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We'll begin with a gentle overview, discussing some of the background context. This will include some words on Monstrous Moonshine, modular functions, boundary conformal field theory, and string theory. Then we'll turn to fusion rings and the basic partition functions of the theory, where much of the data of the theory is conveniently encoded. We'll focus on the partition function of the torus relevant to closed string theory, or the CFT of the bulk and that of the cylinder which is relevant to open string theory and boundary CFT.

References: Review and Lecture Notes: T. We review the topological calculations in supersymmetric field theories in 0,1 and 2 dimensions, with special emphasis on the non-linear s model case. Then we define the A-model on general symplectic manifolds and the B-model on Calabi-Yau manifolds. We describe selected results in mirror symmetry on CY manifolds, such as the mirror symmetry predictions for the Gromow-Witten invariants and the definition of integer invariants for closed an open topological strings in non-compact Calabi-Yau backgrounds with branes.

Finally we describe aspects of the large N relation between topological gauge systems and the topological string and review what these theories calculate in the topological sector of 4d supersymmetric theories. The goal of these lectures is to give an introduction to the relations between enumerative geometry of non-compact, toric Calabi-Yau manifolds, and Chern-Simons theory. I will start with an introduction to the relevant aspects of Chern-Simons theory and its large N expansion. Then, I will explain the realization of Chern-Simons theory on the three-sphere in terms of closed and open topological strings, as well as the important idea of large N or geometric transition.

Finally, I will extend this framework in order to compute topological string amplitudes of general toric manifolds using Chern-Simons ingredients. References Lecture Notes: Hand written notes for the course.

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Gopakumar and C. Aganagic, M. In these talks I will describe mathematical models of D-branes as sheaves and, more generally, derived categories. I will begin with a gentle introduction to some relevant mathematics sheaves, Ext groups. After that, I will discuss mathematical models of D-branes on large-radius Calabi-Yau manifolds as sheaves, describing how sheaves can be used to calculate open string spectra, and how such sheaf-theoretic models can be extended in various directions to take into account B field backgrounds, orbifold structures, and nontrivial Higgs vevs.

Finally, I will give a short introduction to derived categories, their physical realization as boundary states in the B model topological field theory, and stability issues. References: Lecture Notes: E. Basic references on Ext groups of sheaves: Griffiths and Harris, same text as above, sections 5. Hermitian-holomorphic classes and tame symbols related to uniformization, the dilogarithm, and the Liouville Action.

One possible approach to the uniformization of compact Riemann surfaces of genus greater than on is to look at metrics of constant negative curvature. Such metrics can be characterized as critical points of the classical Liouville action functional. We present an algebraic construction of this functional as the square of the metrized holomorphic tangent bundle in a suitably defined hermitian-holomorphic Deligne cohomology group. For a pair of line bundles on the Riemann surface, this construction generalizes Deligne's tame symbol and recovers the algebraic approach to the determinant of cohomology as pursued by Deligne, Brylinski, Gabber and others.

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We will also interpret the above results in terms of group cohomology for Kleinian groups and volume calculations in hyperbolic 3-space involving certain secondary classes, notably the Chern-Simons one. We will outline how this framework relates to recent results in Mathematics and Physics where a compact Riemann Surface is considered as the "hologram" of an associated hyperbolic 3-manifold.

References: Ettore Aldrovandi: On hermitian-holomorphic classes related to uniformization, the dilogarithm, and the Liouville action, math. Ettore Aldrovandi, Leon A. Homological aspects, Comm. Leon A. Takhtajan and Lee-Peng Teo: Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography, math. Jean-Luc Brylinski: Geometric construction of Quillen line bundles. Hautes Etudes Sci. Yuri I. Manin, Matilde Marcolli: Holography principle and arithmetic of algebraic curves. Study of Hurwitz stacks parametrizing branched covers of algebraic curves, and some aspects to Gromov Witten invariants.

Let M be an oriented n -dimensional manifold. We study the causal relations between the wave fronts W 1 and W 2 that originated at some points of M. We introduce a numerical topological invariant CR W 1 , W 2 the so-called causality relation invariant that, in particular, gives the algebraic number of times the wave front W 1 passed through the point that was the source of W 2 before the front W 2 originated.

This invariant can be easily calculated from the current picture of wave fronts on M without the knowledge of the propagation law for the wave fronts. Moreover, in fact we even do not need to know the topology of M outside of a part bar- M of M such that W 1 and W 2 are null-homotopic in bar- M. We also construct the Affine winding number invariant win which is the generalization of the winding number to the case of nonzero-homologous shapes and manifolds other than R 2.

