Abstract:
Modular forms appear in many facets of mathematics, and have played important roles in geometry, mathematical physics, number theory, representation theory, topology, and other areas. Around 1994, motivated by technical issues in homotopy theory, Mark Mahowald, Haynes Miller and I constructed a topological refinement of modular forms, which we call {\em topological modular forms}. At the Zurich ICM I sketched a program designed to relate topological modular forms to invariants of manifolds, homotopy groups of spheres, and ordinary modular forms. This program has recently been completed and new directions have emerged. In this talk I will describe this recent work and how it informs our understanding of both algebraic topology and modular forms.

Abstract:
In this paper the notion of modular cone metric space is introduced and some properties of such spaces are investigated. Also we define convex modular cone metric which takes values in CR(Y) where Y is a compact Hausdorff space. Then a fixed point theorem is proved for contractions in these spaces. Furthermore, we make a remark on paper [9] and it will be proved that their fixed point result in modular metric spaces is not true.

Abstract:
Molodtsov initiated the concept of soft sets in Molodtsov D. Maji et al. defined some operations on soft sets in Maji P. K., Bismas R., Roy A. R. The concept of soft topological space was introduced by some authors. In this paper, we introduce the concept of the pointwise topology of soft topological spaces and the properties of soft mappings spaces. Finally, we investigate the relationships between some soft mappings spaces.

Abstract:
In this note we consider a question of Ono, concerning which spaces of classical modular forms can be generated by sums of $\eta$-quotients. We give some new examples of spaces of modular forms which can be generated as sums of $\eta$-quotients, and show that we can write all modular forms of level $\Gamma_0(N)$ as rational functions of $\eta$-products.

Abstract:
We construct modular spaces of all 6-dimensional real semisimple Drinfeld doubles, i.e. the sets of all possible decompositions of the Lie algebra of the Drinfeld double into Manin triples. These modular spaces are significantly different from the known one for Abelian Drinfeld double, since some of these Drinfeld doubles allow decomposition into several non-isomorphic Manin triples and their modular spaces are therefore written as unions of homogeneous spaces of different dimension. Implications for Poisson-Lie T-duality and especially Poisson-Lie T-plurality are mentioned.

Abstract:
We present two models for the space of knots which have endpoints at fixed boundary points in a manifold with boundary, one model defined as an inverse limit of spaces of maps between configuration spaces and another which is cosimplicial. These models build on the calculus of isotopy functors and are weakly homotopy equivalent to knot spaces when the ambient dimension is greater than three. The mapping space model, and the evaluation map on which it builds, is suitable for analysis through differential topology. The cosimplicial model gives rise to spectral sequences which converge to cohomology and homotopy groups of spaces of knots when they are connected. We explicitly identify and establish vanishing lines in these spectral sequences.

Abstract:
We extend the structure of string topology from mapping spaces to embedding spaces $Emb(S^n,M)$. This extension comes from an action of the cleavage operad, a coloured $E_{n+1}$-operad. For all values of $n \in \mathbb{N}$, this gives an action of certain Thom spectra that provides an action on the level homology. We provide an analysis of the string topology structure derived from the embedding $Emb(S^n,M) \hookrightarrow Map(S^n,M)$. In the $1$-dimensional case it specializes a morphism of BV-algebras $\mathbb{H}_*(Emb(S^1,M)) \to \mathbb{H}_*(Map(S^1,M)$.

Abstract:
We show, using a theorem of Milnor and Margulis, that string theory on compact negatively curved spaces grows new effective dimensions as the space shrinks, generalizing and contextualizing the results in hep-th/0510044. Milnor's theorem relates negative sectional curvature on a compact Riemannian manifold to exponential growth of its fundamental group, which translates in string theory to a higher effective central charge arising from winding strings. This exponential density of winding modes is related by modular invariance to the infrared small perturbation spectrum. Using self-consistent approximations valid at large radius, we analyze this correspondence explicitly in a broad set of time-dependent solutions, finding precise agreement between the effective central charge and the corresponding infrared small perturbation spectrum. This indicates a basic relation between geometry, topology, and dimensionality in string theory.

Abstract:
Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.