Math Day
Mathematics Day in Shahrekord
Shahrekord University
January 4, 2020
Moderator: Masoud Sabzevari (sebzevari@ipm.ir)
Lecture 1
Title: On Characteristic Classes of Manifold Bundles
Instructor: Sam Nariman (Purdue Univeristy, Copenhagen, Denmark)
Date: Saturday, January 4, 2020
Time: 09:00 - 10:00
Venue: Faculty of Mathematical Sciences, Sh. Chamran Lecture Hall
Abstract:

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 torsion-free 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 K-theory. To be more precise, let $G$ be a finite-dimensional 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 infinite-dimensional 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 finite-dimensional 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.

Lecture 2
Title: Coded Load Balancing in Cache Networks
Instructor: Farzad Parvaresh (University of Isfahan and IPM, Iran)
Date: Saturday, January 4, 2020
Time: 10:00 - 11:00
Venue: Faculty of Mathematical Sciences, Sh. Chamran Lecture Hall
Abstract:
We consider load balancing problem in a cache network consisting of storage-enabled 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 non-asymptotic regimes as well.
Lecture 3
Title: Euclidean Lattices: Foundations and Applications
Instructor: Amin Sakzad (Monash University, Australia)
Date: Saturday, January 4, 2020
Time: 11:30 - 12:30
Venue: Faculty of Mathematical Sciences, Sh. Chamran Lecture Hall
Abstract:
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 well-known structured and unstructured lattice-based encryption schemes.