Institute for Advanced Study

Salvatore Pace discusses the order of anomalies in global symmetries: the number of identical copies needed before their diagonal symmetry becomes anomaly-free. Familiar lattice examples often have finite order, whereas perturbative continuum anomalies have infinite order. The seminar examines Onsager symmetry as a lattice counterpart of chiral symmetry for a massless Dirac fermion in one spatial dimension. Its anomaly has order two, but imposing CPT symmetry on the lattice changes it to infinite order, bringing the lattice description closer to the continuum. The work was carried out with Elijah Lew-Smith and Shu-Heng Shao and is reported in arXiv:2606.12510. Attendance is at the IAS Bloomberg Lecture Hall; the organizer also offers Zoom access by request through the contact listed on the event page.

Michael Hutchings introduces Reeb vector fields and explains why their periodic orbits are substantially better understood in three dimensions through Seiberg-Witten theory. The colloquium develops Irie's closing lemma, which creates a periodic orbit through a selected region after a small perturbation, and discusses a quantitative refinement connecting the perturbation size to an upper bound on the orbit period.

This seminar discusses a proof-of-concept gravitational-waveform model described in arXiv:2511.10522. It augments post-Newtonian equations using physics-informed neural-network methods and universal differential equations. The talk also covers ongoing development of a precessing surrogate model designed to evaluate in less than one millisecond while covering a broader portion of the LVK frequency band.

JWST observations reveal sources earlier and brighter than anticipated. This talk examines their implications for cosmological and galaxy-formation models. It considers the unexpectedly numerous galaxies during the first few hundred million years, at redshifts above nine, and how cosmological clustering measurements can distinguish competing explanations. It then presents an initial empirical measurement of the dependence of star-formation variability, or burstiness, on galaxy mass, which may help explain the excess. Finally, it discusses tensions between JWST measurements of reionization and evidence from the cosmic microwave background and Lyman-alpha forest, including implications for DESI’s recent dark-energy results.

Planets form from dust embedded in gaseous protoplanetary disks. Hydrodynamic instabilities can hinder growth by stirring turbulence, yet also help by creating structures that collect solids and promote planetesimal formation. Min-Kai Lin will discuss analytical and numerical studies of the vertical shear instability and convective overstability, their interactions with dust, and generalisations of the streaming instability to disks with accretion or magnetic fields. The seminar examines how these processes alter the conditions for concentrating solids and the challenges that remain in connecting disk dynamics to planet formation.

Zander Kelley presents joint work with Xin Lyu separating randomized and deterministic three-player communication in the number-on-forehead model. Earlier work by Kelley, Lovett and Meka gave an explicit function with an efficient randomized protocol but a deterministic lower bound of Ω(n^(1/3)). The argument studies whether its yes-instances can be covered efficiently by small cylinder intersections. A sifting lemma finds a denser induced subgraph inside a bipartite graph with large grid norm. An improved version raises the communication lower bound to Ω(n^(1/2)). The key structural result covers small cylinder intersections by a few reasonably small slice functions; the talk compares this simplification with Szemerédi’s triangle removal lemma.

Chris Skinner introduces the Birch and Swinnerton-Dyer conjecture and surveys major advances toward it before and after the turn of the millennium. The presentation revisits the substance of his plenary lecture at the 2025 Clay Research Conference on the Millennium Prize Problems.

Linquan Ma introduces recent progress on singularities in mixed characteristic. Starting from singular points of algebraic varieties, such as the cusp y² = x³, the talk explains their algebraic study through quotient rings and the mixed-characteristic setting of integer-coefficient polynomial rings. A central tool is the big Cohen–Macaulay algebra: a typically non-Noetherian ring with strong homological properties whose existence connects to advances in p-adic Hodge theory. The presentation also discusses applications of this singularity theory to birational geometry.

Philip Engel explains a counterexample to the integral Hodge conjecture for very general abelian varieties of dimension at least four. Degenerations of principally polarized abelian varieties are associated with regular matroids, and a new matroid invariant obstructs algebraicity of the minimal curve class on a very general fiber. Combining this obstruction with a theorem of Voisin and the intermediate Jacobian gives stable irrationality of a very general cubic threefold. The work is joint with Olivier de Gaay Fortman and Stefan Schreieder.

Gregorio Baldi surveys recent results on how the Hodge locus is distributed in mixed variations of Hodge structures, including the pure setting. The presentation also examines concrete applications of these results to moduli spaces.

Mikaela Iacobelli introduces kinetic theory and the mathematical description of charged particles in plasmas through Vlasov-type equations. The colloquium examines stability and instability, well-posedness, and how solutions behave in singular limits. It also presents a new family of Wasserstein-type distances that offers tools for studying the stability of kinetic equations. The exposition is intended for a broad mathematical audience.

