The Sperner property

Speaker: 

Richard Stanley

Institution: 

MIT

Time: 

Thursday, January 24, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

In 1927 Emanuel Sperner proved that if $S_1,\dots,S_m$ are distinct 
subsets of an $n$-element set such that we never have $S_i\subset S_j$, 
then $m\leq \binom{n}{\lfloor n/2\rfloor}$. Moreover, equality is achieved 
by taking all subsets of $S$ with $\lfloor n/2\rfloor$ elements. This 
result spawned a host of generalizations, most conveniently stated in the 
language of partially ordered sets. We will survey some of the highlights 
of this subject, including the use of linear algebra and the cohomology of 
certain complex projective varieties. An application is a proof of a 
conjecture of Erd\H{o}s and Moser, namely, for all integers $n\geq 1$ and 
real numbers $\alpha\geq 0$, the number of subsets with element sum 
$\alpha$ of an $n$-element set of positive real numbers cannot exceed the 
number of subsets of $\{1,2,\dots,n\}$ whose elements sum to $\lfloor 
\frac 12\binom n2\rfloor$. We will conclude by discussing two recent 
proofs of a 1984 conjecture of Anders Bj\"orner on the weak Bruhat order 
of the symmetric group $S_n$.

Higher algebra and arithmetic

Speaker: 

Lars Hesselholt

Institution: 

Nagoya University and Copenhagen University

Time: 

Thursday, February 21, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

This talk concerns a twenty-thousand-year old mistake: The natural numbers record only the result of counting and not the process of counting. As algebra is rooted in the natural numbers, the higher algebra of Joyal and Lurie is rooted in a more basic notion of number which also records the process of counting. Long advocated by Waldhausen, the arithmetic of these more basic numbers should eliminate denominators. Notable manifestations of this vision include the Bökstedt-Hsiang-Madsen topological cyclic homology, which receives a denominator-free Chern character, and the related Bhatt-Morrow-Scholze integral p-adic Hodge theory, which makes it possible to exploit torsion cohomology classes in arithmetic geometry. Moreover, for schemes smooth and proper over a finite field, the analogue of de Rham cohomology in this setting naturally gives rise to a cohomological interpretation of the Hasse-Weil zeta function by regularized determinants, as envisioned by Deninger.

Boundary rigidity and the local inverse problem for the geodesic X-ray transform on tensors

Speaker: 

Andras Vasy

Institution: 

Stanford University

Time: 

Thursday, January 17, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

In this talk, based on joint work with Plamen Stefanov and Gunther Uhlmann, I discuss the boundary rigidity problem on manifolds with boundary (for instance, a domain in Euclidean space with a perturbed metric), i.e. determining a Riemannian metric from the restriction of its distance  function to the boundary. This corresponds to travel time tomography, i.e. finding the Riemannian metric from the time it takes for solutions of the corresponding wave equation to travel between boundary points. A version of this relates to finding the speed of seismic waves inside the Earth from travel time data, which in turn permits a study of the structure of the inside of the Earth.

This non-linear problem in turn builds on the geodesic X-ray transform on such a Riemannian manifold with boundary. The geodesic X-ray transform on functions associates to a function its integral along geodesic curves, so for instance in domains in Euclidean space along straight lines. The X-ray transform on symmetric tensors is similar, but one integrates the tensor contracted with the tangent vector of the geodesics. I will explain how, under suitable convexity assumptions, one can invert the geodesic X-ray transform on functions, i.e. determine the function from its X-ray transform, in a stable manner, as well as the analogous tensor result, and the connection to the full boundary rigidity problem.

Pages

Subscribe to RSS - Colloquium