Department of

# Mathematics

Seminar Calendar
for Differential Geometry events the year of Friday, October 22, 2021.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Wednesday, March 10, 2021

6:00 pm in Zoom | Register at http://bit.ly/MathWithoutBorders,Wednesday, March 10, 2021

#### “Structure-preserving discretization of differential calculus and geometry”

###### Anil Hirani, associate professor (University of Illinois Urbana-Champaign, Department of Mathematics)

Abstract: Two important aspects of the structure of differential calculus are the chain and product rules. On manifolds, chain rule generalizes to the exterior derivative commuting with pullback by smooth maps. The product rule generalizes to a rule for the exterior derivative of wedge product of differential forms. These properties, especially the product rule, may be used as one of the defining properties for a covariant derivative in differential geometry. What if one is developing calculus and geometry for non-smooth spaces. One such application is the development of a discrete exterior calculus and discrete differential geometry for simplicial complexes (e.g. triangle meshes and tetrahedral meshes). What should play the role of the smooth maps, the exterior derivative, the wedge product and the covariant derivative so that we can speak of the above structural properties of calculus and geometry in this discrete setting? I will use examples to describe such a discrete calculus and geometry we have been developing. This will be a survey of some old ideas and some new developments. The newer developments are joint work with Mark Schubel (Apple Inc.) and Daniel Berwick-Evans (UIUC).

Monday, September 20, 2021

3:00 pm in 347 Altgeld Hall,Monday, September 20, 2021

#### Differential Geometry over C-infinity rings (part 1)

###### Eugene Lerman (UIUC)

Abstract: (joint work with Yael Karshon) We develop some basic tools of differential geometry for singular spaces. There is a variety of approaches to geometry on singular spaces going back to 1960's. The spaces we work with lie in the intersection of differential spaces of Sikorski and C-infinity schemes of Dubuc. We integrate vector fields to flows and we construct a C-infinity ring analogue of Grothendieck's algebraic de Rham complex. (part 1 of a two part talk, part 2 will be next week)

Friday, September 24, 2021

2:00 pm in 347 Altgeld Hall,Friday, September 24, 2021