Millennium Prize Problems Lecture - Javier Gómez-Serrano: Navier-Stokes Existence or Breakdown
2026-03-11 17:00 - 18:00 (America/New_York) | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Date: March 11, 2026
Time: 5:00–6:00 pm
Location: Harvard Science Center Hall C, 1 Oxford St., Cambridge MA & via Zoom Webinar
Speaker: Javier Gómez-Serrano, Brown University
Title: Navier-Stokes Existence or Breakdown
Abstract: The Navier-Stokes equations have been the cornerstone of fluid dynamics for over a century, accurately describing the motion of viscous fluids such as water and air. However, despite their fundamental importance to mathematics and physics, a profound question remains unanswered: do solutions to these equations always exist for all time, or can they break down and develop singularities (points where the equations lose their validity)? In this Millennium Prize Problems Lecture, I will explore the current mathematical landscape surrounding the Navier-Stokes and related equations. The talk will discuss the historical context, the ongoing search for global regularity versus finite-time blowup, and the latest analytical and computational breakthroughs pushing the boundaries of what we know about fluid behavior.
Read more about the Navier-Stokes Equation at the Clay Math website.

Organizers: Martin Bridson, Clay Mathematics Institute | Dan Freed, Harvard University and CMSA | Mike Hopkins, Harvard University

Millennium Prize Problems Lecture Series

Swampland and our Universe
2026-04-15 08:00 - 2026-04-16 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Swampland and our Universe
Dates: April 15–16, 2026
Location: Harvard CMSA, Room G10, 20 Garden Street, Cambridge MA
The swampland program has inspired a range of new ideas in both cosmology and neutrino physics. This workshop brings together experts in neutrino physics, dark energy, dark matter, early-universe cosmology, and string theory to share insights on these developments and to discuss current and future experimental tests.
Speakers

Ignatios Antoniadis, IAS, Princeton
Alek Bedroya, Princeton
Mike Boylan-Kolchin, UT Austin
M.C. Gonzalez-Garcia, YITP Stony Brook & ICREA U. Barcelona
Mustapha Ishak-Boushaki, UT Dallas
Marc Kamionkowski, Johns Hopkins
Miguel Montero, Institute of Theoretical Physics, Madrid
Georges Obied, U Chicago
Matt Reece, Harvard
Tracy Slatyer, MIT

Organizers: Luis Anchordoqui (CUNY Lehman College), Sonia Paban (Harvard Physics), and Cumrun Vafa (Harvard Physics)

