Mathematics Colloquia and Seminars

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We revisit the classical linear Diophantine problem A x = b, x >= 0, x integral, and the associated affine semigroup of all non-negative linear combinations of the columns of A, a d-by-n integer matrix. We develop a new theory of colorful affine semigroups, where the generators receive different colors and the elements of the semigroup take into account the colors used. The classical theory of affine semigroups is the case of all generators have the same color.

In the first part of the talk, I will talk about the integer analogs of two classical (colorful) theorems in convex geometry. Specifically, we will look at semigroup versions of Colorful Carathéodory and Helly theorems for cones.

In the second part, I will talk about the most special affine semigroups, numerical semigroups. In numerical semigroups, we will study a generalization of Frobenius number and count the number of distinct "special" solutions x for a given b.