Egor Shelukhin presents cases of the Hilbert–Smith conjecture for group actions by homeomorphisms with symplectic structure. The results exclude faithful actions of the additive p-adic group in this setting and give additional restrictions on group actions in symplectic topology. The argument combines a new approach to these action problems with power operations in Floer cohomology and quantitative methods in symplectic topology.

James McKernan discusses the log canonical threshold in birational geometry and Mori theory, and its extension to the classification of foliations on algebraic varieties. He introduces a foliated version of the threshold, proposes that it satisfies the ascending chain condition, and verifies this in dimension two. The approach introduces an interpolated canonical divisor and establishes that the minimal model program can be run for these divisors.

Raghu Meka and Zander Kelley examine how large a subset of {1, …, N} must be to contain a three-term arithmetic progression. Writing its size as at least N/C, the talk places the problem between Roth’s classical guarantee at C approximately log log N and Behrend’s progression-free construction at an exponential scale in the square root of log N. It recalls Bloom and Sisask’s 2020 improvement to C = (log N)^(1+c), for some c > 0, before giving an in-depth account of the Kelley–Meka proof reaching C approximately 2^((log N)^0.09), moving closer to Behrend’s scale.

Kannan Soundararajan surveys the distribution of values of the Riemann zeta function and related L-functions. The emphasis is on moments of zeta along the critical line and on central L-values. The talk describes the conjectural moment asymptotics developed over the preceding twenty-five years and advances toward upper and lower bounds across a range of settings.

Sylvia Serfaty studies the many-particle limit of systems driven by Coulomb or Riesz interactions through gradient flows or analogous conservative dynamics, possibly with added random diffusion. The Riesz interaction has inverse-distance exponent s between d−2 and d in dimension d. The talk presents convergence, as the number of particles tends to infinity, to the expected mean-field evolution. It also discusses deriving the Vlasov–Poisson equation from Newtonian dynamics in the monokinetic case and related results for Ginzburg–Landau vortex dynamics.

Michael Chapman examines the gap between the refutation of Connes’ embedding conjecture by MIP*=RE and the construction of non-hyperlinear groups. The talk introduces non-local games, their values and Tsirelson’s problem, then associates a finitely presented *-algebra to a synchronous game. It explains how representations of this algebra relate to different game values, and why the relevant von Neumann closure cannot embed into an ultraproduct of the hyperfinite II₁ factor. A route to a non-hyperlinear group would require an appropriate counterexample algebra to be a complex group ring. Chapman introduces linear constraint system games, their associated game groups, and explains how the MIP*=RE construction differs from such a game.

Thomas Bloom surveys quantitative bounds for sets of integers without three-term arithmetic progressions, beginning with Roth’s zero-density theorem. Work with Olof Sisask establishes that a set with divergent reciprocal sum must contain a three-term progression, resolving the first nontrivial case of Erdős’s conjecture. The proof combines harmonic analysis with elementary combinatorics. The second part develops a structural theorem for additively non-smoothing sets: sets whose first sumset grows, while subsequent addition reveals no additional structure. Building on Bateman and Katz’s cap-set work, Bloom and Sisask extend this kind of structural analysis to arbitrary groups, including the integers. The talk outlines the elementary proof and its potential applications elsewhere in additive combinatorics.

Alexander Lubotzky examines whether infinite groups can be approximated by asymptotic homomorphisms into finite symmetric or unitary groups. He describes finitely presented groups that admit no unitary approximation in the Frobenius norm and several other norms. The argument links vanishing of higher cohomology to stability, then uses Garland’s method and high-dimensional expanders arising from Bruhat–Tits buildings. Certain non-residually finite central extensions of lattices in p-adic Lie groups are stable, preventing such approximations. These extensions come from a p-adic analogue of a result of Deligne. The talk explains the relevant notions and draws on joint work with M. De Chiffre, L. Glebsky, A. Thom and I. Oppenheim.

Aram Harrow connects the Unique Games conjecture, especially small-set expansion, with the quantum separability problem. The Lasserre hierarchy used for Unique Games and the k-extendible relaxation of quantum separability turn out to be closely equivalent algorithmic approaches. The talk examines how quantum-information conjectures could yield quasipolynomial algorithms or hardness results for Unique Games, and discusses promising research directions. It introduces the necessary quantum mechanics, Unique Games, and hierarchy background rather than presupposing it. The presentation draws partly on the papers arXiv:1205.4484 and arXiv:1210.6367.