In 1942 Kelly conjectured that any graph having at least 3 vertices is uniquely determined by the multiset of all its subgraphs obtained by deleting a vertex and all edges adjacent to it. In 1964 Harary conjectured analogously that any graph having at least 4 edges is uniquely determined by all its subgraphs obtained by deleting a single edge, which is known as the edge reconstruction conjecture. As of today, both conjectures are still open.

In the talk I will discuss some of the classical results about the conjectures and some evidence in favor of them. Also I will explicitly show that the edge reconstruction conjecture holds for a specific type of graphs.

In this talk, we compare and contrast a few finite element h-adaptive
and hp-adaptive algorithms. We test these schemes on three example PDE
problems and we utilize and evaluate an a posteriori error estimate.

In the process, we introduce a new framework to
study adaptive algorithms and a posteriori error estimators. Our innovative
environment begins with a solution u and then uses interpolation to
simulate solving a corresponding PDE. As a result, we always know the
exact error and we avoid the noise associated with solving.

Using an effort indicator, we evaluate the relationship between accuracy
and computational work. We report the order of convergence of different
approaches. And we evaluate the accuracy and effectiveness of an
a posteriori error estimator.

Differential Harnack estimates, also called Li-Yau-Hamilton estimates, play an important part in geometric analysis, especially geometric flows. Firstly, I will review the developments of differential Harnack estimates. Then I will talk about the Constrained matrix differential Harnack estimates for the heat equation on Kaehler Manifolds.

In a series of papers, E. Hecke described the representation of the group
$SL(2,p)$ on the regular differentials of the modular curve $X$ of level $p$. This was one of the
first applications of character theory outside of finite group theory, and one of the first
constructions of representations using cohomology. I will review Hecke's results, and
interpret them in the modern language of automorphic representations.