The win invariant gives the algebraic number of times the wave front has passed through a given point between two different time moments without the knowledge of the wave front propagation law. The invariants described above are particular cases of the general affine linking invariant al of nonzero homologous submanifolds N 1 and N 2 in M introduced by us. To construct al we introduce a new pairing on the bordism groups of space of mappings of N 1 and N 2 into M. Duality between commutative and non-commutative gauge fields is encoded by the Seiberg-Witten equations [1], which have been solved in the U 1 case [2,3,4,5].

The solution leads to predictions for derivative corrections to the gauge sector of open-string effective actions, in presence of a large background magnetic field [6]. These predictions have been checked by explicit string-computations [7,8]. Going beyond the limit of a large background magnetic field involves taking the open-string metric into account, thus deforming the corrections [9,10].

The "combinatorial triangulation conjecture" stated that the first case could not occur, for M compact. Siebenmann's example showed the third case is also possible. From Wikipedia, the free encyclopedia. Mathematics timeline. The Real Projective Plane. Retrieved 16 January Topics on Riemann Surfaces and Fuchsian Groups. Cambridge University Press.

Retrieved 17 January History of Topology. Retrieved 30 June Retrieved 6 January A History of Algebraic and Differential Topology, - Retrieved 4 January Hamiltonian Submanifolds of Regular Polytopes. Logos Verlag Berlin GmbH. Retrieved 15 June Thibaut - Zycha. Walter de Gruyter. Elsevier Science. Elements of the History of Mathematics. Intersection Theory. Differential Topology.

Topology and Its Applications. Retrieved 1 January Canadian Mathematical Bulletin. Canadian Mathematical Society. Retrieved 6 July Homotopy, Homology, and Manifolds. American Mathematical Soc. Retrieved 2 July Instantons and Four-Manifolds. World Scientific. Retrieved 7 July Categories : Manifolds Historical timelines. Namespaces Article Talk. Views Read Edit View history. Languages Add links. By using this site, you agree to the Terms of Use and Privacy Policy.

Euler's theorem on polyhedra "triangulating" the 2-sphere. The subdivision of a convex polygon with n sides into n triangles, by means of any internal point, adds n edges, one vertex and n - 1 faces, preserving the result. So the case of triangulations proper implies the general result. Develops non-Euclidean geometry , in particular the hyperbolic plane.

Reconstructs real projective geometry , including the real projective plane. Geometric properties of the complex projective plane. Gauss—Bonnet theorem for the differential geometry of closed surfaces. Introduction of the Riemann surface into the theory of analytic continuation. Riemannian metrics give an idea of intrinsic geometry of manifolds of any dimension. The Lie group concept is developed, using local formulae. Klein's Erlangen program puts an emphasis on the homogeneous spaces for the classical groups , as a class of manifolds foundational for geometry.

Charles Rezk's papers and preprints.

Dini develops the implicit function theorem , the basic tool for constructing manifolds locally as the zero sets of smooth functions. Formulation of Hamiltonian mechanics in terms of the cotangent bundle of a manifold, the configuration space. Fundamental group of a topological space. Fundamental work Analysis situs , the beginning of algebraic topology. Hilbert's fifth problem posed the question of characterising Lie groups among transformation groups , an issue partially resolved in the s. Hilbert's fifteenth problem required a rigorous approach to the Schubert calculus , a branch of intersection theory taking place on the complex Grassmannian manifolds.

Tentative axiomatisation topological spaces are not yet defined of two-dimensional manifolds. As a conjecture, the Dehn-Somerville equations relating numerically triangulated manifolds and simplicial polytopes. The uniformization theorem for simply connected Riemann surfaces. Survey article Analysis Situs in Klein's encyclopedia gives the first proof of the classification of surfaces, conditional on the existence of a triangulation, and lays the foundations of combinatorial topology.

Heinrich Franz Friedrich Tietze. Habilitationschrift for the University of Vienna, proposes another tentative definition, by combinatorial means, of "topological manifold". The Hauptvermutung , a conjecture on the existence of a common refinement of two triangulations. This was an open problem, for manifolds, to More precisely, we show the stronger statement that every symmetric monoidal left adjoint functor between presentably symmetric monoidal infinity-categories is represented by a strong symmetric monoidal left Quillen functor between simplicial, combinatorial and left proper symmetric monoidal model categories.

Abstract: We define and discuss lax and weighted colimits of diagrams in infinity-categories and show that the coCartesian fibration associated to a functor is given by its lax colimit. A key ingredient, of independent interest, is a simple characterization of the free Cartesian fibration associated to a a functor of infinity-categories.

