence, I argue, is a unifying principle for scalable robotics—bridging dexterous manipulation and whole-body control. However, even a force-aware robot that cannot learn from its own experience will remain brittle. Today’s systems are effectively frozen after training, unable to adapt once deployed. Real-world autonomy instead demands learning in deployment: the ability to improve continuously from interactions, failures, and successes. In this talk, I will present our lab’s recent work on lifelong learning and outline a future path for how combining it with force-centric design could enable reliable, useful robots in the real world.
10:15–10:30 am: Discussion
10:30–11:00 am: Tea Break
11:00–11:45 am: Antonio C. Costa, Sorbonne University/Paris Brain Institute
Title: Unraveling the structure of behavioral variation: a dynamical approach to naturalistic data
Abstract: Animal behavior varies widely, both within the same individual over time and between individuals. While often overlooked, this variation reflects hidden control variables and mechanisms that were shaped by evolution. For example, variation in behavioral traits can help populations withstand environmental change, while atypical motor patterns in neurological disorders may offer clues for personalized therapies. Comparing such complex behaviors is difficult. When dynamics are nonlinear and unfold over multiple timescales, standard metrics based on summary statistics often miss meaningful differences. To address this, we introduce a framework that encodes multiscale dynamics to compare behavior from data. By modeling nonlinear dynamics probabilistically (using transfer operators inferred from time-series data), we define a distance metric that captures behavioral differences across timescales. Tailored to finite, noisy datasets, our approach identifies principal axes of variation and enables rigorous clustering of individual trajectories. We demonstrate this framework in various biological systems, including bacterial chemotaxis and larval zebrafish locomotion, where the inferred axes of behavioral variation reflect underlying physiological variables and developmental histories.
11:45 am–12:00 pm: Discussion
12:00–1:30 pm: Catered Lunch
1:30–2:15 pm: Elizabeth Tibbetts, University of Michigan
Title: What paper wasps can teach us about the evolution of animal minds
Abstract: Why do animals differ in their cognitive abilities? Some animals fail at apparently simple tasks, while others have a remarkable capacity to collect, retain, and use information from the environment to guide their behavior. Although paper wasps brains are smaller than a grain of rice, Tibbetts will show that wasps can perform seemingly complex behaviors like individual face recognition, transitive inference, social eavesdropping, and concept learning. She will also describe experiments that take advantage of natural variation in behavior within and among wasp species to test how social interactions shape the development and evolution of cognitive abilities.
2:15–2:30 pm: Discussion
2:30–3:15 pm: Robert Wood, Harvard University
Title: The Mechanical Side of Artificial Intelligence
Abstract: Artificial Intelligence research typically focuses on perception, learning, and control methods to enable autonomous agents, including robots, to make and act on decisions in real-world scenarios. However, even the most capable AI without a well-designed physical structure is of minimal use for canonical robotics tasks. Our research is focused on the design, mechanics, materials, and manufacturing of novel robot platforms that make perception, control, or action easier or more robust for natural, unstructured, and often unpredictable environments. Key principles in this pursuit include bioinspired designs, smart materials for novel sensors and actuators, and the development of multi-scale, multi-material manufacturing methods. This talk will illustrate this philosophy by highlighting the creation of three classes of robots with unique hardware challenges: bioinspired microrobots, soft-bodied robots for manipulation, and robots for interacting with delicate marine life.
3:15–4:00 pm: Discussion
4:00–5:00 pm: George Lauder, Harvard University
Title: Fish schooling behavior from kinematics to hydrodynamics to energetics
Abstract: Do fish moving in a school reduce their energetic costs compared to swimming alone? If so, how does collective motion reduce the energy needed to move? Only within the last two years have experimental studies directly demonstrated that fish swimming in a group have lower energy expenditure than solitary locomotion. Most studies of how fish move in a collective have focused on understanding the potential benefits of swimming in fixed relative positions. But recent experiments on fish schooling behavior have revealed that fish within the school are nearly constantly rearranging their relative positions. In this talk I will show how fish in a school can save energy even if they do not maintain fixed positions. Analyses of water flow patterns within fish schools have been used to resolve this “paradox” and show that fish movement within a school creates hydrodynamic shelters with zones of reduced flow velocity that nearby fish can take advantage of.

Higher current algebras and chiral algebras
2026-04-27 15:00 - 16:00 | CMSA G102, 20 Garden Street, Cambridge, MA, 02138
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Charles Young (University of Hertfordshire)
Title: Higher current algebras and chiral algebras
Abstract: Vertex algebras capture physicists’ notion of OPEs in chiral CFTs, in complex dimension one. For various motivations, one would like to have analogs of vertex algebras in higher dimensions. Chiral algebras, in the sense of Beilinson-Drinfeld and Francis-Gaitsgory, provide a natural framework here, because they re-express the vertex algebra axioms (which are rather sui generis, and therefore hard to generalize) as something more recognizable (a chiral algebra is a Lie algebra, of a sort).
I will review this, and then go on to introduce a certain concrete model of the unit chiral algebra in higher dimensions. In higher dimensions one is forced to work up to coherent homotopy in some fashion; in this model it turns out to be in the mildest fashion one could hope for: namely, one moves from Lie algebras to their homotopy analogs, L-infinity algebras, and from chiral algebras to homotopy chiral algebras in a sense introduced by Malikov-Schechtman.
The main tool in the talk will be a strict cdga model — the polysimplicial model — of derived global sections of the structure sheaf on configuration space. The hope is that this model will prove well-adapted to doing concrete calculations, and in that direction, I will gesture towards a homotopy version of the usual Arnold/Orlik-Solomon relations for broken circuits.
This is joint work with Zhengping Gui and Laura Felder and is based largely on the preprint 2506.09728

Enacted collective cognition: Brainless problem-solving in weaver ants
2026-04-27 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Ofer Feinerman, Weizmann Institute of Science
Title: Enacted collective cognition: Brainless problem-solving in weaver ants
Abstract: Unlike most ants, weaver ants construct their nests by pulling together leaves. Because individual ants are small relative to the leaves, they assemble their bodies into temporary tools that bend the leaves into a hollow structure, later stabilized with larval silk. Remarkably, they achieve functional nests across a wide range of leaf shapes and configurations, suggesting that this distributed system is capable of solving complex, open-ended problems.
To understand how this is possible, we performed laboratory experiments using controlled leaf configurations. In simple cases, we show that ants can rely on a zipping heuristic that produces closed nests, and we use differential geometry to demonstrate how flexible leaves are transformed into rigid structures. Crucially, this zipping behavior forms a feedback loop in which ants continuously read and modify the evolving structure. In this sense, the nest itself functions as a shared physical information system.
This suggests that cognition in this system is not located within individual ants, but is enacted through the co-dynamics of the colony and the structure it builds. We present preliminary experiments with more complex leaf configurations, showing that this process can solve increasingly challenging construction problems. Together, these results point to a distributed, brainless, and enactive form of cognition.

