SMRI Seminar: Lewicka -- Convex integration for the system of isometric immersions and the Monge-Ampere system

Dear All, 

I'm delighted to let you know that Prof Marta Lewicka (University of Pittsburgh) will
give a short course of 6 lectures in Analysis and PDEs, including an SMRI lecture during
her stay in Sydney (15 June -30 June 2026).  The first lecture functions as a general
research seminar, giving an overview of the field and of the main results and would
be accessible to a broader audience, while the continuation would be more specialised
and would function more as a course. The subsequent lectures will then focus on the
details and proofs.  

All lectures will be held in the SMRI Seminar Room (A12 Macleay). The dates and times
of the lectures are as follows: 

Thursday 18/06/2026 SMRI seminar, 3:00 PM -4:00 PM 

Tuesday 23/06/2026, 1:00 PM - 3:00 PM 

Monday 29/06/2026, 2:00 PM - 3:00 PM 

Tuesday 30/06/2026 1:00 PM - 3:00 PM 

You are warmly invited to attend the lectures and join us for lunch with the speaker
beforehand.  We will gather at 11:30am on the ground floor (Level 2 from the lift), next
to the Learning Hub East (Rm250).  

Seminar : SMRI Seminar: Lewicka 

Day/time : Thursday, 18 June at 3:00 PM, 

Location : SMRI seminar room (A12 Macleay Room 301) 

Speaker : Prof Marta Lewicka (University of Pittsburgh) 

Title: Convex integration for the system of isometric immersions and the Monge-Ampere
system 

Abstract: In 1956, N.  Kuiper showed that every C^1-regular short immersion of a
2-dimensional Riemannian metric into R^3 can be uniformly approximated by its exact
isometric immersions with regularity C^1.  Kuiper's result improved upon a prior
construction by J.  Nash (thus nowadays known as the Nash-Kuiper algorithm) valid for
immersions into R^4.  In that fundamental work, Nash predicted that: "apparently
rigidity disappears completely when the imbedding space has enough dimensions".  

We show that for two-dimensional metrics, rigidity indeed disappears completely already
in R^4, in the sense that every C^1-regular short immersion of such a metric into R^4
can be uniformly approximated by its exact isometric immersions with regularity
C^{1,\alpha} for any \alpha<1.  We obtain analogous flexibility result for the
Monge-Ampere system, which is a multidimensional version of the two-dimensional
Monge-Ampere equation, and extend both results to arbitrary dimension d and the
respective codimension d_*-d+1 (where d_*=d(d+1)/2 is the Janet dimension).