Title: Berezin integral as a limit of Riemann sum

Abstract: Berezin integral is widely used in physics, but unfortunately its properties contradict the usual properties of Riemann integral. As a result of this, it was normally assumed that it can not be described as a Riemann sum and, instead, it was viewed as a formal algebraic operation. However, Roman Sverdlov wrote a paper in collaboration with Thomas Scanlon showing that Berezin integral can be modeled as a Riemann sum, after all. Accordingly, its properties were reconciled with the intuition we have from calculus. This was done in two steps. First, a single anticommuting product, was replaced with an algebra consisting of two separate products: one is Clifford product and the other is anticommuting wedge product. While wedge product is used in the "finite" part of the integral, the Clifford product is used between infinitesimal and the finite part. And, secondly, a usual integral over the whole space, was replaced with a choice between three possibilities. One possibility is an integral over the closed surface (aka "surface integral"). The other is integral over space where usual measure is being replaced by weighted directed measure (aka "directed volume integral"). The third is an integral being replaced with a finite sum (aka "minimalist model"). In this talk some of those constructions will be outlined. The talk will be based on the first half of the paper https://aip.scitation.org/doi/full/10.1063/1.5144877 which is also available online at https://math.berkeley.edu/~scanlon/papers/BIRS-9May2020.pdf.

About the speaker: Roman Sverdlov received his Ph.D. in Physics from University of Michigan in 2009 under the supervision of Luca Bombelli and Marc Ross. He spent five years in India first as a postdoc at the Raman Research Institute (Bangalore), second with a visitting position at Institute of Mathematical Sciences (Chennai), and finally as postdoc at IISER Mohali. In 2014 he returned to the US with a visitting position at the Department of Physics at the University of Mississippi for a year and started a Ph.D. in Mathematics before transferring to New Mexico in 2016. He is now on his sixth year as a PhD student in Mathematics at UNM working under the direction of Terry Loring.