The classifying space of a Morse flow category
2026-04-28 16:15 - 18:00 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Joint Math/CMSA Geometry and Quantum Theory Seminar
Speaker: Lorenzo Riva, Harvard CMSA
Title: The classifying space of a Morse flow category

Abstract: Following a paper of Calle and Liu we show that, under suitable tameness assumptions, the classifying space of the Morse category associated to a manifold M with a Morse function recovers the homotopy type of M, thereby addressing a gap in a preprint of Cohen-Jones-Segal. We also give a counterexample when those assumptions are not satisfied.

Transcendental Epsilon Multiplicity via Divisor Volumes
2026-04-30 16:00 - 17:00 | CMSA Room G02, 20 Garden Street, Cambridge, MA, 02138
Algebra Seminar
Speaker: Sudipta Das, Tata Institute of Fundamental Research
Title: Transcendental Epsilon Multiplicity via Divisor Volumes
Abstract: In this talk, our goal is to establish a structural bridge between asymptotic commutative algebra and transcendence theory to show that there exists an ideal in a Noetherian local ring whose epsilon multiplicity is transcendental. By equating the local-cohomological definition of epsilon multiplicity to a global divisorial volume integral on a projective bundle, we apply Baker’s theorem on linear forms in logarithms to prove that the resulting arithmetic invariant falls strictly outside the field of algebraic numbers. This talk is based on collaborative work with Vinh Pham and Stephen Landsittel.

note room change to G02

Lifting F-split surfaces to the Witt vectors
2026-05-01 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Iacopo Brivio, CMSA
Title: Lifting F-split surfaces to the Witt vectors

Abstract: Algebraic varieties in positive characteristic are ill behaved compared to characteristic zero ones. Several important tools available over the complex numbers, such as the Hodge decomposition theorem, are either not available or straight-away false. There are two important classes of positive characteristic varieties which have better behavior: Witt liftable varieties and Frobenius split varieties. A folklore conjecture predicts that the latter class is contained in the former. In a joint work with Bernasconi, Kawakami and Witaszek we proved that this is the case for surfaces. I will give an overview of this result as well as some applications thereof.

Twisted D-branes and TQFTs valued in Calabi-Yau categories
2026-05-04 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Surya Raghavendran, Yale University
Title: Twisted D-branes and TQFTs valued in Calabi-Yau categories
Abstract: Recently, Bozec–Calaque–Scherotzke have articulated a noncommutative version of the AKSZ construction, which associates to a smooth Calabi–Yau category a fully extended TQFT valued in a category of iterated Calabi–Yau cospans. In this talk, I will study a class of examples of such theories which arise in the context of conjectures of Costello and Li, which describe Type II strings in certain Ramond–Ramond backgrounds as topological strings. These TQFTs capture structural features of the BPS physics of D-branes that are universal in Chan–Paton factors. Conjecturally commutative limits of the values of such theories on closed manifolds can sometimes be geometrically quantized to yield algebraic structures with Hall-type products. Examples of this paradigm include CoHAs associated to complex 3-folds, CoHAs attached to local systems on 3-manifolds, and the categorified Hall algebras of Porta–Sala.

Dynamics as intersection problem
2026-05-04 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Nikita Nekrasov, Simons Center
Title: Dynamics as intersection problem
Abstract: Most classical and quantum field theories are based on an action principle. However, there are important exceptions to this — hydrodynamics and the theory of self-dual fields. In this talk we formulate the covariant relativistic fluid dynamics, with or without magnetic fields, as well as the theory of chiral boson in 1+1 dimensions, self-dual tensor in 1+5 dimensions, and self-dual four-form of IIB supergravity, in terms of intersection theory of an auxiliary phase space. This provides a common covariant geometric framework for systems without a conventional action, while at the same time laying the groundwork for quantization via the Kontsevich approach. Joint work with Paul Wiegmann.

New directions in synthetic data
2026-05-06 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Tatsunori Hashimoto, Stanford
Title: New directions in synthetic data
Abstract: Synthetic data has been an effective, if boring set of techniques: prompt some language model to restructure your corpus to match some downstream task, with occasionally some distillation. In this talk, we will take a more expansive view of synthetic data as a general algorithmic tool for generative modeling, arguing that the design space and possibilities of synthetic data are much bigger than it might seem. Through a few recent works, we will show that synthetic data has major benefits beyond transforming the data – improving in-domain perplexities, and enabling unique algorithmic primitives, such as neighborhood smoothing and concatenated ‘mega’ documents. With this broader view, we will point towards a nascent but interesting possibility of treating data itself as an algorithmic object to be engineered and optimized end-to-end.

From Poincaré/Koszul duality to (twisted) AdS/CFT correspondence
2026-05-08 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Keyou Zeng
Title: From Poincaré/Koszul duality to (twisted) AdS/CFT correspondence
Abstract: Poincaré duality is a fundamental result in the (co)homology theory of manifolds. It has many applications in topology and vast generalizations to other types of “spaces,” such as singular/stratified spaces and schemes. In this talk, I will discuss a variant of Poincaré duality for factorization algebras, also known as Koszul duality. At the end of the talk, I will relate this notion to a mathematical formulation of what physicists call the AdS/CFT correspondence, as proposed by Costello and Li.

