Asymptotics: the unified transform, a new approach to the Lindelöf Hypothesis, and the ultra-relativistic limit of the Minkowskian approximation of general relativity

Speaker: 

Athanassios S. Fokas

Institution: 

University of Cambridge/USC

Time: 

Thursday, April 11, 2019 - 4:00pm to 5:00pm

Location: 

RH 306

Employing standard, as well as novel techniques of asymptotics, three different problems will be discussed: (i) The computation of the large time asymptotics of initial-boundary value problems via the unified transform (also known as the Fokas Method, www.wikipedia.org/wiki/Fokas_method)[1]. (ii) The evaluation of the large t-asymptotics to all orders of the Riemann zeta function [2], and the introduction of a new approach to the Lindelöf Hypothesis [3]. (iii) The proof that the ultra-relativistic limit of the Minkowskian approximation of general relativity [4] yields a force with characteristics of the strong force, including confinement and asymptotic freedom [5].

[1] J. Lenells and A. S. Fokas. The Nonlinear Schrödinger Equation with t-Periodic Data: I. Exact Results, Proc. R. Soc. A 471, 20140925 (2015).
J. Lenells and A. S. Fokas, The Nonlinear Schrödinger Equation with t-Periodic Data: II. Perturbative Results, Proc. R. Soc. A 471, 20140926 (2015).
[2] A.S. Fokas and J. Lenells, On the Asymptotics to All Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function, Mem. Amer. Math. Soc. (to appear).
[3] A.S. Fokas, A Novel Approach to the Lindelof Hypothesis, Transactions of Mathematics and its Applications (to appear).
[4] L. Blanchet and A.S. Fokas, Equations of Motion of Self-Gravitating N-Body Systems in the First Post-Minkowskian
Approximation, Phys. Rev. D 98, 084005 (2018).
[5] A.S. Fokas, Super Relativistic Gravity has Properties Associated with the Strong Force, Eur. Phys. J. C (to appear).

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.

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