Electric-Magnetic Duality And The Geometric Langlands Program

The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions. The key ingredients are electric-magnetic duality of gauge theory, mirror symmetry of sigma-models, branes, Wilson and 't Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlands program, such as Hecke eigensheaves and D-modules, arise naturally from the physics.Comment: 225 pp; further clarification

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

On the elimination of inessential points in the smallest enclosing ball problem

International audienceWe consider the construction of the smallest ball B * enclosing a set Xn formed by n points in R d. We show that any probability measure on Xn, with mean c and variance matrix V , provides a lower bound b on the distance to c of any point on the boundary of B * , with b having a simple expression in terms of c and V. This inequality permits to remove inessential points from Xn, which do not participate to the definition of B * , and can be used to accelerate algorithms for the construction of B *. We show that this inequality is, in some sense, the best possible. A series of numerical examples indicates that, when d is reasonable small (d ≤ 10, say) and n is large (up to 10 5), the elimination of inessential points by a suitable two-point measure, followed by a direct (exact) solution by quadratic programming, outperforms iterative methods that compute an approximate solution by solving the dual problem

Mediation, reality and reason: an examination of Hegel's phenomenology of spirit

This thesis examines Hegel's attempt to mediate the opposition of subect and object. It is an explication and an interpretation of "Consciousness", "Self-Consciousness" and "Reason". Hegel attempted to develop a system of philosophy whose conclusion was demonstrated to be true and one in which all that falls within experience became rationally comprehended. Since, according to Hegel, consciousness contains within it the two elements "subject" and "object" he analyzed the experience of consciousness in its relation to these modes. The procedure of the Phenomenology is to examine the claims of objectivity and those of subjectivity to be the essence of the true. Hegel shows that complete philosophical knowledge requires both sides to be equally essential. Part II discusses the unification of subject and object through the examination of knowledge and shows that this examination must also be of the object which is known. Therefore, in Hegel's theory, a true epistemology must also be an ontology. His conclusion is that non-sceptical knowledge is possible and that it is co-extensive with the actual. Part III shows that Hegel's conclusion is possible because Reason is what is real. So his rationalism is a metaphysical claim. This distinguishes it from other forms of rationalism and it is therefore immune to usual criticisms. However, his position requires a rational necessity in the world and, since contingency is an element of experience, he failed to give the complete account of experience he himself demanded

Fundamental Theorem of Algebra: A Survey of History and Proofs

Higher Educatio

Computational methods in protein structure comparison and analysis of protein interaction networks

Proteins are versatile biological macromolecules that perform numerous functions in a living organism. For example, proteins catalyze chemical reactions, store and transport various small molecules, and are involved in transmitting nerve signals. As the number of completely sequenced genomes grows, we are faced with the important but daunting task of assigning function to proteins encoded by newly sequenced genomes. In this thesis we contribute to this effort by developing computational methods for which one use is to facilitate protein function assignment. Functional annotation of a newly discovered protein can often be transferred from that of evolutionarily related proteins of known function. However, distantly related proteins can still only be detected by the most accurate protein structure alignment methods. As these methods are computationally expensive, they are combined with less accurate but fast methods to allow large-scale comparative studies. In this thesis we propose a general framework to define a family of protein structure comparison methods that reduce protein structure comparison to distance computation between high-dimensional vectors and therefore are extremely fast. Interactions among proteins can be detected through the use of several mature experimental techniques. These interactions are routinely represented by a graph, called a protein interaction network, with nodes representing the proteins and edges representing the interactions between the proteins. In this thesis we present two computational studies that explore the connection between the topology of protein interaction networks and protein biological function. Unfortunately, protein interaction networks do not explicitly capture an important aspect of protein interactions, their dynamic nature. In this thesis, we present an automatic method that relies on graph theoretic tools for chordal and cograph graph families to extract dynamic properties of protein interactions from the network topology. An intriguing question in the analysis of biological networks is whether biological characteristics of a protein, such as essentiality, can be explained by its placement in the network. In this thesis we analyze protein interaction networks for Saccharomyces cerevisiae to identify the main topological determinant of essentiality and to provide a biological explanation for the connection between the network topology and essentiality