When do anomalous finite symmetries in (3+1)d enforce gaplessness?
2026-05-11 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Matthew Yu (University of Oxford)
Title: When do anomalous finite symmetries in (3+1)d enforce gaplessness?
Abstract: I will explain a comprehensive framework for characterizing the infrared (IR) phases of a fermionic QFTs in (3+1)d, based on their quantum anomalies associated with a finite symmetry. We uncover a fundamental dichotomy among these anomalies: the first class of anomalies can always be realized by symmetric gapped states, while the second class can never be realized by gapped states without breaking the given symmetry, establishing the phenomenon of symmetry-enforced gaplessness in these settings. Using the construction of symmetry extension afforded to us by new developments in fusion 2-categories, we construct the candidate gapped states that theories with the first class of anomalies can flow to in the IR. As an application, I will provide examples of concrete predictions for the candidate IR phases of (3+1)d gauge theories based on our results.

Statistical Shape Analysis of Complex Natural Structures
2026-05-11 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Anuj Srivastava, Johns Hopkins University
Title: Statistical Shape Analysis of Complex Natural Structures
Abstract: Statistical modeling and analysis of structured data is a fast-growing field in Statistics and Data Science. Rapid advances in imaging techniques have led to tremendous amounts of data for analyzing imaged objects across several scientific disciplines. Examples include shapes of cancer cells, botanical trees, human biometrics, 3D genome, brain anatomical structures, crowd videos, nano-manufacturing, and so on. Shapes are relevant even in non-imaging data contexts, e.g., the shapes of COVID rate curves or the shapes of activity cycles in lifestyle data. Imposing statistical models and inferences on shapes seems daunting because the shape is an abstract notion and one requires precise mathematical representations to quantify shapes. This talk has two parts. In the first part, I will present some recent developments in “elastic representations” of structures such as functions, curves, surfaces, and graphs. In the second part, I will focus on statistical analyses: computing shape summaries, estimation under shape constraints, hypothesis testing, time-series models, and regression models involving shapes.

Polynomial invariants of conjugation over finite fields
2026-05-14 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebra Seminar
Speaker: Aryaman Maithani, University of Utah
Title: Polynomial invariants of conjugation over finite fields

Abstract: Consider the conjugation action of GL₂(K) on the polynomial ring K[X₂ₓ₂].
When K is an infinite field, the ring of invariants is a polynomial ring generated by the trace and the determinant.
We describe the ring of invariants when K is a finite field, and show that it is a hypersurface.

Workshop on Calabi-Yau metrics and optimal transport
2026-05-18 09:00 - 2026-05-22 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Workshop on Calabi-Yau metrics and optimal transport
Dates: May 18–22, 2026
Location: Harvard CMSA, 20 Garden Street, Cambridge MA
Recent advances in the study of Calabi-Yau metrics have revealed an interesting connection with optimal transport, and the regularity theory for optimal transport is expected to play an increasingly important role in the study of Kähler geometry. The goal of this workshop is to bring together the optimal transport and complex geometry communities to investigate problems arising from these exciting developments.
Videos of selected talks are now available on the Youtube Playlist
Minicourse Speakers

Robert McCann, University of Toronto
Yang Li, Cambridge University

Workshop Speakers

Rolf Andreasson, Chalmers University, Sweden
Benjy Firester, MIT
Jakob Hultgren, Umea University, Sweden
Young-Heon Kim, University of British Columbia
Nam Le, Indiana University
Jiakun Liu, University of Sydney
Arghya Rakshit, University of Toronto
Gabor Szekelyhidi, Northwestern University
Yueqiao Wu, Johns Hopkins University

Organizers:

Tristan Collins, University of Toronto
Mattias Jonsson, University of Michigan
Connor Mooney, University of California, Irvine
Freid Tong, University of Toronto

Schedule (pdf)
Monday, May 18, 2026
9:00–9:30 am
Breakfast
9:30–10:45 am
Tutorial: Yang Li, Cambridge University (via Zoom Webinar)
Title: On the metric SYZ conjecture
Abstract: For a polarised degeneration family of Calabi-Yau manifolds near the large complex structure limit, the metric SYZ conjecture asks for a special Lagrangian torus fibration on the generic region of the Calabi-Yau manifolds. I will summarize the progress on the metric SYZ conjecture so far, emphasizing on some more recent progress.
10:45–11:15 am
Break
11:15 am–12:30 pm
Tutorial: Robert McCann, University of Toronto
Title: A geometric approach to apriori estimates for optimal transport maps
Abstract: A key inequality which underpins the regularity theory of optimal transport for costs satisfying the Ma-Trudinger-Wang condition is the Pogorelov second derivative bound. This translates to an a priori interior modulus of the differential estimate for smooth optimal maps. We describe a new derivation of this estimate with Brendle, Leger and Rankin which relies in part on Kim, McCann, and Warren’s observation that the graph of an optimal map becomes a volume maximizing non-timelike submanifold when the product of the source and target domains is endowed with a suitable pseudo-Riemannian geometry that combines both the marginal densities and the cost. This unexpected links optimal transport to the plateau problem in geometry with split signature, and shows the key difficulty is showing the maximizing non-timelike submanifold is in fact (uniformly) spacelike. J. Reine Angew. Math. 817 (2024) 251-266 doi.org/10.1515/crelle-2024-0071 arXiv 2311.10208
12:30–2:00 pm
Lunch (catered)
2:00–3:15 pm
Talk: Nam Le, Indiana University
Title: Variational approach to degenerate Monge-Ampère equations with mixed measures and monotonicity
Abstract: In this talk, we will discuss the solvability and uniqueness for several degenerate Monge-Ampère equations including the Monge-Ampère eigenvalue problem in real Euclidean spaces that involve singular Borel measures. Our approach systematically analyzes the Monge-Ampère energy from the variational point of view and appropriately exploits monotonicity arguments. We will examine several essential tools: the mixed Monge-Ampère measure, Aleksandrov-Blocki-Jerison type maximum principles, convex envelope, comparison principles for subcritical equations, and integration by parts whose failure leads to symmetry breaking and nonuniqueness phenomena.
3:15–3:45 pm
Break
3:45–5:00 pm
Talk: Yueqiao Wu, Johns Hopkins University
Title: Valuative aspects of complete Calabi-Yau metrics of Euclidean volume growth
Abstract: The search of a complete Calabi-Yau metric on an affine variety X amounts to solving a complex Monge-Ampère equation subject to nice “boundary conditions” at infinity. In the case where X is the complement of an SNC anticanonical divisor on a Fano manifold, generalizing the work of Tian-Yau, Collins-Li showed that such boundary data can be extracted from solutions to certain real Monge-Ampère equations. If we require the metric to have Euclidean volume growth, however, it is understood that the boundary conditions should come from prescribing a Calabi-Yau asymptotic cone at infinity. This is the same as giving the algebro-geometric data of a valuation which induces a degeneration of X to a K-stable affine cone. In this talk, we will explain that such valuations in fact always come from Fano type compactifications of X, similar to the ones considered by Tian-Yau and Collins-Li. In addition, K-semistability of the affine cone can be characterized intrinsically by a valuative criterion on X. Based on joint work with Mattias Jonsson.

