Soonsik Kwon, Paul Jung
-Speaker: Terence Tao (UCLA)
-Date: 15~16 June, 2017
-Time: 16:00 pm
-Place: Fusion Hall, KI B/D, KAIST
[ 15 June 2017 ]
Title: The Erdős discrepancy problem
Abstract: The discrepancy of a sequence f(1), f(2), ... of numbers is defined to be the largest value of |f(d) + f(2d) + ... + f(nd)| as n,d range over the natural numbers. In the 1930s, Erdos posed the question of whether any sequence consisting only of +1 and -1 could have bounded discrepancy. In 2010, the collaborative Polymath5 project showed (among other things) that the problem could be effectively reduced to a problem involving completely multiplicative sequences. Finally, using recent breakthroughs in the asymptotics of completely multiplicative sequences by Matomaki and Radziwill, as well as a surprising application of the Shannon entropy inequalities, the Erdos discrepancy problem was solved in 2015. In this talk I will discuss this solution and its connection to the Chowla and Elliott conjectures in number theory.
[ 16 June 2017 ]
Title: Finite time blowup constructions for supercritical equations
Abstract: Many basic PDE of physical interest, such as the three-dimensional Navier-Stokes equations, are "supercritical" in that the known conserved or bounded quantities for these equations allow the nonlinear components of the PDE to dominate the linear ones at fine scales. Because of this, almost none of the known methods for establishing global regularity for such equations can work, and global regularity for Navier-Stokes in particular is a notorious open problem. We present here some ways to show that if one allows some modifications to these supercritical PDE, one can in fact construct solutions that blow up in finite time (while still obeying conservation laws such as conservation of energy). This does not directly impact the global regularity question for the unmodified equations, but it does rule out some potential approaches to establish such regularity.
email@example.com / +82-42-350-8545http://home.kias.re.kr/MKG/h/dls2017
Summer School on Probabilistic Methods
- 06/26/2017 - 06/30/2017
10 Lectures + 5 Exercise Sessions for 5 Days
Tentative Schedule of Regular Days
2 Morning Lectures + Long Break + 1 Exercise Session before the dinner
(Monday: Starting at 11am)
The Probabilistic Method is a powerful technique developed by the legendary Paul Erdős. To show the existence of a combinatorial object one defines a random object and shows that with positive probability it has the desired properties. Erdős deduces that the object must exist. In homage, we refer to this idea as Erdős Magic. Closely aligned is the study of random structures perse. Throughout we deal with large structures and asymptotic analysis permeates all of the lectures.
The specific lectures should be regarded as tentative. We will have two lectures each day but the material given may take more or less than one lecture period. Also, new material may be added and old material deleted. Expect the actual lectures to be these descriptions times 1 ± o(1).
Lecture I: What is Erdős Magic?
Lecture II: More Erdős Magic
Lecture III: Asymptopia
Lecture IV: Random Graphs
Lecture V and VI: The Erdős-Rényi Phase Transition
Lecture VII: Games Mathematicians Play
Lecture VIII: Needles in Exponential Haystacks
Lecture IX: Counting Connected Graphs
Lecture X: Gems