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As an application of these results, we prove that lax representable functors are preserved under exponentiation, and also that the total space of a presentable Cartesian fibration between infinity-categories is presentable, generalizing a theorem of Makkai and Pare to the infinity-categorical setting. Lastly, in the appendix, we observe that pseudofunctors between 2,1 -categories give rise to functors between infinity-categories via the Duskin nerve. Abstract: The main goal of the present paper is the construction of twisted generalized differential cohomology theories and the comprehensive statement of its basic functorial properties.

Technically it combines the homotopy theoretic approach to untwisted generalized differential cohomology developed by Hopkins-Singer and later by the first author and D. Gepner with the oo-categorical treatement of twisted cohomology by Ando-Blumberg-Gepner. We introduce the notion of a differential twist for a given generalized cohomology theory and construct twisted differential cohomology groups resp. The main technical results of the paper are existence and uniqueness statements for differential twists. These results will be applied in a variety of examples, including K-theory, topological modular forms and other cohomology theories.

In: Algebr. Abstract: We establish a canonical and unique tensor product for commutative monoids and groups in an infinity-category C which generalizes the ordinary tensor product of abelian groups. In the case that C is the infinity-category of spaces this produces a multiplicative infinite loop space machine which can be applied to the algebraic K-theory of rings and ring spectra.

The main tool we use to establish these results is the theory of smashing localizations of presentable infinity-categories. In particular, we identify preadditive and additive infinity-categories as the local objects for certain smashing localizations. Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in infinity-categories.

In: Rev.

We observe that such a reduction is exactly the additional datum needed for the construction of a T-dual pair. We illustrate the theory by working out the example of the canonical lifting gerbe on a compact Lie group which is a torus bundles over the associated flag manifold. It was a recent observation of Daenzer and van Erp arXiv In: J. Homotopy Relat. Abstract: We discuss two aspects of the presentation of the theory of principal infinity-bundles in an infinity-topos, introduced in [NSSa], in terms of categories of simplicial pre sheaves.

First we show that over a cohesive site C and for G a presheaf of simplicial groups which is C-acyclic, G-principal infinity-bundles over any object in the infinity-topos over C are classified by hyper-Cech-cohomology with coefficients in G. Then we show that over a site C with enough points, principal infinity-bundles in the infinity-topos are presented by ordinary simplicial bundles in the sheaf topos that satisfy principality by stalkwise weak equivalences. Finally we discuss explicit details of these presentations for the discrete site in discrete infinity-groupoids and the smooth site in smooth infinity-groupoids, generalizing Lie groupoids and differentiable stacks.

In the companion article [NSSc] we use these presentations for constructing classes of examples of twisted principal infinity-bundles and for the discussion of various applications. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fiber bundles in terms of associated bundles. The induced associated infinity-bundles subsume the notions of gerbes and higher gerbes in the literature. We discuss here this general theory of principal infinity-bundles, intimately related to the axioms of Giraud, Toen-Vezzosi, Rezk and Lurie that characterize infinity-toposes.

We show a natural equivalence between principal infinity-bundles and intrinsic nonabelian cocycles, implying the classification of principal infinity-bundles by nonabelian sheaf hyper-cohomology. We observe that the theory of geometric fiber infinity-bundles associated to principal infinity-bundles subsumes a theory of infinity-gerbes and of twisted infinity-bundles, with twists deriving from local coefficient infinity-bundles, which we define, relate to extensions of principal infinity-bundles and show to be classified by a corresponding notion of twisted cohomology, identified with the cohomology of a corresponding slice infinity-topos.

In a companion article [NSSb] we discuss explicit presentations of this theory in categories of simplicial pre sheaves by hyper-Cech cohomology and by simplicial weakly-principal bundles; and in [NSSc] we discuss various examples and applications of the theory. K-Theory Abstract: The theory of dendroidal sets has been developed to serve as a combinatorial model for homotopy coherent operads by Moerdijk and Weiss.

An infinity-operad is a dendroidal set D satisfying certain lifting conditions. These groups generalize the K-theory of symmetric monoidal resp. We establish some useful properties like invariance under the appropriate equivalences and long exact sequences which allow us to compute these groups in some examples. Abstract: We show that every sheaf on the site of smooth manifolds with values in a stable infinity,1 -category like spectra or chain complexes gives rise to a differential cohomology diagram and a homotopy formula, which are common features of all classical examples of differential cohomology theories.

These structures are naturally derived from a canonical decomposition of a sheaf into a homotopy invariant part and a piece which has a trivial evaluation on a point.