Tuesday, May 19, 2026
9:00–9:30 am
Breakfast
9:30–10:45 am
Tutorial: Robert McCann, University of Toronto
Title: Trading linearity for ellipticity: A low regularity Lorentzian splitting theorem
Abstract: While Einstein’s theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually breakdown. It is thus of great interest to study the theory in low regularity settings. In the lecture, we establish a low regularity splitting theorem by sacrificing linearity of the d’Alembertian to recover ellipticity. We exploit a negative homogeneity $p$-d’Alembert operator for this purpose. The same technique yields a simplified proof of Eschenberg (1988) Galloway (1989) and Newman’s (1990) confirmation of Yau’s (1982) conjecture, bringing all three Lorentzian splitting results into a framework closer to the Cheeger-Gromoll splitting theorem from Riemannian geometry. Based on joint work with Mathias Braun, Nicola Gigli, Argam Ohanyan, and Clemens Saemann: 1) arXiv 2501.00702 2) arXiv 2408.15968 3) arXiv 2410.12632 4) arXiv 2507.06836
10:45–11:15 am
Break
11:15 am–12:30 pm
Tutorial: Yang Li, Cambridge University (via Zoom Webinar)
Title: On the metric SYZ conjecture
Abstract: For a polarised degeneration family of Calabi-Yau manifolds near the large complex structure limit, the metric SYZ conjecture asks for a special Lagrangian torus fibration on the generic region of the Calabi-Yau manifolds. I will summarize the progress on the metric SYZ conjecture so far, emphasizing on some more recent progress.
12:30–2:00 pm
Lunch Break
2:00–3:15 pm
Talk: Young-Heon Kim, University of British Columbia
Title: Trajectory Inference via Multi-marginal Schrödinger Bridges
Abstract: Trajectory inference arises in important scientific problems. In particular, biological development can be interpreted as a curve in the space of gene-expression distributions, and the goal is to infer this trajectory from observed data. There has been progress by using optimal transport (OT) as a way to interpolate between distributions. More recently, Schrödinger bridges, a stochastic generalization of OT, have been considered. In this talk, we discuss stability of such OT-based methods. This is joint work with Geoffrey Schiebinger and Rentian Yao.
3:15
Break
6:30 pm
Dinner

Wednesday, May 20, 2026
9:00–9:30 am
Breakfast
9:30–10:45 am
Tutorial: Yang Li, Cambridge University (via Zoom Webinar)
Title: On the metric SYZ conjecture
Abstract: For a polarised degeneration family of Calabi-Yau manifolds near the large complex structure limit, the metric SYZ conjecture asks for a special Lagrangian torus fibration on the generic region of the Calabi-Yau manifolds. I will summarize the progress on the metric SYZ conjecture so far, emphasizing on some more recent progress.
10:45–11:15 am
Break
11:15 am–12:30 pm
Tutorial: Robert McCann, University of Toronto
Title: The monopolist’s free boundary problem in the plane: an excursion into the economic value of private information
Abstract: The principal-agent problem is an important paradigm in economic theory for studying the value of private information: the nonlinear pricing problem faced by a monopolist is one example; others include optimal taxation and auction design. For multidimensional spaces of consumers (i.e. agents) and products, Rochet and Chone (1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences are bilinear in the product and agent parameters. This optimization corresponds mathematically to a convexity-constrained obstacle problem. The solution is divided into multiple regions, according to the rank of the Hessian of the optimizer.
If the monopolists costs grow quadratically with the product type we show that a partially smooth free boundary delineates the region where it becomes efficient to customize products for individual buyers. We give the first complete solution of the problem on square domains, and discover new transitions from unbunched to targeted and from targeted to blunt bunching as market conditions become more and more favorable to the seller.
Based on works with Kelvin Shuangjian Zhang, Cale Rankin, and Lucas O’Brien in various combinations:
1) Math. Models Methods Appl. Sci. 34 (2024) 2351-2394; 2) J. Convex Anal. (Rockafellar 90 Issue), 32 (2) (2025) 579-584; 3) arXiv 2303.04937; 4) arxiv 2412.15505; 5) arXiv 2603.14100.