Videos are available on the CMSA Youtube Swampland Playlist
Schedule
(download pdf)
Wednesday, Apr. 15, 2026
8:00–9:00 am
Breakfast
9:00–10:00 am
Marc Kamionkowski, Johns Hopkins: Dark-matter dynamics and new physics
Abstract: Galactic halos that are spherical, stationary, and composed of collisionless dark matter are easy to describe mathematically. If dark matter decays or interacts or there is some departure from equilibrium or time evolution of the system, all bets are off. In this case costly N-body simulations are required. If, however, one retains the assumption of spherical symmetry, these systems can be evolved numerically with a far simpler algorithm that is easily coded run in a matter of minutes on a laptop, rather than a day on a supercomputer. I will describe this approach and illustrate with simulations of self-interacting dark matter, decaying dark matter (with and without anisotropic velocity distributions, supermassive-black-hole growth, tidal stripping, mixed SIDM/CDM models. Come prepared with your own non-standard dark-matter model; we’ll see if we can simulate it during the talk!
10:00–10:30 am
Coffee Break
10:30–11:30 am
Tracy Slatyer, MIT: What (more) the CMB can teach us about dark matter
Abstract: Observations of the cosmic microwave background have already provided critical evidence for dark matter, but can also stringently constrain a range of dark matter properties. I will outline CMB constraints on dark matter properties based on purely gravitational effects, and then discuss in more detail how both CMB anisotropies and the blackbody spectrum can be used to test dark matter interactions with the Standard Model.
11:30 am–1:00 pm
Lunch Break (catered)
1:00–2:00 pm
Alek Bedroya, Princeton: How Quantum Gravity Constrains Physics on the Largest Length Scales
Abstract: I will review the hierarchy of energy scales in quantum gravity, from the Hubble scale in the IR to the quantum-gravity cutoff and the Planck scale in the UV, and emphasize the nontrivial UV/IR relations that connect them. I will discuss conjectures constraining scalar potentials and explain how they are related to the behavior of the other energy scales, together with bottom-up arguments based on general principles of quantum gravity such as holography. In particular, I will explain how well-motivated holographic assumptions lead to strong restrictions on scalar potentials, ruling out parametrically prolonged accelerated expansion for positive potentials and obstructing parametric scale separation for negative potentials associated with AdS vacua. Title: How Quantum Gravity Constrains Physics on the Largest Length Scales
Abstract: I will review the hierarchy of energy scales in quantum gravity, from the Hubble scale in the IR to the quantum-gravity cutoff and the Planck scale in the UV, and emphasize the nontrivial UV/IR relations that connect them. I will discuss conjectures constraining scalar potentials and explain how they are related to the behavior of the other energy scales, together with bottom-up arguments based on general principles of quantum gravity such as holography. In particular, I will explain how well-motivated holographic assumptions lead to strong restrictions on scalar potentials, ruling out parametrically prolonged accelerated expansion for positive potentials and obstructing parametric scale separation for negative potentials associated with AdS vacua.
2:00–2:30 pm
Coffee Break
2:30–3:30 pm
Mustapha Ishak-Boushaki, UT Dallas: Persistent and serious challenge to the ΛCDM throne: Evidence for dynamical dark energy rising from combinations of different types of datasets
Abstract: We derive multiple constraints on dark energy and compare dynamical dark energy models with a time-varying equation of state (w0waCDM) versus a cosmological constant model (LCDM). We use Baryon Acoustic Oscillation (BAO) from DESI and DES, Cosmic Microwave Background from Planck with and without lensing from Planck and ACT (noted CMBL and CMB, respectively), supernovae(SN), and cross-correlations between galaxy positions and galaxy lensing from DES. We use pairs or triplets of datasets where we exclude one type of dataset each time and categorize them as “NO SN”, “NO CMB” and “NO BAO” combinations. In all cases, we find that the combinations favor the w0waCDM model over LCDM, with significance ranging from 2.0 to 3.0-sigma. The persistence of this pattern across various dataset combinations even when any of the datasets is excluded supports an overall validation of this trending result regardless of any specific dataset. Next, we use larger combinations of these datasets after verifying their mutual consistency within the w0waCDM model. We find combinations that give robust significance levels, with DESI+DESY6BAO+CMBL+SN giving 3.4-sigma. In sum, while we need to remain cautious, the trend and pattern of these results beyond any single type of dataset and their associated systematics presents a compelling overall portrait not in favor of the LCDM and constitutes a serious challenge to the model’s reign. A few other cosmological results will be provided.
3:30–4:00 pm
Coffee Break
4:00–5:00 pm
Georges Obied, U Chicago: The Dark Dimension and its interplay with DESI data
Abstract: In this talk, I will discuss the motivation for considering an extra mesoscopic Dark Dimension of length l ~ 1 – 10 microns, taking into account theoretical and observational arguments. I will then talk about cosmological aspects of the Dark Dimension. In particular this scenario leads, by the universal coupling of the Standard Model sector to bulk gravitons, to massive spin 2 KK excitations of the graviton in the Dark Dimension (the “dark gravitons”) as an unavoidable dark matter candidate. Observations allow such an extra dimension of size in the micron range. Finally, I will discuss how this scenario can naturally accommodate features recently observed by the DESI survey such as an effective dark energy equation of state that is smaller than -1.

