Rutgers Geometry/Topology seminar: Fall 2018 - Spring 2019

Tuesdays 3:40-4:40 in Hill 525

Past seminars: 2017-2018

Fall 2018

Date Speaker Title (click for abstract)
Sep. 4th No Seminar
Sep. 11th
Sep. 18th
Sep. 25th Maggie Miller (Princeton)
Oct. 2nd Nick Salter (Columbia)
Oct. 9th Ary Shaviv (Weizmann institute in Israel) Schwartz functions on sub-analytic manifolds
Oct. 16th
Oct. 23rd Francesco Lin (Princeton)
Oct. 30th
Nov. 6th
Nov. 13th
Nov. 20th No Seminar Happy Thanksgiving!
Nov. 27th
Dec. 4th
Dec. 11th

Spring 2019

Date Speaker Title (click for abstract)
Jan. 22nd No Seminar
Jan. 29th
Feb. 5th
Feb. 12th
Feb. 19th
Feb. 26th
March 5th
March 12th
March 19th No Seminar Spring break
March 26th
April 2nd
April 9th
April 16th
April 23rd
April 30rd

Abstracts

Schwartz functions on sub-analytic manifolds

Schwartz functions are classically defined on R^n as smooth functions such that they, and all their (partial) derivatives, decay at infinity faster than the inverse of any polynomial. The space of Schwartz functions is a Frechet space, and its continuous dual space is called the space of tempered distributions. A third space that plays a key role in the Schwartz theory is the space of tempered functions – a function is said to be tempered if point-wise multiplication by it preserves the space of Schwartz functions. This theory was formulated on R^n by Laurent Schwartz, later on Nash manifolds (smooth semi-algebraic varieties) by Fokko du Cloux and by Avraham Aizenbud and Dmitry Gourevitch, and on singular algebraic varieties by Boaz Elazar and myself.

The goal of this talk is to present the recently developed Schwartz theory on sub-analytic manifolds. I will first explain how one can attach a Schwartz space to an arbitrary open subset of R^n. Then, I will define (globally) sub-analytic manifolds – loosely speaking these are manifolds that locally look like sub-analytic open sub-sets of R^n (I will explain what are these too) and have some ”finiteness” property. Model theorists may think of definable manifolds in R^n. I will prove that one can intrinsically define the space of Schwartz functions (as well as the spaces of tempered functions and of tempered distributions) on these manifolds, and prove that these spaces are ”well behaved” (in the sense that they form sheaves and co-sheaves on the Grothendieck sub-analytic topology). Along the way we will see where sub-analyticity is used, and why this theory is ill-defined in the category of smooth (not necessarily sub-analytic) manifolds. Mainly, some ”polynomially bounded behaviour” (that holds in the sub-analytic case thanks to Lojasiewicz’s inequality) is required. As time permits I will describe some possible applications.

Organizers: Feng Luo, Xiaochun Rong, Hongbin Sun