Thursday, May 21, 2026
9:00–9:30 am
Breakfast
9:30–10:45 am
Talk: Gabor Szekelyhidi, Northwestern University
Title: Nondegenerate Neck Pinches along the 2d Lagrangian mean curvature flow
Abstract: The Thomas-Yau-Joyce conjecture predicts that the mean curvature flow can be used to decompose Lagrangian submanifolds in Calabi-Yau manifolds into special Lagrangian building blocks. The basic mechanism for this decomposition is given by neck pinches. I will discuss work on the behavior of such neck-pinch singularities, in particular the class of nondegenerate neck pinches, which satisfy certain properties conjectured by Joyce. The construction also relates to work of Neves on finite time singularity formation.
10:45–11:15 am
Break
11:15 am–12:30 pm
Talk: Rolf Andreasson, Chalmers University, Sweden
Title: Optimal transport between boundaries of dual reflexive polytope
Abstract: I will present an optimal transport problem between the boundaries of a pair of reflexive polytopes. Under a certain structural condition on its solution, this problem is related the study of metric degenerations of families of Calabi–Yau hypersurfaces in the corresponding toric Fano variety. A better understanding of such solutions and their regularity would shed light on several aspects of the degeneration and conjectural Gromov–Hausdorff limit, and I will present some open directions of research. This is based on joint work with Jakob Hultgren, Mattias Jonsson, Enrica Mazzon and Nicholas McCleerey.
12:30–2:00 pm
Lunch Break
2:00–3:15 pm
Talk: Jakob Hultgren, Umea University, Sweden
Title: Affine mondoromy, cost functions and real Monge-Ampère equations
Abstract: Recent results of Blum-Liu and Y. Li show that the metric SYZ conjecture holds. The solution hinges on the existence of valuatively independent bases for spaces of sections of the polarising bundle. These bases induce a cost function, providing a link between the non-Archimedean Monge-Ampère equation and optimal transport. The resulting SYZ-fibration is constructed on a large but non-explicit set. In order to better understand this set, more information about the cost function is arguably needed. We propose a cost function explicitly computable from the monodromy of an affine structure on the essential skeleton. In the case of the Fermat family of cubic curves, this cost function (as opposed to the one attained from the ambient projective space) can be shown to agree with the one attained from a valuatively independent basis. We conjecture that this equality of cost functions holds in general, and demonstrate in examples how the explicit cost function can be used to directly produce solutions to real Monge-Ampère equations on the essential skeleton. Joint work with Sohaib Khalid.
3:15–3:45 pm
Break
3:45–5:00 pm
Talk: Benjy Firester, MIT
Title: Free boundary Monge-Ampere equations with applications to optimal transport and Calabi-Yau geometry
Abstract: I will present a variational framework to solve a general class of free-boundary Monge-Ampère equations. This approach combines the classical first and second boundary value problems by imposing both the boundary data and the gradient image of the solution. I will explore applications to the Monge-Ampère eigenvalue problem, convex reconstruction theorems, and geometric problems including a hemispherical Minkowski problem, Calabi-Yau metrics, and free boundary toric Kähler–Einstein/Kähler-Ricci soliton metrics. I will also discuss the connection to the boundary regularity of optimal transport.

Friday, May 22, 2026
9:00–9:30 am
Breakfast
9:30–10:45 am
Talk: Jiakun Liu, University of Sydney
Title: Free boundary problems in optimal transportation
Abstract: In this talk, I will present some recent results on the regularity of free boundaries in optimal transportation, including higher-order regularity, global regularity, and a model case involving multiple targets. These results are based on a series of joint works with Shibing Chen, Xianduo Wang, and Xu-Jia Wang.
10:45–11:15 am
Break
11:15 am–12:30 pm
Talk: Arghya Rakshit, University of Toronto
Title: Solutions to the Monge–Ampère equation with singular structures
Abstract: We construct examples of solutions to the Monge–Ampère equation with point masses exhibiting polyhedral singular structures. We further analyze the stability of these singular sets under small perturbations of the data. In addition, we construct solutions whose Monge–Ampère measure contains a singular component supported on lower-dimensional sets and we study the regularity of such solutions.

Abelian duality via derived geometry
2026-05-18 15:00 - 16:00 | CMSA Room G02, 20 Garden Street, Cambridge, MA, 02138
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Owen Gwilliam, UMass Amherst
Title: Abelian duality via derived geometry
Abstract: We discuss how to synthesize differential cohomology and the BV formalism to describe generalized Maxwell theories (or abelian p-form gauge theories), and how this framework allows a succinct formulation of abelian duality. Given time, we will discuss how these methods apply to the 6d self-dual 2-form gauge theory that appears as part of the 6d N=(1,0) and (2,0) superconformal theories known as abelian tensor multiplets. This is joint work in progress with Chris Elliott, Ingmar Saberi, and Brian Williams.

Geometry and Quantum Theory Seminar
2026-05-19 16:15 - 18:00 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Joint Math/CMSA Geometry and Quantum Theory Seminar
Speaker: Vasily Krylov, Harvard

Separation of timescales controls feature learning and overfitting in large neural networks
2026-05-20 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Pierfrancesco Urbani, Universite Paris-Saclay, CNRS, CEA, Institut de physique theorique
Title: Separation of timescales controls feature learning and overfitting in large neural networks
Abstract: To understand the inductive bias and generalization capabilities of large, overparameterized machine learning models, it is essential to analyze the dynamics of their training algorithms. Using dynamical mean field theory we investigate the learning dynamics of large two-layer neural networks. Our findings reveal that, for networks with a large width, the training process exhibits a separation of timescales phenomenon. This leads to several key observations:
1. The emergence of a slow timescale linked to the growth in Gaussian/Rademacher complexity of the network;
2. An inductive bias favoring low complexity when the initial model complexity is sufficiently small;
3. A dynamical decoupling between feature learning and overfitting phases;
4. A non-monotonic trend in test error, characterized by a “feature unlearning” regime at later stages of training.
Joint work with Andrea Montanari.

Introduction to First Proof: A conversation
2026-06-03 13:00 - 14:00 | Virtual
Introduction to First Proof: A conversation
Date: June 3, 2026
Time: 1:00–2:00 pm
Location: via Webinar
Harvard CMSA Director Dan Freed will lead a dialogue with First Proof Editors Mohammed Abouzaid (Stanford), Nikhil Srivastava (UC Berkeley), Rachel Ward (UT Austin), and Lauren Williams (Harvard) to explore the origins and goals of First Proof, sample some of their current work, and discuss their future plans. This hour-long conversation will give a window into the fast-moving and deep interaction of Mathematics and AI.

