Moderator:
Masoud Sabzevari (sebzevari@ipm.ir)

Around 1955, Chern conjectured that affine manifolds have zero as their Euler number. One way to think about affine structures on manifolds is to construct a flat torsionfree
connection on the tangent bundle. Benzecri and Milnor studied this question in dimension 2 and later in 80's
Milnor studied how characteristic classes of vector bundles change when we have certain geometric structures on the bundle,
in particular, flat structure.
In this direction, he formulated a conjecture in algebraic Ktheory. To be more precise,
let $G$ be a finitedimensional Lie group and $G^{\delta}$ be the same group with the discrete
topology. The classifying space $BG$ classifies principal $G$bundles and the classifying space
$BG^{\delta}$ classifies flat principal $G$bundles (i.e. those bundles that admit a connection
whose curvature vanishes). The natural homomorphism from $G^{\delta}$ to $G$ induces a continuous
map from $BG^{\delta}$ to $BG$. Milnor and Friedlander conjectured that this map induces an
isomorphism on cohomology with finite coefficients. In this talk, we discuss the same map for
infinitedimensional Lie groups, in particular for diffeomorphism groups and symplectomorphisms.
In these cases, we use techniques from homotopy theory and the moduli space of manifolds to show that
similar to finitedimensional Lie groups, the map from $BG^{\delta}$ to $BG$ induces a split
surjection on cohomology with integer coefficients in the stable range. I will also discuss
applications of these results in foliation theory, in particular, characteristic classes of surface
bundles. 
We consider load balancing problem in a cache network consisting of storageenabled servers forming a distributed
content delivery scenario. Previously proposed load balancing solutions cannot perfectly balance out requests among
servers, which is a critical issue in practical networks. Therefore, we investigate a coded cache content placement
where coded chunks of original files are stored in servers based on the files popularity distribution. In our scheme,
upon each request arrival at the delivery phase, by dispatching enough coded chunks to the request origin from the nearest
servers, the requested file can be decoded. Here, we show that if $n$ requests arrive randomly at $n$ servers, the proposed
scheme results in the maximum load of $O(1)$ in the network. This result is shown to be valid under various assumptions for
the underlying network topology. Our results should be compared to the maximum load of two baseline schemes, namely, nearest
replica and power of two choices strategies, which are $\Theta(\text{log}\hspace{2pt}n)$ and
$\Theta(\text{log}\hspace{2pt}\text{log}\hspace{2pt}n),$ respectively. This finding shows that using coding, results
in a considerable load balancing performance improvement, without compromising communications cost performance. This is confirmed
by performing extensive simulation results, in nonasymptotic regimes as well. 
In this talk, I will briefly introduce Euclidean lattices. I will then summarize the (hard) problems related to lattices in
Mathematics, Computer Science, and Wireless Communications. I will then present some wellknown structured and unstructured
latticebased encryption schemes. 
