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1. Stable homotopy groups of spheres

   https://doi.org/10.1073/pnas.2012335117  

   Isaksen, Daniel C. ; Wang, Guozhen ; Xu, Zhouli ( September 2020 , Proceedings of the National Academy of Sciences)

   We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a computational method using motivic homotopy theory, viewed as a deformation of classical homotopy theory. This yields a streamlined computation of the first 61 stable homotopy groups and gives information about the stable homotopy groups in dimensions 62 through 90. As an application, we determine the groups of homotopy spheres that classify smooth structures on spheres through dimension 90, except for dimension 4. The method relies more heavily on machine computations than previous methods and is therefore less prone to error. The main mathematical tool is the Adams spectral sequence.
2. On the classification of 5d SCFTs

   https://doi.org/10.1007/JHEP09(2020)007  

   Bhardwaj, Lakshya ( September 2019 , ArXivorg)

   We determine all 5d SCFTs upto rank three by studying RG flows of 5d KK theories. Our analysis reveals the existence of new rank one and rank two 5d SCFTs not captured by previous classifications. In addition to that, we provide for the first time a systematic and conjecturally complete classification of rank three 5d SCFTs. Our methods are based on a recently studied geometric description of 5d KK theories, thus demonstrating the utility of these geometric descriptions. It is straightforward, though computationally intensive, to extend this work and systematically classify 5d SCFTs of higher ranks (greater than or equal to four) by using the geometric description of 5d KK theories.
3. Two Applications of Nilpotent Higgsing in Class-S

   Distler, Jacques ; Elliot, Grant ( March 2022 , ArXivorg)

   We introduce a class of Higgs-branch RG flows in theories of class-S, which flow between d = 4 N = 2 SCFTs of the same ADE type. We discuss two applications of this class of RG flows: 1) determining the current-algebra levels in SCFTs where they were previously unknown — a program we carry out for the class-S theories of type E6 and E7 — and 2) constructing a multitude of examples of pairs of N = 2 SCFTs whose “conventional invariants” coincide. We disprove the conjecture of [1] that the global form of the flavour symmetry group is a reliable diagnostic for determining when two such theories are isomorphic.
4. Full Tilt: Universal Constructors for General Shapes with Uniform External Forces

   https://doi.org/10.1137/1.9781611975482.167  

   Balanza-Martinez, Jose ; Luchsinger, Austin ; Caballero, David ; Reyes, Rene ; Cantu, Angel ; Schweller, Robert ; Garcia, Luis ; Wylie, Tim ( January 2019 , Proceedings of the annual ACM-SIAM Symposium on Discrete Algorithms)

   We investigate the problem of assembling general shapes and patterns in a model in which particles move based on uniform external forces until they encounter an obstacle. In this model, corresponding particles may bond when adjacent with one another. Succinctly, this model considers a 2D grid of “open” and “blocked” spaces, along with a set of slidable polyominoes placed at open locations on the board. The board may be tilted in any of the 4 cardinal directions, causing all slidable polyominoes to move maximally in the specified direction until blocked. By successively applying a sequence of such tilts, along with allowing different polyominoes to stick when adjacent, tilt sequences provide a method to reconfigure an initial board configuration so as to assemble a collection of previous separate polyominoes into a larger shape. While previous work within this model of assembly has focused on designing a specific board configuration for the assembly of a specific given shape, we propose the problem of designing universal configurations that are capable of constructing a large class of shapes and patterns. For these constructions, we present the notions of weak and strong universality which indicate the presence of “excess” polyominoes after the shape is constructed.more »In particular, for given integers h, w, we show that there exists a weakly universal configuration with O(hw) 1 × 1 slidable particles that can be reconfigured to build any h × w patterned rectangle. We then expand this result to show that there exists a weakly universal configuration that can build any h × w-bounded size connected shape. Following these results, which require an admittedly relaxed assembly definition, we go on to show the existence of a strongly universal configuration (no excess particles) which can assemble any shape within a previously studied “drop” class, while using quadratically less space than previous results. Finally, we include a study of the complexity of deciding if a particle within a configuration may be relocated to another position, and deciding if a given configuration may be transformed into a second given configuration. We show both problems to be PSPACE-complete even when no particles stick to one another and movable particles are restricted to 1 × 1 tiles and a single 2 × 2 polyomino.« less
5. Anomaly inflow methods for SCFT constructions in type IIB

   https://doi.org/10.1007/JHEP02(2021)116  

   Bah, Ibrahima ; Bonetti, Federico ; Minasian, Ruben ; Weck, Peter ( February 2021 , Journal of High Energy Physics)

   A bstract We extend the anomaly inflow methods developed in M-theory to SCFTs engineered via D3-branes in type IIB. We show that the ’t Hooft anomalies of such SCFTs can be computed systematically from their geometric definition. Our procedure is tested in several 4d examples and applied to 2d theories obtained by wrapping D3-branes on a Riemann surface. In particular, we show how to analyze half-BPS regular punctures for 4d $$ \mathcal{N} $$ N = 4 SYM on a Riemann surface. We discuss generalizations of this formalism to type IIB configurations with F 3 , H 3 fluxes, as well as to F-theory setups.