First Proof, Second Batch: Results
2026-06-10 13:00 - 14:00 | Virtual
First Proof, Second Batch: Results
Date: June 10, 2026
Time: 1:00–2:00 pm ET
Location: via Webinar
The First Proof Editors present the results of their Second Batch benchmark testing on AI systems.
First Proof Editors

Mohammed Abouzaid (Stanford)
Nikhil Srivastava (UC Berkeley)
Rachel Ward (UT Austin)
Lauren Williams (Harvard)

Big Data Conference 2026
2026-09-03 09:00 - 2026-09-04 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Big Data Conference 2026
Dates: Sep. 3–4, 2026
Location: Harvard University CMSA, 20 Garden Street, Cambridge MA & via Zoom
The Big Data Conference features speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
Register to attend in person
Register for Zoom Webinar

Confirmed Speakers

SueYeon Chung, Harvard
Bailey Flanigan, MIT
Sergey Ovchinnikov, MIT
Ariel Procaccia, Harvard
Adit Radhakrishnan, MIT
Andrew Sutherland, MIT
Chris Wiggins, Columbia
Rex Ying, Yale
Wei Zhou, Harvard

Organizers

Michael Desai, Harvard
Michael R. Douglas, CMSA
Yannai Gonczarowski, Harvard
Melanie Weber, Harvard

Schedule (download pdf)
Thursday, Sep. 3, 2026
8:45–9:10 am
Breakfast
9:10–9:15 am
Introductions
9:15–10:15 am
Rex Ying, Yale
A Riemannian Geometry Perspective on Foundation Models
Abstract: In development of foundation models and LLMs, Euclidean space has been the de facto geometric setting for machine learning architectures. However, at a large scale, real-world data often exhibit inherently non-Euclidean structures, such as multi-way relationships, hierarchies, symmetries, and non-isotropic scaling, in a variety of domains, such as languages, vision, and the natural sciences. It is challenging to effectively capture these structures within the constraints of Euclidean spaces. This talk will cover my research that moves beyond Euclidean geometry to maintain the scaling law for the next generation of foundation models. By adopting these geometries, foundation models could more efficiently leverage the structures. Task-aware adaptability that dynamically reconfigures embeddings to match the geometry of downstream applications, could further enhance efficiency and expressivity. This talk will demonstrate the advantages of hyperbolic geometry in building foundation models from individual non-Euclidean components, LLM pre-training / fine-tuning, retrieval-augmented generation, and mixture of expert models. We further present a comprehensive library that greatly simplifies the process of building and adapting foundation models in non-Euclidean. Finally, the talk will enumerate a few promising directions that are being actively explored in this domain.
10:15–10:30 am
Break
10:30–11:30 am
Adit Radhakrishnan, MIT
Toward universal steering and monitoring of AI models
Abstract: Artificial intelligence (AI) models contain much of human knowledge. Understanding the representation of this knowledge will lead to improvements in model capabilities and safeguards. Building on advances in feature learning, we developed an approach for extracting linear representations of semantic notions or concepts in AI models. We showed how these representations enabled model steering, through which we exposed vulnerabilities and improved model capabilities. We demonstrated that concept representations were transferable across languages and enabled multiconcept steering. Across hundreds of concepts, we found that larger models were more steerable and that steering improved model capabilities beyond prompting. We showed that concept representations were more effective for monitoring misaligned content than for using judge models. Our results illustrate the power of internal representations for advancing AI safety and model capabilities.
11:30 am–12:45 pm
Lunch
12:45–1:45 pm
Andrew Sutherland, MIT
Crowdsourcing mathematical databases
Abstract: The L-functions and Modular Forms Database (LMFDB) is one of the largest online repositories of mathematical research data. It contains comprehensive catalogs of mathematical objects that arise in the context of the Langlands program, including number fields, elliptic curves, modular forms, and higher-dimensional analogs of these objects. In cooperation with the Foundation for Science and AI Research (SAIR), we recently ran a competition aimed at solving the inverse Galois problem over Q in degree 24 (IGP24). This goal was achieved by collecting more than 50 million candidate number fields submitted by more than 100 teams, including examples that realize all 25,000 transitive permutation groups of degree 24 as Galois groups. AI tools and computer algebra systems played a central role in this project, both in building the infrastructure to run the contest and in enhancing the capabilities of participants.
1:45–2:00 pm
Break
2:00–3:00 pm
Bailey Flanigan, MIT
Algorithmic Tools for Trading Off Sortition Ideals
Abstract: Citizens’ assemblies and other deliberative minipublics — representative groups of everyday people convened to deliberate on a policy issue and then make recommendations — are now used by governments around the world. Choosing who sits on these panels is the problem of sortition: randomly selecting a small group of citizens that represents the broader population. Sortition has been the subject of substantial computer science research in recent years, and the resulting algorithms are now widely used in practice. This talk will describe the key challenges that arise in the practice of sortition, and the algorithmic tools that have been developed to navigate them optimally.
3:00–3:15 pm
Break
3:15–4:15 pm
Chris Wiggins, Columbia
Prescriptive Learning at Scale: Applied and Deployed Decision-Making
Abstract: Many applications in health and industry require making decisions while learning how the world responds to them: e.g., which article to recommend, which treatment to assign, or when to show a paywall. This problem is prescriptive, what to do rather than what is true. Statistical decision theory has taken up the problem repeatedly since Neyman’s interwar work; recent decades have brought a new wave of results, motivated in large part by the opportunity to make and evaluate decisions at scale online. I will highlight a thread of 21st-century results running through adaptive experimentation, contextual bandits, and off-policy evaluation: they combine advances in supervised learning with fundamental ideas from statistical sampling, and now shape digital products, from content recommendation to marketing, as well as adaptive interventions in health.

