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SUMMARY:Camillo De Lellis (IAS)
DTSTART;VALUE=DATE-TIME:20200413T150000Z
DTEND;VALUE=DATE-TIME:20200413T160000Z
DTSTAMP;VALUE=DATE-TIME:20230925T222535Z
UID:IASanalysis/1
DESCRIPTION:Title: Flows of vector fields: classical and modern\nby Camillo De Lellis
(IAS) as part of IAS Analysis Seminar\n\n\nAbstract\nConsider a (possibly
time-dependent) vector field $v$ on the Euclidean space. The classical Ca
uchy-Lipschitz (also named Picard-Lindel\\"of) Theorem states that\, if th
e vector field $v$ is Lipschitz in space\, for every initial datum $x$ the
re is a unique trajectory $\\gamma$ starting at $x$ at time $0$ and solvin
g the ODE $\\dot{\\gamma} (t) = v (t\, \\gamma (t))$. The theorem looses i
ts validity as soon as $v$ is slightly less regular. However\, if we bundl
e all trajectories into a global map allowing $x$ to vary\, a celebrated t
heory put forward by DiPerna and Lions in the 80es show that there is a un
ique such flow under very reasonable conditions and for much less regular
vector fields. A long-standing open question is whether this theory is the
byproduct of a stronger classical result which ensures the uniqueness of
trajectories for {\\em almost every} initial datum. I will give a complete
answer to the latter question and draw connections with partial different
ial equations\, harmonic analysis\, probability theory and Gromov's $h$-pr
inciple.\n
LOCATION:https://researchseminars.org/talk/IASanalysis/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felix Otto (Max Planck Institute Leipzig)
DTSTART;VALUE=DATE-TIME:20200420T150000Z
DTEND;VALUE=DATE-TIME:20200420T160000Z
DTSTAMP;VALUE=DATE-TIME:20230925T222535Z
UID:IASanalysis/2
DESCRIPTION:Title: A variational approach to the regularity theory for the Monge-Ampère
equation\nby Felix Otto (Max Planck Institute Leipzig) as part of IAS
Analysis Seminar\n\n\nAbstract\nWe present a purely variational approach t
o the regularity theory for the\nMonge-Ampère equation\, or rather optim
al transportation\, introduced with M. Goldman. Following De Giorgi’s ph
ilosophy for the regularity theory of minimal surfaces\, it is based on th
e approximation of the displacement by a harmonic gradient\, which leads t
o a One-Step Improvement Lemma\, and feeds into a Campanato iteration on t
he $C^{1\,\\alpha}$-level for the displacement\, capitalizing on affine in
variance.\nOn the one hand\, this allows to reprove the $C^{1\,\\alpha}$-
regularity result (Figalli-Kim\, De Philippis-Figalli) bypassing Caffarell
i’s celebrated theory. This also extends to boundary regularity (Chen-Fi
galli)\, which is joint work with T. Miura.\nOn the other hand\, due to it
s robustness\, it can be used as a large-scale regularity theory for the p
roblem of matching the Lebesgue measure to the Poisson measure in the ther
modynamic limit. This is joint work with M. Goldman and M. Huesmann.\n
LOCATION:https://researchseminars.org/talk/IASanalysis/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven Zelditch (Northwestern University)
DTSTART;VALUE=DATE-TIME:20200428T150000Z
DTEND;VALUE=DATE-TIME:20200428T160000Z
DTSTAMP;VALUE=DATE-TIME:20230925T222535Z
UID:IASanalysis/3
DESCRIPTION:Title: Ellipses of small eccentricity are determined by their Dirichlet (or\,
Neumann) spectra\nby Steven Zelditch (Northwestern University) as par
t of IAS Analysis Seminar\n\n\nAbstract\nIn 1965\, M. Kac proved that disc
s were uniquely determined by their Dirichlet (or\, Neumann) spectra. Unt
il recently\, disks were the only smooth plane domains known to be determi
ned by their eigenvalues. Recently\, H. Hezari and I proved that ellipses
of small eccentricity are also determined uniquely by their Dirichlet (or\
, Neumann) spectra. The proof uses recent results of Avila\, de Simoi\, an
d Kaloshin\, proving that nearly circular plane domains with rationally i
ntegrable billiards must be ellipses. It also uses a ``bounce decompositi
on'' for the wave trace\, representing the wave trace as a sum of q-bounce
oscillatory integrals. It is shown that for nearly circular domains\, eac
h is a spectral invariant and that the ellipse is uniquely determined by i
ts q-bounce invariants.\n
LOCATION:https://researchseminars.org/talk/IASanalysis/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhiyuan Zhang (Université Paris 13)
DTSTART;VALUE=DATE-TIME:20200504T150000Z
DTEND;VALUE=DATE-TIME:20200504T160000Z
DTSTAMP;VALUE=DATE-TIME:20230925T222535Z
UID:IASanalysis/4
DESCRIPTION:Title: Exponential mixing of 3D Anosov flows\nby Zhiyuan Zhang (Universit
é Paris 13) as part of IAS Analysis Seminar\n\n\nAbstract\nWe show that a
topologically mixing $C^{\\infty}$ Anosov flow on a 3 dimensional compact
