For an integer m, let N(m) denotes the number of integer solutions to the equation x^3 + y^3 =m. The sum of cubes problem concerns with the question of how large N(m) can be. Analogously, for any cubic form f(x,y), let N_f(m) denote the number of integer solutions for f(x,y)=m. In 1983, Silverman obtained large values for N_f(m) by connecting this to construction of higher rank elliptic curves. In this talk, we will discuss an explicit lower bound attained by N(m) for infinitely many m.
We'll discuss a result of Fischler and coauthors that proves a high level of linear independence of values of the zeta function at (odd) integers. We will take this opportunity to outline approaches to Irrationality questions that can be detailed at a later stage.
Interaction between biomolecules, like protein and DNA are important for various various cellular processes. Understanding these processes requires molecular level insights into the underlying interactions.The cellular processes are often disrupted by external agents or endogenous factors, necessitating the development of drugs to effectively counter their deleterious effects. Monoclonal antibodies are a class of therapeutic proteins with applications in cancers and autoimmune diseases. Certain applications require high concentration antibody formulations leading to the issue of protein aggregation, which results in a loss of drug efficacy and initiation of immune response upon administration. Controlling aggregation requires molecular insights into protein-protein and protein-environment interactions occurring in therapeutic formulations.By capturing dynamic motions and structural changes at an atomic level, molecular dynamics simulations can help us decipher the mechanisms underlying such phenomena. In this talk, I will discuss application of molecular dynamics simulations in understanding biopolymer interactions inside the cells and within therapeutic formulations.