Thursday, Apr. 16, 2026
8:00–8:30 am
Breakfast
8:30–9:30 am
MC Gonzalez-Garcia, YITP Stony Brook & ICREA U. Barcelona: Massive Neutrinos in 2026: What we know, what we do not know (yet?), and what we do not understand
Abstract: In this talk I will present an update of the current understanding (and some not understanding) of the neutrino masses and the lepton mixing and some other minimal SM extensions as derived from direct scrutiny of the results of neutrino flavour oscillation experiments, some other laboratory probes, and the cosmos.
9:30–10:00 am
Coffee Break
10:00–11:00 am
Miguel Montero, IFT, Madrid: Neutrinos and B-L symmetry in the Dark Dimension scenario
Abstract: The Dark Dimension proposes the existe of a micrometer-sized large extra dimension, whose size is tied to the observed small vacuum energy. I will review the scenario, and then discuss how to embed the B-L global symmetry of the SM, focusing on one possibility which leads to an explanation of the observed coincidence between neutrino mass scale and the vacuum energy scale, while leading to 3 light species of right-handed neutrinos. I will also briefly discuss potential opportunities for detection of the resulting neutrino oscillations.
11:00–11:30 am
Coffee Break
11:30 am–12:30 pm
Ignatios Antoniadis, IAS, Princeton: Searching for the dark dimension in neutrino experiments
Abstract: Micron size extra dimensions offer a possibility to explain the smallness of neutrino masses if the right-handed neutrino propagates in the higher dimensional bulk. I will discuss the theoretical framework and the experimental signatures of this proposal in present and future experiments of KATRIN prototype, aiming to measure the magnitude of neutrino masses and to search for extra sterile-type species.
12:30–1:30 pm
Lunch Break (catered)
1:30–2:30 pm
Mike Boylan-Kolchin, UT Austin: Galaxies as Tracers of the Matter Density Field
Abstract: Galaxy formation is often (rightly) thought of as involving a complex interplay of messy astrophysical processes, but it also traces the nonlinear evolution of the matter density in the Universe. Remarkably, it appears that properties of this nonlinear field are intimately connected to properties of the initial linear fluctuations and some basic physics of dark matter interactions. I will explore some of these connections, with applications that include the surprisingly fast evolution of early galaxy formation as revealed by JWST and properties of the lowest-mass dark matter clumps capable of hosting galaxies in the local Universe.
2:30–3:00 pm
Coffee Break
3:00–4:00 pm
Matt Reece, Harvard: Axions from String Theory, and String Theory from Axions
Abstract: String theory compactifications contain the right ingredients to produce axion fields that might solve the Strong CP problem or contribute to dark matter or dynamical dark energy in our universe. After briefly reviewing some of these ingredients, I will frame the inverse question: suppose that an axion is discovered, and its decay constant is measured in an experiment. Could this help us to locate ourselves in the string landscape? In particular, I will discuss how an axion could give us clues about the fundamental string scale and the scale of supersymmetry breaking.

Millennium Prize Problems Lecture - Peter Sarnak: Riemann Hypothesis
2026-04-15 17:00 - 18:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Date: April 15, 2026
Time: 5:00–6:00 pm
Location: Harvard Science Center Hall C, 1 Oxford St., Cambridge MA
Speaker: Peter Sarnak, Institute for Advanced Study
Title: The Riemann Hypothesis
Abstract: After reviewing the hypothesis as put forth by Riemann we discuss its generalizations and analogues. We highlight a few of their implications and workarounds, and probing their truths.
Read more about the Riemann Hypothesis at the Clay Math website.
Organizers: Martin Bridson, Clay Mathematics Institute | Dan Freed, Harvard University and CMSA | Mike Hopkins, Harvard University