manifold is exponential mixing with respect to any equilibrium measure wi
th Hölder potential. This is a joint work with Masato Tsujii.\n
LOCATION:https://researchseminars.org/talk/IASanalysis/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guy C. David (Ball State University)
DTSTART;VALUE=DATE-TIME:20200512T150000Z
DTEND;VALUE=DATE-TIME:20200512T160000Z
DTSTAMP;VALUE=DATE-TIME:20230925T222535Z
UID:IASanalysis/5
DESCRIPTION:Title: Quantitative decompositions of Lipschitz mappings\nby Guy C. David
(Ball State University) as part of IAS Analysis Seminar\n\n\nAbstract\nGi
ven a Lipschitz map\, it is often useful to chop the domain into pieces on
which the map has simple behavior. For example\, depending on the dimensi
ons of source and target\, one may ask for pieces on which the map behaves
like a bi-Lipschitz embedding or like a linear projection. For many issue
s\, it is even more useful if this decomposition is quantitative\, i.e.\,
with bounds independent of the particular map or spaces involved. After su
rveying the question of bi-Lipschitz decomposition\, we will discuss the m
ore complicated case in which dimension decreases\, e.g.\, for maps from $
\\mathbb{R}^3$ to $\\mathbb{R}^2$. This is recent joint work with Raanan S
chul\, improving a previous result of Azzam-Schul.\n
LOCATION:https://researchseminars.org/talk/IASanalysis/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hong Wang (Institute for Advanced Study)
DTSTART;VALUE=DATE-TIME:20200518T150000Z
DTEND;VALUE=DATE-TIME:20200518T160000Z
DTSTAMP;VALUE=DATE-TIME:20230925T222535Z
UID:IASanalysis/6
DESCRIPTION:Title: Square function estimate for the cone in R^3\nby Hong Wang (Instit
ute for Advanced Study) as part of IAS Analysis Seminar\n\n\nAbstract\nWe
prove a sharp square function estimate for the cone in R^3 and consequentl
y the local smoothing conjecture for the wave equation in 2+1 dimensions.
The proof uses induction on scales and an incidence estimate for points an
d tubes.\n\nThis is joint work with Larry Guth and Ruixiang Zhang.\n
LOCATION:https://researchseminars.org/talk/IASanalysis/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Max Engelstein (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20200601T150000Z
DTEND;VALUE=DATE-TIME:20200601T160000Z
DTSTAMP;VALUE=DATE-TIME:20230925T222535Z
UID:IASanalysis/7
DESCRIPTION:Title: Winding for Wave Maps\nby Max Engelstein (University of Minnesota)
as part of IAS Analysis Seminar\n\n\nAbstract\nWave maps are harmonic map
s from a Lorentzian domain to a Riemannian target. Like solutions to many
energy critical PDE\, wave maps can develop singularities where the energy
concentrates on arbitrary small scales but the norm stays bounded. Zoomin
g in on these singularities yields a harmonic map (called a soliton or bub
ble) in the weak limit. One fundamental question is whether this weak limi
t is unique\, that is to say\, whether different bubbles may appear as the
limit of different sequences of rescalings.\n\nWe show by example that un
iqueness may not hold if the target manifold is not analytic. Our constru
ction is heavily inspired by Peter Topping's analogous example of a ``wind
ing" bubble in harmonic map heat flow. However\, the Hamiltonian nature of
the wave maps will occasionally necessitate different arguments. This is
joint work with Dana Mendelson (U Chicago).\n
LOCATION:https://researchseminars.org/talk/IASanalysis/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alik Mazel (AMC Health)
DTSTART;VALUE=DATE-TIME:20200525T150000Z
DTEND;VALUE=DATE-TIME:20200525T160000Z
DTSTAMP;VALUE=DATE-TIME:20230925T222535Z
UID:IASanalysis/8
DESCRIPTION:Title: An application of integers of the 12th cyclotomic field in the theory
of phase transitions\nby Alik Mazel (AMC Health) as part of IAS Analys
is Seminar\n\n\nAbstract\nThe construction of pure phases from ground stat
es is performed for $ u > u_*(d)$ for all values of $d$ except for 39 spec
ial ones. For values $d$ with a single equivalence class all periodic grou
nd states generate the corresponding pure phase which provides a complete
description of extreme Gibbs measures (complete phase diagram). For a gene
ral $d$ we prove that at least one class of ground states generates pure p
hases and propose an algorithm that decides\, after finitely many iteratio
ns\, which classes of ground states generate pure phases. We conjecture th
at in case of several classes only one of them generates pure phases which
is confirmed by (numerical) application of our algorithm to several (rela
tively small) values of $d$.\n
LOCATION:https://researchseminars.org/talk/IASanalysis/8/
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