Explore the construction of k-regular maps using finite local schemes in this 58-minute lecture by Jarosław Buczyński. Delve into the historical problem of determining minimal dimensions for linearly independent point images, dating back to Chebyshev and Borsuk. Learn how algebraic geometry methods can be applied to construct k-regular maps and relate upper bounds to Hilbert scheme dimensions. Discover explicit examples for k ≤ 5 and upper bounds for arbitrary m and k. Examine the connection to interpolation theory and its implications for continuous functions on topological spaces. Follow the progression from manifolds and k-regular maps to secant varieties, Hilbert schemes, and scheme-theoretic methods for interpolation. Gain insights into various types of k-regularity and their applications in algebraic topology and geometry.