Abstract:
Let $R$ be an $E_\infty$-ring spectrum. Given a map $\zeta$ from a space $X$ to $BGL_1R$, one can construct a Thom spectrum, $X^\zeta$, which generalises the classical notion of Thom spectrum for spherical fibrations in the case $R=S^0$, the sphere spectrum. If $X$ is a loop space ($\simeq \Omega Y$) and $\zeta$ is homotopy equivalent to $\Omega f$ for a map $f$ from $Y$ to $B^2GL_1R$, then the Thom spectrum has an $A_\infty$-ring structure. The Topological Hochschild Homology of these $A_\infty$-ring spectra is equivalent to the Thom spectrum of a map out of the free loop space of $Y$. This paper considers the case $X=S^1$, $R=K_p^\wedge$, the p-adic $K$-theory spectrum, and $\zeta = 1-p \in \pi_1BGL_1K_p^\wedge$. The associated Thom spectrum $(S^1)^\zeta$ is equivalent to the mod p $K$-theory spectrum $K/p$. The map $\zeta$ is homotopy equivalent to a loop map, so the Thom spectrum has an $A_\infty$-ring structure. I will compute $\pi_*THH^{K_p^\wedge}(K/p)$ using its description as a Thom spectrum.

Abstract:
Given a simply connected, closed four manifold, we associate to it a simply connected, closed, spin five manifold. This leads to several consequences : the stable and unstable homotopy groups of such a four manifold is determined by its second Betti number, and the ranks of the homotopy groups can be explicitly calculated. We show that for a generic metric on such a smooth four manifold with second Betti number at least three, the number of geometrically distinct periodic geodesics of length at most l grow exponentially as a function of l. The number of closed Reeb orbits of length at most l on the spherization of the cotangent bundle also grow exponentially for any Reeb flow.

Abstract:
In this paper we give a formula for the homotopy groups of $(n-1)$-connected $2n$-manifolds as a direct sum of homotopy groups of spheres in the case the $n^{th}$ Betti number is larger than $1$. We demonstrate that when the $n^{th}$ Betti number is $1$ the homotopy groups might not have such a decomposition. The techniques used in this computation also yield formulae for homotopy groups of connected sums of sphere products and CW complexes of a similar type. In all the families of spaces considered here, we establish a conjecture of J. C. Moore.

Abstract:
We compute the cohomology of the right generalised projective Stiefel manifolds and use it to find bounds on the rank of the complementary bundle for certain vector bundles. Further the cohomology computations are also used to find bounds on the span and on the immersibility of the manifolds in certain cases.

Abstract:
For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension 4n+1, these groups are related to computations in stable cohomotopy. Using stable homotopy theory, we make explicit computations to show that the inertia group is non-trivial in many cases. In complex dimension 9, we deduce some results on geometric structures on homotopy complex projective spaces and complex hyperbolic manifolds.

Abstract:
Using equivariant obstruction theory we construct equivariant maps from certain classifying spaces to representation spheres for cyclic groups and dihedral groups. Restricting them to finite skeleta constructs equivariant maps between spaces which are related to the topological Tverberg conjecture. This answers negatively a question of \"Ozaydin posed in relation to weaker versions of the same conjecture. Further, it also has consequences for Borsuk-Ulam properties of representations of cyclic and dihedral groups.

Abstract:
Let $\widetilde{I}_{2n,k}$ denote the space of $k$-dimensional, oriented isotropic subspaces of $\mathbb{R}^{2n}$, called the oriented isotropic Grassmannian. Let $f \colon \widetilde{I}_{2n,k} \rightarrow \widetilde{I}_{2m,l} $ be a map between two oriented isotropic Grassmannians of the same dimension, where $k,l \geq 2$. We show that either $(n,k) = (m,l)$ or the degree of $f$ must be zero. Let $\mathbb{R}\widetilde{G}_{m,l}$ denote the oriented real Grassmann manifold. For $k,l \geq 2$ and $\dim{\widetilde{I}_{2n,k}} = \dim{\mathbb{R}\widetilde{G}_{m,l}}$, we also show that the degree of maps $g \colon \mathbb{R} \widetilde{G}_{m,l} \rightarrow \widetilde{I}_{2n,k} $ and $h \colon \widetilde{I}_{2n,k} \rightarrow \mathbb{R} \widetilde{G}_{m,l}$ must be zero.

Abstract:
For a discrete group G, we represent the Bredon cohomology with local coefficients as the homotopy classes of maps in the category of equivaraint crossed complexes. Subsequently, we construct a naive parametrized G-spectrum, such that the cohomology theory defined by it reduces to the Bredon cohomology with local coefficients when restricted to suspension spectra.

Abstract:
Given a multifunction from $X$ to the $k-$fold symmetric product $Sym_k(X)$, we use the Dold-Thom Theorem to establish a homological selection Theorem. This is used to establish existence of Nash equilibria. Cost functions in problems concerning the existence of Nash Equilibria are traditionally multilinear in the mixed strategies. The main aim of this paper is to relax the hypothesis of multilinearity. We use basic intersection theory, Poincar\'e Duality in addition to the Dold-Thom Theorem.

Abstract:
Compositional design of systems on chip from preverified components helps to achieve shorter design cycles and time to market. However, the design process is affected by the issue of protocol mismatches, where two components fail to communicate with each other due to protocol differences. Convertibility verification, which involves the automatic generation of a converter to facilitate communication between two mismatched components, is a collection of techniques to address protocol mismatches. We present an approach to convertibility verification using module checking. We use Kripke structures to represent protocols and the temporal logic to describe desired system behavior. A tableau-based converter generation algorithm is presented which is shown to be sound and complete. We have developed a prototype implementation of the proposed algorithm and have used it to verify that it can handle many classical protocol mismatch problems along with SoC problems. The initial idea for -based convertibility verification was presented at SLA++P '07 as presented in the work by Roopak Sinha et al. 2008.