Advances in R-matrices and their applications (after Maulik-Okounkov, Kang-Kashiwara-Kim-Oh,...)

R-matrices are the solutions of the Yang-Baxter equation. At the origin of the quantum group theory, they may be interpreted as intertwining operators. Recent advances have been made independently in different directions. Maulik-Okounkov have given a geometric approach to R-matrices with new tools in symplectic geometry, the stable envelopes. Kang-Kashiwara-Kim-Oh proved a conjecture on the categorification of cluster algebras by using R-matrices in a crucial way. Eventually, a better understanding of the action of transfer-matrices obtained from R-matrices led to the proof of several conjectures about the corresponding quantum integrable systems.Comment: This is an English translation of the Bourbaki seminar 1129 (March 2017). The French version will appear in Ast\'erisqu

Elliptic Determinantal Process of Type A

We introduce an elliptic extension of Dyson's Brownian motion model, which is a temporally inhomogeneous diffusion process of noncolliding particles defined on a circle. Using elliptic determinant evaluations related to the reduced affine root system of types $A$, we give determinantal martingale representation (DMR) for the process, when it is started at the configuration with equidistant spacing on the circle. DMR proves that the process is determinantal and the spatio-temporal correlation kernel is determined. By taking temporally homogeneous limits of the present elliptic determinantal process, trigonometric and hyperbolic versions of noncolliding diffusion processes are studied.Comment: v5: AMS-LaTeX, 35 pages, no figure, references updated for publication in Probab. Theory Relat. Field

Number Theory, Analysis and Geometry: In Memory of Serge Lang

Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future. In the spirit of Lang’s vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas, namely number theory, analysis and geometry, representing Lang’s own breadth of interests. A special introduction by John Tate includes a brief and engaging account of Serge Lang’s life

Spin/Pin-Structures and Real Enumerative Geometry

The present, partly expository, monograph consists of three parts. The first part treats Spin- and Pin-structures from three different perspectives and shows them to be suitably equivalent. It also introduces an intrinsic perspective on the relative Spin- and Pin-structures of Fukaya-Oh-Ohta-Ono and Solomon, establishes properties of these structures in both perspectives, and again shows them to be suitably equivalent. The second part uses the intrinsic perspective on (relative) Spin- and Pin-structures to detail constructions of orientations on the determinants of real Cauchy-Riemann operators and study their properties. The final part applies the results of the first two parts to the enumerative geometry of real curves and obtains an explicit comparison between the curve signs in the intrinsic definition of Welschinger and later Pin-structure dependent definitions. This comparison makes use of both the classical and instrinisc perspectives on Pin-structures and thus of the equivalence between them established in this monograph. The preface and the introductions to the three parts describe the present work in more detail.Comment: 280 pages; Chapter 3 re-organized and expanded to symplectic sixfold

Invariant Source Forms, Conservation Laws, and the Inverse Problem of the Calculus of Variations

We prove that under certain assumptions a partial differential equation (PDE) is necessarily variational, i.e. that it can be written as the Euler-Lagrange equation for some Lagrangian. Instead of investigating the PDE directly, we consider the so-called source form which is the natural object in this problem. We show that if a source form satisfies certain symmetries and corresponding conservation laws then it must necessarily be variational. Conservation laws describe important properties of physical differential equations, like the conservation of energy, momentum, angular-momentum, mass, charge and so on. It is well-known from Noether's first and second theorems that symmetries of a variational functional lead to conservation laws. In some sense, we want to reverse this statement and prove that symmetries and corresponding conservation laws lead to a variational functional

Streaming SPHINCS+ for Embedded Devices Using the Example of TPMs

We present an implementation of the hash-based post-quantum signature scheme SPHINCS + that enables heavily memory-restricted devices to sign messages by streaming-out a signature during its computation and to verify messages by streaming-in a signature. We demonstrate our implementation in the context of Trusted Platform Modules (TPMs) by proposing a SPHINCS + integration and a streaming extension for the TPM specification. We evaluate the overhead of our signature-streaming approach for a stand-alone SPHINCS + implementation and for its integration in a proof-of-concept TPM with the proposed streaming extension running on an ARM Cortex-M4 platform. Our streaming interface greatly reduces the memory requirements without introducing a performance penalty. This is achieved not only by removing the need to store an entire signature but also by reducing the stack requirements of the key generation, sign, and verify operations. Therefore, our streaming interface enables small embedded devices that do not have sufficient memory to store an entire SPHINCS + signature or that previously were only able to use a parameter set that results in smaller signatures to sign and verify messages using all SPHINCS + variants.</p