Prime, Knots and the Adele Class Space

• Alain Connes (Collège de France & IHES)

Room: Centre de Conférences Marilyn et James Simons We show that the scaling site and its periodic orbits of length log p offer a geometric framework for the well-known analogy between primes and knots. The role of the maximal abelian cover of the scaling site is played by the adele class space which is the quotient of adeles by the action of rational numbers by multiplication. The inverse image of the periodic orbit $C_p$ is canonically isomorphic to the mapping torus of the multiplication by the Frobenius at $p$ in the abelianized étale fundamental group of the spectrum of the ring $Z$ localized at $p$, thus exhibiting the linking of p with all other primes. We give a functorial construction of finite covers of the scaling site associated to finite abelian extension of $Q$. These covers share the same ramification as the field extension, and the monodromy of the periodic orbit $C_p$ in the cover corresponds to the Frobenius$(p)$ element of the Galois group. This is joint work with C. Consani.

Inclusive Scattering Matrix

• Albert Schwarz (University of California at Davis & IHES)

Room: Centre de Conférences Marilyn et James Simons Inclusive scattering matrix closely related to inclusive cross-sections is defined in much more general situations than conventional scattering matrix. It contains the same information when the latter is well defined. It seems that the most natural description of scattering in quantum electrodynamics is based on inclusive scattering matrix. I'll discuss the general definition of inclusive scattering matrix in the framework of geometric approach to quantum theory and its expression in terms of generalized Green functions that appear in Keldysh formalism. I'll briefly explain the formalism of $L$-functionals and the definition of inclusive scattering matrix in terms of adiabatic $S$-matrix in this formalism.

Modularity of Donaldson-Thomas Invariants on Calabi-Yau Threefolds

• Boris Pioline (LPTHE - Sorbonne Université)

Room: Centre de Conférences Marilyn et James Simons Donaldson-Thomas invariants are the mathematical incarnation of BPS indices counting black hole micro-states in string compactifications. They are notoriously difficult to compute, and subject to wall-crossing phenomena. String dualities predict that generating series of DT invariants counting D4-D2-D0 black holes should have modular (or more generally mock modular) behavior. For one-parameter CY threefolds such as the quintic, one may compute the first few terms in the generating series using vanishing theorems and wall-crossing formulae,and find a unique modular completion. This in turn allows to predict new Gopakumar-Vafa invariants, and determine the topological string amplitude to higher genus than hitherto possible. Based on work in collaboration with Sergey Alexandrov, Soheyla Feyzbakhsh, Albrecht Klemm and Thorsten Schimmanek.

Quantum Periods for Complements

• Maxim Kontsevich (IHES)

Room: Centre de Conférences Marilyn et James Simons A polynomial P in two variables defines a one-parameter family of spectral curves which are level sets of P, and the corresponding variation of Hodge structures on 1st cohomology groups of these curves. What happens if one quantizes the algebra of polynomials, i.e. deforms it to the Weyl algebra? I'll explain an approach based on second cohomology of complements to the level sets. In particular, one obtains a cohomological description of WKB series for Bohr-Sommerfeld quantization rules. This is joint work with A.Soibelman.

Topological Invariants of Gapped States and ’t Hooft Anomalies

• Anton Kapustin (Caltech)

Room: Centre de Conférences Marilyn et James Simons Recently, an approach to constructing topological invariants of gapped ground-states of lattice systems has been developed in our joint work with N. Sopenko. It applies to arbitrary gapped states of infinite-volume lattice spin systems with rapidly decaying interactions and employs C*-algebraic techniques. In this talk I will explain an interpretation of these invariants as obstructions to gauging, i.e. to promoting a symmetry to a local symmetry. The key observation is that locality on a lattice is an asymptotic notion sensitive only to the large-scale geometry of the support set. Following Kashiwara and Schapira, one can encode locality using a natural Grothendieck topology on a category of semilinear subsets of Eucludean space. Infinitesimal symmetries of a gapped state form a cosheaf over the corresponding site, and the topological invariants are encoded in its Cech complex.