Millennium Prize Problems Lecture Series

CMSA/Tsinghua Math-Science Literature Lecture: Nicolai Reshetikhin (Tsinghua): Asymptotic representation theory
2026-04-22 09:00 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA/Tsinghua Math-Science Literature Lecture
Date: April 22, 2026
Time: 9:00 – 10:30 am ET
Location: via Zoom Webinar
Speaker: Nicolai Reshetikhin, Yau Mathematical Sciences Center, Tsinghua University
Title: Asymptotic representation theory
Abstract: Loosely speaking asymptotic representation theory studies representations of “large” groups or algebras. One of the first results in this direction is the study of Plancherel measures on the symmetric group $S_N$ in the limit $Nto infty$ by Vershik and Kerov and Logan and Shepp. The first part of the talk will be an overview of results on statistics of irreducible representations in large tensor products. Then we focus on more modern results on statistics of tilting and projective modules in large tensor products and on how some problems in asymptotic representation theory are related to dimer models in statistical mechanics.

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

Sixth Annual Yip Lecture | Regina Barzilay, MIT: Can machine learning methods design drugs?
2026-04-23 16:00 - 17:00
Sixth Annual Yip Lecture
Date: April 23, 2026
Time: 4:00–5:00 pm ET
Location: Harvard Science Center Hall A & via Zoom Webinar
Speaker: Regina Barzilay, MIT
Title: Can ML methods design drugs?
Abstract: Today, life sciences are driven by prohibitively expensive wet lab experimentations, which limit the pace and scope of discovery. This talk focuses on AI algorithms that enable in-silico modeling of biological processes. Specifically, I will focus on algorithms for molecular and cellular modeling. I will highlight several successful examples where these algorithms have already transformed drug discovery. In the second part of the talk, I want to focus on problems where current methods fail to deliver as expected, motivating the need for algorithmic innovations.
Bio: Regina Barzilay is a School of Engineering Distinguished Professor for AI and Health in the Department of Electrical Engineering and Computer Science (EECS) at MIT. Since 2018, she has been the AI faculty lead for the MIT Jameel Clinic and a member of MIT CSAIL.
She is a member of three national academies, including the National Academy of Engineering, the National Academy of Medicine, and the American Academy of Arts & Sciences.
She is also a recipient of various awards, including a 2017 MacArthur fellowship “Genius Grant.” In 2020, she was awarded the Squirrel AI Award for Artificial Intelligence for the Benefit of Humanity. More recently, she has been recognized in the 2025 TIME100 AI List and awarded with the IEEE Frances E. Allen Medal for her development of innovative machine learning algorithms that have significantly advanced human language technology and transformed medical diagnostics and drug discovery.
She completed her PhD in Computer Science from Columbia University, and spent a year as a postdoc at Cornell University. Barzilay received her undergraduate degree from Ben-Gurion University of the Negev, Israel.
The Yip Lecture takes place thanks to the support of Dr. Shing-Yiu Yip.

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.
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

Videos of selected talks are now available on the Youtube Playlist

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.

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 & 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
Adit Radhakrishnan, MIT
Andrew Sutherland, MIT
Chris Wiggins, Columbia
Rex Ying, Yale
Wei Zhou, Harvard

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
Register to attend in person
Register for Zoom Webinar

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
Dimitry Krotov, IAS
Slava Krushkal, Virginia
Mike Mulligan, UCR, Logical Intelligence
Gabriele Poesia, Stanford
Mathew Vanherreweghe, Logical Intelligence
Sean Welleck, CMU (via Zoom)
Mattiew Wyart, JHU

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

Details TBA

Support provided by Logical Intelligence.

Ahlfors Lectures: Mohammed Abouzaid
2026-09-16 - 2026-09-17 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Ahlfors Lectures
Dates: September 16 & 17, 2026
Time: 4:00–5:00 pm
Speaker: Mohammed Abouzaid (Stanford)
Register to attend in person or online at: https://www.math.harvard.edu/event/ahlfors-lecture-series-mohammed-abouzaid/

September 16, 2026
Title: Framed bordism and nearby Lagrangians
Abstract: The problem of understanding Lagrangian embeddings in symplectic manifolds remains a mystery. The simplest class of symplectic manifolds are the cotangent bundles of smooth manifolds, which is the setting for the most prominent open problem in the subject: Arnold’s nearby Lagrangian conjecture which asserts the connectedness of the space of Lagrangians for which the relative class of the symplectic form vanishes. In particular, the conjecture asserts that all such submanifolds are diffeomorphic. In view of recent work with Alvarez-Gavela, Courte, and Kragh, as well as work of Procelli and Smith, I will use the question as a motivation for joint work with Andrew Blumberg on a bordism-theoretic approach to define bordism-valued refinements of Floer homology.

