Durante a primeira semana do evento, serão ministrados minicursos pela/os supervisora/es, para revisão dos conceitos necessários e introdução aos problemas a serem abordados. Não é esperado nenhum conhecimento prévio na área específica do problema.
Os projetos confirmados até o momento são os seguintes.
Supervisão: Ali Tahzibi (ICMC - USP) e Javid Validashti (DePaul University, EUA)
This project explores the relationship between lattice points in 3-dimensional simplexes and the corresponding volumes of these geometric objects, with potential applications to optimization and linear programming.
Given three positive integers \(a\), \(b\), and \(c\), the simplex \(S(a, b, c)\) is defined in the positive orthant of 3-space with vertices at the origin \((0, 0, 0)\), and the points \((a, 0, 0)\), \((0, b, 0)\), and \((0, 0, c)\). We define \(L(a, b, c)\) as the number of lattice points contained in \(S(a, b, c)\), including its boundaries, but excluding those lying on the hyperplane \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\).
The primary goal of this project is to prove that if \(L(a, b, c) = L(n, n, n)\) for some positive integer \(n\), then the volume of \(S(a, b, c)\) is less than or equal to the volume of \(S(n, n, n)\), with equality if and only if \(a = b = c = n\). This generalizes a similar question in 2-dimensional simplexes, making it an excellent combinatorial exercise.
This problem is also deeply connected to optimization and linear program- ming, as \(L(a, b, c)\) is the number of integer solutions of the system of linear inequalities \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\leq 1\). Understanding lattice point structures within simplexes and their volumes is a key aspect of combinatorial optimization.
In addition to theoretical exploration, participants are encouraged to use computer programming to enumerate lattice points and calculate volumes, providing practical experience with algorithms used in optimization and linear programming. The project will allow students to engage with discrete geometry, combinatorics, and computational mathematics, with a strong emphasis on applications to concrete problems in optimization.