Oscar Lautsch and Golrokh Nouri Awarded 2022 Undergraduate Research Prizes

In algebraic geometry, an open problem is to understand which geometric objects can be degenerated to combinatorial objects called "toric varieties". This problem has connections to and applications in a number of areas including mirror symmetry, birational geometry, and numerical algebraic geometry. In his 2022 USRA project, Oscar addressed the problem of toric degenerations for general hypersurfaces. After much testing and experimentation, he arrived at a very clean result: a general homogeneous polynomial of degree d in n+1 variables has an irreducible binomial as its initial term after some linear change of coordinates if and only if d<2n. This implies the existence of a toric degeneration when d<2n, and suggests that for larger d such a degeneration might not exist. Oscar's result formed the basis for a coauthored paper which has appeared in the Canadian Mathematical Bulletin.

Many substances, used either therapeutically or recreationally, induce tolerance, meaning that their effects decrease with repeated use, often requiring unhealthy increases in dosage to achieve the same results. To address this, starting in the Summer of 2021, Golrokh conducted a study to design dosing schedules that maximize the desired effect while factoring in tolerance, utilizing a differential equations model that considered the effect of a substance on an individual over time. The model was fit to data on caffeine and nicotine, and optimization was used to determine an optimal consumption schedule. The approach was illustrated with a caffeine consumption plan for users who need varying levels of alertness on different days of the week. The research, conducted collaboratively with her supervisor, has been written up and is being prepared for submission to a journal.