Friday, Sep. 4, 2026
8:45–9:15 am
Breakfast
9:15–10:15 am
Ariel Procaccia, Harvard
No Generation Without Representation
Abstract: AI systems and democratic processes are confronting similar challenges around representation. I examine two related questions that cut across both domains. First, how can AI enable democratic processes that handle vast spaces of opinions or statements while ensuring proportional representation of a population’s views? Second, when AI systems themselves provide normative guidance, whose viewpoints do they reflect, and can we make this precise? Drawing on social choice theory, I present formal frameworks and algorithms for both problems, showing that meaningful representation guarantees are feasible and practical.
10:15–10:30 am
Break
10:30–11:30 am
Wei Zhou, Harvard
Biobank-scale genetic discovery: from association testing to global meta-analysis
Abstract: Biobanks linking genomic data with electronic health records provide unprecedented opportunities for genetic discovery for complex human diseases, but they also pose analytical challenges that extend well beyond sample size. Within a biobank, association studies must account for population structure and relatedness, highly unbalanced case–control ratios, rare genetic variants, longitudinal and censored outcomes, and the computational demands of analyzing hundreds of thousands of individuals and millions of genetic variants. Across biobanks, additional challenges arise from differences in ancestry, phenotype definitions, recruitment strategies and genetic effects.In this talk, I will discuss statistical and computational methods developed to address these challenges at successive stages of biobank analysis. These include scalable generalized linear mixed models for binary traits, survival mixed models for censored time-to-event outcomes, and gene- and region-based tests that aggregate rare variants. I will describe statistical approximations and computational strategies that make these analyses feasible at biobank scale while maintaining calibration in the presence of relatedness and highly unbalanced phenotypes. I will then introduce the Global Biobank Meta-analysis Initiative (GBMI) and describe how genetic evidence can be combined across biobanks without sharing individual-level data. Examples from GBMI will illustrate how combining evidence across biobanks can increase statistical power through larger sample sizes and broaden genetic discovery through greater ancestral diversity.
11:30 am–12:45 pm
Lunch
12:45–1:45 pm
Sergey Ovchinnikov, MIT
Using AI for protein structure modeling and design
Abstract: In this talk, I’ll describe recent work in model interpretability, focusing on decomposing protein language models and structure prediction models like AlphaFold. The goal is to understand what they are learning, their limitations and how we can use this information to develop better models and design proteins.
1:45–2:00 pm
Break
2:00–3:00 pm
SueYeon Chung, Harvard/Flatiron Institute
Computing with Neural Manifolds: A Multi-Scale Framework for Understanding Biological and Artificial Neural Networks
Recent breakthroughs in experimental neuroscience and machine learning have opened new frontiers in understanding the computational principles governing neural circuits and artificial neural networks (ANNs). Both biological and artificial systems exhibit an astonishing degree of orchestrated information processing capabilities across multiple scales – from the microscopic responses of individual neurons to the emergent macroscopic phenomena of cognition and task functions. At the mesoscopic scale, the structures of neuron population activities manifest themselves as neural representations. Neural computation can be viewed as a series of transformations of these representations through various processing stages of the brain. The primary focus of my lab’s research is to develop theories of neural representations that describe the principles of neural coding and, importantly, capture the complex structure of real data from both biological and artificial systems.
In this talk, I will present three related approaches that leverage techniques from statistical physics, machine learning, and geometry to study the multi-scale nature of neural computation. First, I will introduce new theories based on statistical physics and convex geometry that connect complex geometric structures that arise from neural responses (i.e., neural manifolds) to the efficiency of neural representations in implementing a task. Second, I will employ these theories to analyze how these representations evolve across scales, shaped by the properties of single neurons, learning dynamics, and the transformations across distinct brain regions. Finally, I will show how these insights extend efficient coding principles beyond early sensory stages, linking representational geometry to efficient task implementations. This framework not only help interpret and compare models of brain data but also offers a principled approach to designing ANN models for higher-level vision. This perspective opens new opportunities for using neuroscience-inspired principles to guide the development of intelligent systems.

Fall CMSA Welcome Event
2026-09-03 16:15 - 17:30 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Fall CMSA Welcome Event
Date: September 3, 2026
Time: 4:15 pm
Location: CMSA Common Room, 20 Garden Street, Cambridge MA

All CMSA and Math affiliates are invited.

The Geometry of Machine Learning 2026
2026-09-08 09:00 - 2026-09-11 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
The Geometry of Machine Learning 2026
Dates: September 8–11, 2026
Location: Harvard CMSA, Room G10, 20 Garden Street, Cambridge MA 02138 & via Zoom Webinar
Register to attend in person
Register for Zoom Webinar
Large language models are presently, and will increasingly, be complemented by other dimensions of intelligence: formal verification and energy-based optimizers, becoming parts of larger ecosystems. Can AIs reason geometrically and can we use geometry to reveal how data is currently processed in NNs? Can AIs reveal the geometry of mathematics, as well as studying geometry as a subject within math. This conference is intended to continue the discussion of these topics.
Confirmed Speakers:

Nada Amin, Harvard
Randall Balestriero, Brown
Michael Brenner, Harvard and Google
Bennet Chow, UCSD
Surya Ganguli, Stanford
Boris Hanin, Princeton
Roi Holtzman, Oxford
Robert Koirala, UCSD
Dmitry Krotov, Dynamical Mind
Slava Krushkal, Virginia
Jared Duker Lichtman, Stanford
Mike Mulligan, UCR, Logical Intelligence
Luca Pesce, Harvard
Gabriel Poesia, U Michigan (via Zoom)
Mathew Vanherreweghe, Logical Intelligence
Sean Welleck, CMU (via Zoom)
Mattiew Wyart, JHU (via Zoom)

Organizers: Michael R. Douglas (CMSA) and Mike Freedman (CMSA)

Schedule
Tuesday, Sep. 8, 2026
8:15–8:45 am
Breakfast
8:45–9:30 am
Mike Mulligan, UCR, Logical Intelligence
Compression is all you need: Modeling mathematics
Abstract: The mathematics humans discover and value (“human math”) is a vanishingly small subset of all valid deductions (“formal math”). I’ll argue that human math is distinguished by its compressibility through hierarchically nested definitions and theorems, like a polynomial-growth space rather than the exponential-growth space one might expect when proofs are viewed as strings of symbols. The argument combines toy monoid models with an empirical analysis of MathLib, a large Lean library of formalized mathematics we treat as a proxy for human math. I’ll close with how compression itself can serve as a measure of mathematical interest, giving agents a sense of direction toward where human math lives.
9:45–10:30 am
Boris Hanin, Princeton
10:30–11:00 am
Break
11:00–11:45 am
Slava Krushkal, University of Virginia
Using AI to study 4-manifold topology
Abstract: Central open questions in geometric classification theory of topological 4-manifolds have a reformulation in terms of the Round Handle Problem. It asks whether a given link in the 3-sphere is slice (bounds disjoint disks) in the 4-manifold obtained by attaching certain round handles to the 4-ball. I will discuss an algebraic-combinatorial formulation of the problem, ongoing AI-assisted work on it, and the results obtained to date.
12:00–12:45 pm
Robert Koirala and Bennett Chow, UCSD
AI for Ricci flow: Discovery and Formalization

