Workshop on Higher Geometry and Field Theory

Report (2015)

Mathematician - Job of the Year 2014

Speeches/Talks (2015)

Z_2^n Supergeometry

Presentation (2014, December 17)

Z_2^n-Supergeometry II: Batchelor-Gawedzki Theorem

E-print/Working paper (2014)

Derived Geometry and Applications

Presentation (2014)

Algebraic geometry over differential operators

Presentation (2014)

The supergeometry of Loday algebroids

in Journal of Geometric Mechanics (2013), 5(2), 185--213

Higher Algebras and Lie Infinity Homotopy Theory

Report (2013)

Abstracts and list of participants of the Workshop on "Higher Algebras and Lie Infinity Homotopy Theory" organized in June 2013 at the University of Luxembourg by Vladimir Dotsenko and Norbert Poncin

Differential Geometry

Learning material (2012)

Higher trace and Berezinian of matrices over a Clifford algebra

in Journal of Geometry and Physics (2012), 62(11), 22942319

We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^n-graded commutative associative algebra A. The applications include a new approach to the classical ... [more ▼]

We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^n-graded commutative associative algebra A. The applications include a new approach to the classical theory of matrices with coefficients in a Clifford algebra, in particular of quaternionic matrices. In a special case, we recover the classical Dieudonné determinant of quaternionic matrices, but in general our quaternionic determinant is different. We show that the graded determinant of purely even (Z_2)^n-graded matrices of degree 0 is polynomial in its entries. In the case of the algebra A = H of quaternions, we calculate the formula for the Berezinian in terms of a product of quasiminors in the sense of Gelfand, Retakh, and Wilson. The graded trace is related to the graded Berezinian (and determinant) by a (Z_2)^n-graded version of Liouville’s formula. [less ▲]