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.
Mathematics Seminar | Alladi Ramakrishnan Hall
Mar 20 14:00-15:00
Ann Mary Mathew | Assumption College, Changanassery, Kerala
Complex systems set in the context of competitive resource allocation often exhibit special emergent features. The classic El Farol bar problem traces the emergence of cooperation in a competitive environment and its binary version, Minority Game, is often used to analyse the impact of information on emergence. The situations in which agents compete for physical space as a limited resource while displaying crowd-avoidance behavior, are particularly intriguing due to the promising yet unexplored nature of the area.
In this talk, I introduce a specific model where the agents are dispersed on a lattice and follow a simple relocation mechanism (win-stay-lose-shift) as part of the spatial competition to enhance personal payoff. Two parameters - neighborhood size and tolerance threshold - define the level of local crowding experienced by the agents. Agents access information about the occupancy of sites up to a distance called the information radius. The evolutionary dynamics originate from the repeated relocation of agents to vacant sites within the information radius in search of more comfortable spatial locations. Some of the relevant questions are about the level of self-organization achieved by the competing agents, the carrying capacity of the system and the influence of control parameters on the emergence within the model. I answer some of these questions based on the numerical results obtained using agent-based modeling.
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.