Kolokviální přednáška proběhne 22.3.2023 od 16:00 v posluchárně M1.

Jon Noel

Title: Squaring the Circle with Simple Pieces

Abstract:
Is it possible to partition a disk in the plane into finitely many pieces and re-assemble those pieces via isometries to yield a partition of a square? This question was asked by Tarski back in 1925 and answered in the affirmative by Laczkovich some 65 years later.
Laczkovich's proof uses the Axiom of Choice in a strong way; as a  result, the pieces of his partition are very hard to imagine. In 2017, two new proofs emerged which achieve pieces that are Lebesgue measurable or even Borel; the latter result is fully "constructive." We improve on these results by constructively achieving pieces which have (a) lower Borel complexity and (b) "small" boundaries. A benefit of the second condition is that the pieces of our partition are, in some sense, "visualizable." The proof uses basic concepts in graph theory, such as Eulerian tours, matchings and network flows. Based on joint work with András Máthé and Oleg Pikhurko.

Kolokviální přednáška proběhne 12.10.2022 od 16:00 v posluchárně M1.

Péter Pál Pach

Title: The Alon-Jaeger-Tarsi conjecture via group ring identities

Abstract:
The Alon-Jaeger-Tarsi conjecture states that for any finite field F of size at least 4  and any nonsingular  matrix A over F there exists a  vector x such that neither x nor Ax has a 0 component. In this talk we discuss the proof of this result for primes larger than 61 and show some further applications of our method about coset covers and additive bases. (Joint work with János Nagy and István Tomon.)

Péter Pál Pach is an associate professor, head of the MTA-BME Lendület Arithmetic Combinatorics Research group supported by a Lendület (Momentum) grant, Department of Computer Science and Information Theory, Budapest University of Technology and Economics. His numerous influential papers appear in top journals like Annals of Mathematics, Advances in Mathematics, Journal of Number Theory, etc.