September 17, 2026
Title: Complex bordism and Hamiltonian fibrations
Abstract: As a consequence of his theory of weights, Deligne proved that smooth complex projective varieties, carrying a smooth morphism to the projective line, have rational cohomology groups that are isomorphic to the tensor product of the cohomology of the fibre with that of the base. In joint work with McLean and Smith, we should that the result not only holds for symplectic manifolds, but is in fact a consequence of an analogous result that holds for complex cobordism. The proof identifies bordism of “stably complex derived orbifolds” as the correct framework in which to formulate these results, and I will discuss joint work with Shaoyun Bai to understand these bordism groups.
The Ahlfors Lecture Series is presented by the Harvard University Math Department in memory of Professor Lars Ahlfors.

CMSA/Tsinghua Math-Science Literature Lecture: Robert Gompf: On classifying smoothings of R^4–from the beginning to the present
2026-09-16 09:00 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA/Tsinghua Math-Science Literature Lecture
Date: September 16, 2026
Time: 9:00 – 10:30 am ET
Location: CMSA G10, 20 Garden Street & via Zoom Webinar

Please register to attend
In-person registration

Zoom Webinar registration

Speaker: Robert E. Gompf, University of Texas, Austin
Title: On classifying smoothings of R^4–from the beginning to the present
Abstract: The field of 4-manifold topology was born around 1981 with two independent revolutionary breakthroughs, by Fields medalists Michael Freedman and Simon Donaldson. Freedman showed that topological 4-manifolds obeyed the well-developed theory of higher-dimensional manifolds, while Donaldson used nonlinear analysis (gauge theory) to show that smooth 4-manifolds are quite different from their high-dimensional counterparts. The clash between the two theories makes 4-manifold topology uniquely intricate within manifold theory. One of the most startling consequences is that unlike in other dimensions, Euclidean 4-space R^4 admits exotic smoothings. Equivalently, there are smooth 4-manifolds that are homeomorphic, but not diffeomorphic, to R^4. We will trace the classification theory of these manifolds from their discovery to the most recent development (joint with Aristotelis Panagiotopoulos) that they cannot be classified by any countable family of numerical invariants.

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

Workshop on Lagrangian Floer theory and applications
2026-09-28 08:00 - 2026-10-02 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Workshop on Lagrangian Floer Theory and Applications
Dates: September 28 – October 2, 2026
Location: CMSA G10, 20 Garden St., Cambridge MA 02138
This workshop is part of the Lagrangian Floer Theory and Applications Program

Invited Speakers

Mohammed Abouzaid, Stanford University
Octav Cornea, University of Montréal
Tobias Ekholm, Uppsala University
Sheel Ganatra, USC
Andrew Hanlon, University of Oregon
Yanki Lekili, Imperial College London
Mark Mclean, Stonybrook
Georgios Dimitroglou Rizell, Uppsala University
Paul Seidel, MIT
Egor Shelukhin, University of Montréal
Vivek Shende, Berkeley/Syddansk
Siu-Cheong Lau, Boston University

Organizers: Denis Auroux (Harvard), Jonny Evans (Lancaster), and Chris Woodward (Rutgers)

Math Science Lectures in Honor of Raoul Bott | Daniel Litt, U Toronto
2026-10-19 16:00 - 17:00
TITLE TBA
Dates: October 19 & 21, 2026
Time: 4:00–5:00 pm ET
Location: Science Center Lecture Hall A and via Webinar
In-person registration
Webinar Registration

Speaker: Daniel Litt, University of Toronto & Benedict H. Gross Distinguished Visitor, Harvard Mathematics Department
Lecture 1: Monday, October 19, 2026, followed by a reception in the Math Dept Austine & Chilton McDonnell Common Room, 4th floor

Lecture 2: Wednesday, October 21, 2026

Harvard Mathematics Professor Raoul Bott (1923 – 2005), was a Hungarian-American mathematician known for numerous foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem.

