Marc Hoyois

Graduate Student
Department of Mathematics
Northwestern University
2033 Sheridan Road
Evanston, Illinois 60208, USA
hoyois at math dot northwestern dot edu

My current research:

• From algebraic cobordism to motivic cohomology (revised March 2013) [pdf]
  Proof of the Hopkins–Morel equivalence in motivic homotopy theory.
• The étale symmetric Künneth theorem (draft, June 2012) [pdf]
• A note on étale homotopy (February 2013) [pdf]
  An informal exposition of higher Galois theory for ∞-topoi and its connection to the étale homotopy type of Artin–Mazur–Friedlander.

Some older stuff:

• Notes on the birational classification of surfaces. [pdf]
• Notes on the first reconstruction theorem in Gromov–Witten theory. [pdf]
• Notes on the Nisnevich topology and Thom spaces in motivic homotopy theory, with a proof of the motivic tubular neighborhood theorem. [pdf]
• Chern character and derived algebraic geometry (Master thesis, EPFL) [pdf, slides, poster]
• Sur la cohomologie des schémas (master 2nd semester, EPFL) [pdf]
  Basic scheme theory from the geometric viewpoint. Definition and properties of sheaf cohomology and of Čech cohomology. Computation of the cohomology of affine schemes and of projective spaces.
• The Syntax of First-Order Logic (bachelor 6th semester and master 1st semester, EPFL) [pdf]
  A constructive treatment of various topics in first-order logic: Herbrand's theorem and the epsilon theorems; Craig's interpolation lemma; Gödel's first incompleteness theorem (two proofs); a detailed account of Gödel's second incompleteness theorem with a satisfying level of generality (the proof is complete up to finitely many theorems of Peano arithmetic); basic Zermelo–Fraenkel set theory; the consistency of the axiom of choice and of the generalized continuum hypothesis.
• Décompositions paradoxales (bachelor 5th semester, EPFL) [pdf]
  This is an exploration of the notion of equidecomposability in a set being acted upon by a group. Applications include the generalized Banach–Tarski paradox, the Sierpiński–Mazurkiewicz paradox, and the von Neumann paradox in the plane. The equivalence between the amenability and the nonparadoxality of discrete groups is proved (Tarski's theorem).
• Sur la constante de Khinchin (bachelor 4th semester, EPFL) [pdf]
  Khinchin's theorem states that there exists a real number K, Khinchin's constant, such that the sequence of partial geometric means of the elements of the continuous fraction of almost any real number converges to K. This text contains an elementary discussion of continuous fractions and two different proofs of this result: Khinchin's original proof and a more conceptual proof using ergodic theory.