Wednesday, Sep. 9, 2026
8:15–8:45 am
Breakfast
8:45–9:30 am
Surya Ganguli, Stanford
9:45–10:30 am
Mathew Vanherreweghe, Logical Intelligence
10:30–11:00 am
Break
11:00–11:45 am
Michael Brenner, Harvard and Google
12:00–12:45 pm
Mattiew Wyart, JHU (via Zoom)
Learn from your own latents, not from tokens
Abstract: Language models need more than a hundred thousand times the data a child does. One explanation is that predicting raw tokens is simply the wrong level: methods like data2vec and JEPA instead train a network to predict its own internal representations, with strong empirical results but no theory of why. Using a hierarchical grammar that models that language and images have a hidden hierarchical structure, we quantify the gain exactly. Token-level learning needs a number of examples growing exponentially with the depth of the hierarchy; latent prediction needs a number independent of it. We also show data2vec performs this hierarchical prediction implicitly, which suggests that explicitly stacking levels – as in H-JEPA – yields little.

Thursday, Sep. 10, 2026
8:15–8:45 am
Breakfast
8:45–9:30 am
Dmitry Krotov, Dynamical Mind
Dense Associative Memory: Physical systems for novel AI architectures
Abstract: Dense Associative Memories are recurrent neural networks with fixed-point attractor states that are described by an energy function. In contrast to conventional Hopfield Networks, which were popular in the 1980s, Dense Associative Memories have a very large information storage capacity, making them appealing tools for many problems in AI. In this talk, I will provide an intuitive understanding and mathematical framework for this class of models and give examples of problems in AI that can be tackled using these new ideas. Specifically, I will explore the relationship between Dense Associative Memories and transformers. I will present a neural network called the Energy Transformer, which unifies energy-based modeling, associative memories, and transformers in a single architecture. I will demonstrate how Energy Transformers can be used for challenging tasks in image processing, solve partial differential equations, and serve as computational modules for energy-based language modeling. I will also discuss an exciting possibility of mapping these models onto analog hardware accelerators, which could enable much more energy-efficient inference compared to GPUs.
9:45–10:30 am
Roi Holtzman, Oxford
10:30–11:00 am
Break
11:00–11:45 am
Sean Welleck, CMU (via Zoom)
12:00–12:45 pm
Gabriel Poesia, University of Michigan (via Zoom)
Friday, Sep. 11, 2026
8:15–8:45 am
Breakfast
8:45–9:30 am
Nada Amin, Harvard
Compiling Programs to Neurons
Abstract: Neural networks are ordinarily programmed indirectly: we specify architectures, objectives, and data, and rely on learning to discover a computation. I will focus on Cajal, a typed, higher-order linear programming language whose programs compile correctly to linear neurons and, with iteration, to recurrent neurons. This allows discrete programming structures such as conditionals and iteration to coexist with gradient-based learning. Experiments show that connecting compiled neurons with learned networks can improve learning speed and data efficiency. This work is led by PhD student Joey Velez-Ginorio and is joint with his advisors at UPenn, Konrad Kording and Steve Zdancewic. I will close with a complementary direction from my work: using language models together with formal verification to generate programs and proofs with machine-checkable guarantees. Together, these projects explore how programming languages can provide structure and control at the boundary between programs and learned systems.
9:45–10:30 am
Randall Balestriero, Brown University
Counterfactual World Models for Real World Deployment
10:30–11:00 am
Break
11:00–11:45 am
Luca Pesce, Harvard CMSA
A spiked perspective on feature learning with gradient-based methods
Abstract: Depth is widely believed to give neural networks a clear computational advantage over shallow models, and making this belief precise is a central problem in learning theory. We study a controlled high-dimensional setting where this can be done. The targets are hierarchical: the relevant structure is distributed across latent subspaces of decreasing dimension, so that a shallow model must resolve all of it simultaneously, while a deep network does not. Each layer forms an intermediate representation, allowing learning to proceed in stages, with every stage reducing the effective dimension of the remaining problem. We analyze this staged mechanism through the gradient descent dynamics of a deep network yielding a sharp separation in sample complexity between shallow and deep architectures. The result is a concrete account of why depth allows such functions to be learned from substantially fewer samples than shallow methods require.
12:00–12:45 pm
Jared Duker Lichtman, Stanford

Support provided by Logical Intelligence.

Quasi-local algebras and invertible phases of matter
2026-09-08 16:15 - 18:00 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Nikita Sopenko, CMSA
Title: Quasi-local algebras and invertible phases of matter
Abstract: In the algebraic approach to quantum field theory, observables localized in bounded regions form a net of von Neumann algebras generating what is known as a quasi-local algebra. I will discuss the classification of quasi-local algebras under a notion of equivalence that generalizes Brauer equivalence and captures only large-scale properties. I will argue that this problem coincides with the classification of invertible phases of lattice systems in one dimension higher, providing a way to relate quantum field-theoretic and microscopic descriptions. As a byproduct, I will describe a candidate for the generalized cohomology theory associated with invertible phases, as conjectured by Kitaev.

New Technologies in Mathematics Seminar
2026-09-09 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: tba

Working with LLMs to do high quality math
2026-09-09 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AI for the Working Mathematician
Speaker: Daniel Litt, University of Toronto
Title: Working with LLMs to do high quality math
Abstract: Much hay has been made of the capabilities of LLMs to do math autonomously, and indeed frontier models have resolved some long-standing, interesting open questions. But our goal is not to produce papers, but rather to do high quality mathematics. I’ll discuss some experiments over the past year in trying to get LLMs to help with producing high quality mathematics, non-autonomously, and speculate a bit about the future.

Geometry and Mathematical Physics Seminar
2026-09-10 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Geometry and Mathematical Physics Seminar
Speaker: Brian Williams, Boston University

Member Seminar
2026-09-11 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Elena Kim

Colloquium
2026-09-14 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
