Embeddability of graphs into the Klein surface

Flötotto A. Embeddability of graphs into the Klein surface. Bielefeld (Germany): Bielefeld University; 2010

Structural and Topological Graph Theory and Well-Quasi-Ordering

Στη σειρά εργασιών Ελασσόνων Γραφημάτων, οι Neil Robertson και Paul Seymour μεταξύ άλλων σπουδαίων αποτελεσμάτων, απέδειξαν την εικασία του Wagner που σήμερα είναι γνωστή ως το Θεώρημα των Robertson και Seymour. Σε κάθε τους βήμα προς την συναγωγή της τελικής απόδειξης της εικασίας, κάθε ειδική περίπτωση αυτής που αποδείκνυαν ήταν συνέπεια ενός "δομικού θεωρήματος" το οποίο σε γενικές γραμμές ισχυριζόταν ότι ικανοποιητικά γενικά γραφήματα περιέχουν ως ελάσσονα γραφήματα ή άλλες δομές που είναι χρήσιμα για την απόδειξη, ή ισοδύναμα, ότι η δομή των γραφημάτων τα οποία δεν περιέχουν ένα χρήσιμο για την απόδειξη γράφημα ως έλασσον είναι κατά κάποιο τρόπο περιορισμένη συνάγοντας έτσι και πάλι μια χρήσιμη πληροφορία για την απόδειξη. Στην παρούσα εργασία, παρουσιάζουμε -σχετικά μικρές- αποδείξεις διαφόρων ειδικών περιπτώσεων του Θεωρήματος των Robertson και Seymour, αναδεικνύοντας με αυτό τον τρόπο την αλληλεπίδραση της δομικής θεωρίας γραφημάτων με την θεωρία των καλών-σχεδόν-διατάξεων. Παρουσιάζουμε ακόμα την ίσως πιο ενδιαφέρουσα ειδική περίπτωση του Θεωρήματος των Robertson και Seymour, η οποία ισχυρίζεται ότι η εμβαπτισιμότητα σε κάθε συγκεκριμένη επιφάνεια δύναται να χαρακτηριστεί μέσω της απαγόρευσης πεπερασμένων το πλήθος γραφημάτων ως ελάσσονα. Το τελευταίο αποτέλεσμα συνάγεται ως ένα αποτέλεσμα της θεωρίας των καλών-σχεδόν-διατάξεων αναδεικνύοντας με αυτό τον τρόπο την αλληλεπίδρασή της με την τοπολογική θεωρία γραφημάτων. Τέλος, σταχυολογούμε αποτελέσματα αναφορικά με την καλή-σχεδόν-διάταξη κλάσεων γραφημάτων από άλλες -πέραν της σχέσης έλασσον- σχέσεις γραφημάτων.In their Graph Minors series, Neil Robertson and Paul Seymour among other great results proved Wagner's conjecture which is today known as the Robertson and Seymour's theorem. In every step along their way to the final proof, each special case of the conjecture which they were proving was a consequence of a "structure theorem", that sufficiently general graphs contain minors or other sub-objects that are useful for the proof - or equivalently, that graphs that do not contain a useful minor have a certain restricted structure, deducing that way also a useful information for the proof. The main object of this thesis is the presentation of -relatively short- proofs of several Robertson and Seymour's theorem's special cases, illustrating by this way the interplay between structural graph theory and graphs' well-quasi-ordering. We present also the proof of the perhaps most important special case of the Robertson and Seymour's theorem which states that embeddability in any fixed surface can be characterized by forbidding finitely many minors. The later result is deduced as a well-quasi-ordering result, indicating by this way the interplay among topological graph theory and well-quasi-ordering theory. Finally, we survey results regarding the well-quasi-ordering of graphs by other than the minor graphs' relations

Cluster algebras and tilings for the m=4 amplituhedron

The amplituhedron $A_{n,k,m}(Z)$ is the image of the positive Grassmannian $Gr_{k,n}^{\geq 0}$ under the amplituhedron map $Gr_{k,n}^{\geq 0} \to Gr_{k,k+m}$ induced by a positive linear map $Z:\mathbb{R}^n \to \mathbb{R}^{k+m}$. It was originally introduced in physics in order to give a geometric interpretation of scattering amplitudes. More specifically, one can compute scattering amplitudes in $N=4$ SYM by decomposing the amplituhedron into 'tiles' (closures of images of $4k$-dimensional cells of $Gr_{k,n}^{\geq 0}$ on which the amplituhedron map is injective) and summing up the 'volumes' of the tiles. Such a decomposition into tiles is called a tiling. In this article we deepen our understanding of tiles and tilings of the $m=4$ amplituhedron. We prove the cluster adjacency conjecture for BCFW tiles of $A_{n,k,4}(Z)$, which says that facets of BCFW tiles are cut out by collections of compatible cluster variables for $Gr_{4,n}$. We also give an explicit description of each BCFW tile as the subset of $Gr_{k, k+4}$ where certain cluster variables have particular signs. And we prove the BCFW tiling conjecture, which says that any way of iterating the BCFW recurrence gives rise to a tiling of the amplituhedron $A_{n,k,4}(Z)$. Along the way we construct many explicit seeds for the $Gr_{4,n}$ comprised of high-degree cluster variables, which may be of independent interest in the study of cluster algebras