Math Science Lectures in Honor of Raoul Bott | Daniel Litt, Benedict H. Gross Distinguished Visitor, Harvard Mathematics Department
2026-10-21 16:00 - 17:00
TITLE TBA
Dates: October 19 & 21, 2026
Time: 4:00–5:00 pm ET
Location: Science Center Lecture Hall A and via Webinar
In-person registration
Webinar Registration

Speaker: Daniel Litt, Benedict H. Gross Distinguished Visitor, Harvard Mathematics Department
Lecture 1: Monday, October 19, 2026, followed by a reception in the Math Dept Austine & Chilton McDonnell Common Room, 4th floor

Lecture 2: Wednesday, October 21, 2026

Harvard Mathematics Professor Raoul Bott (1923 – 2005), was a Hungarian-American mathematician known for numerous foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem.

Northeast Conference on Categorical Methods
2026-11-06 09:00 - 2026-11-07 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Northeast Conference on Categorical Methods
Dates: November 6–​7, 2026
Location: CMSA G10, 20 Garden St., Cambridge MA 02138

This conference is intended to bring together researchers in all areas of mathematics and mathematical physics whose work involves the use of methods from categorical algebra, abstract homotopy theory, and higher category theory. Our goal is to showcase new, exciting research and offer an avenue for researchers from the Northeast US to discuss their work and collaborate with people in this ever-growing community.

There will be time for 4 short talks by early-career mathematicians.

REGISTRATION FORM (link)

Invited Speakers

Andrew Blumberg, Columbia University
Soren Galatius, Columbia University
Tony Pantev, University of Pennsylvania
Nick Rozenblyum, University of Toronto
Tomer Schlank, University of Chicago/Harvard University
Brian Williams, Boston University
Inna Zakharevich, Cornell University

Organized by Dan Freed, Harvard Math & CMSA; Owen Gwilliam, UMass Amherst; and Lorenzo Riva, Harvard CMSA

Stable Homotopy Theory and Arithmetic Geometry Boot Camp
2027-02-22 09:00 - 2027-02-26 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Stable Homotopy Theory and Arithmetic Geometry Boot Camp
Dates: February 22-26, 2027

Stable Homotopy Theory and Arithmetic Geometry Workshop I
2027-03-01 09:00 - 2027-03-05 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Stable Homotopy Theory and Arithmetic Geometry Workshop I
Dates: March 1–5, 2027

Seventh Annual Yip Lecture | L. Mahadevan, Harvard
2027-03-03 16:00 - 17:00
Sixth Annual Yip Lecture
Date: Wednesday, March 3, 2027
Time: 4:00–5:00 pm
Location: Harvard Science Center Hall D & via Zoom Webinar
Speaker: L. Mahadevan, Harvard

The Yip Lecture takes place thanks to the support of Dr. Shing-Yiu Yip.

2027 Ding Shum Lecture: Avi Wigderson, IAS
2027-03-31 16:00 - 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
2026 Ding Shum Lecture
Date: Wednedsay, March 31, 2027
Time: 4:00 pm
Location: Harvard Science Center Hall D & via Zoom Webinar
A reception will follow the talk in the Mathematics Department Austine & Chilton McDonnell Common Room, 4th floor, Science Center.
Speaker: Avi Wigderson, IAS

This event is made possible by the generous funding of Ding Lei and Harry Shum.

Stable Homotopy Theory and Arithmetic Geometry Workshop II
2027-05-03 09:00 - 2027-05-07 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Stable Homotopy Theory and Arithmetic Geometry Workshop II
Dates: May 3–7, 2027