Argonne National Laboratory Report ANL-78-70

Report on electrochemical energy development, including development of advanced, high-temperature lithium/metal sulfide batteries for vehicle propulsion and stationary energy storage

Defining Gravity: Effective Field Theory, Entanglement, and Cosmology

Many of the most exciting open problems in high-energy physics are related to the behavior and ultimate nature of gravity and spacetime. In this dissertation, several categories of new results in quantum and classical gravity are presented, with applications to our understanding of both quantum field theory and cosmology. A fundamental open question in quantum field theory is related to ultraviolet completion: Which low-energy effective field theories can be consistently combined with quantum gravity? A celebrated example of the swampland program---the investigation of this question---is the weak gravity conjecture, which mandates, for a U(1) gauge field coupled consistently to gravity, the existence of a state with charge-to-mass ratio greater than unity. In this thesis, we demonstrate the tension between the weak gravity conjecture and the naturalness principle in quantum field theory, generalize the weak gravity conjecture to multiple gauge fields, and exhibit a model in which the weak gravity conjecture solves the standard model hierarchy problem. Next, we demonstrate that gravitational effective field theories can be constrained by infrared physics principles alone, namely, analyticity, unitarity, and causality. In particular, we derive bounds related to the weak gravity conjecture by placing such infrared constraints on higher-dimension operators in a photon-graviton effective theory. We furthermore place bounds on higher-curvature corrections to the Einstein equations, first using analyticity of graviton scattering amplitudes and later using unitarity of an arbitrary tree-level completion, as well as constrain the couplings in models of massive gravity. Completing our treatment of perturbative quantum gravity, outside of the swampland program, we also reformulate graviton perturbation theory itself, finding a field redefinition and gauge-fixing of the Einstein-Hilbert action that drastically simplifies the Feynman diagram expansion. Furthermore, our reformulation also exhibits a hidden symmetry of general relativity that corresponds to the double copy relations equating gravity amplitudes to sums of squares of gluon amplitudes in Yang-Mills theory, a surprising correspondence that yields insights into the structure of quantum field theories. Moving beyond perturbation theory into nonperturbative questions in quantum gravity, we consider the deep relation between spacetime geometry and properties of the quantum state. In the context of holography and the anti-de Sitter/conformal field theory correspondence, we test the proposed ER=EPR correspondence equating quantum entanglement with wormholes in spacetime. In particular, we demonstrate that the no-cloning theorem in quantum mechanics and the no-go theorem for topology change of spacetime are dual under the ER=EPR correspondence. Furthermore, we prove that the presence of a wormhole is not an observable in quantum gravity, rescuing ER=EPR from potential violation of linearity of quantum mechanics. Excitingly, we also prove a new area theorem within classical general relativity for arbitrary dynamics of two collections of wormholes and black holes; this area theorem is the ER=EPR analogue of entanglement conservation. We next turn our attention to the emergence of spacetime itself, placing consistency conditions on the proposed correspondence between anti-de Sitter space and the Multiscale Entanglement Renormalization Ansatz, a special tensor network that constitutes a computational tool for finding the ground state of certain quantum systems. Further examining the role of quantum entanglement entropy in the emergence of general relativity, we ask whether there is a consistent microscopic formulation of the entropy in theories of entropic gravity; we find that our results weaken equation-of-state proposals for entropic gravity while strengthening those more akin to holography, guiding future investigation of theories of emergent gravity. Finally, we examine the consequences of the Hamiltonian constraint in classical gravity for the early universe. The Hamiltonian constraint allows for the Liouville measure on the phase space of cosmological parameters for homogeneous, isotropic universes to be converted into a probability distribution on trajectories, or equivalently, on initial conditions. However, this measure diverges on the set of spacetimes that are spatially flat, like the observable universe. In this thesis, we derive the unique, classical, Hamiltonian-conserved measure for the subset of flat universes. This result allows for distinction between different models of cosmic inflation with similar observable predictions; for example, we find that the measure favors models of large-scale inflation, as such potentials more naturally produce the number of e-folds necessary to match cosmological observations.</p