BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Camillo De Lellis (IAS) DTSTART:20200413T150000Z DTEND:20200413T160000Z DTSTAMP:20260404T094310Z 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 (p ossibly time-dependent) vector field $v$ on the Euclidean space. The class ical Cauchy-Lipschitz (also named Picard-Lindel\\"of) Theorem states that\ , if the vector field $v$ is Lipschitz in space\, for every initial datum $x$ there is a unique trajectory $\\gamma$ starting at $x$ at time $0$ and solving the ODE $\\dot{\\gamma} (t) = v (t\, \\gamma (t))$. The theorem l ooses its validity as soon as $v$ is slightly less regular. However\, if w e bundle all trajectories into a global map allowing $x$ to vary\, a celeb rated theory put forward by DiPerna and Lions in the 80es show that there is a unique such flow under very reasonable conditions and for much less r egular vector fields. A long-standing open question is whether this theory is the byproduct of a stronger classical result which ensures the uniquen ess of trajectories for {\\em almost every} initial datum. I will give a c omplete answer to the latter question and draw connections with partial di fferential equations\, harmonic analysis\, probability theory and Gromov's $h$-principle.\n LOCATION:https://stable.researchseminars.org/talk/IASanalysis/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Felix Otto (Max Planck Institute Leipzig) DTSTART:20200420T150000Z DTEND:20200420T160000Z DTSTAMP:20260404T094310Z UID:IASanalysis/2 DESCRIPTION:Title: A variational approach to the regularity theory for the Monge-A mpère equation\nby Felix Otto (Max Planck Institute Leipzig) as part of IAS Analysis Seminar\n\n\nAbstract\nWe present a purely variational app roach to the regularity theory for the\nMonge-Ampère equation\, or rathe r optimal transportation\, introduced with M. Goldman. Following De Giorgi ’s philosophy for the regularity theory of minimal surfaces\, it is base d on the approximation of the displacement by a harmonic gradient\, which leads to a One-Step Improvement Lemma\, and feeds into a Campanato iterati on on the $C^{1\,\\alpha}$-level for the displacement\, capitalizing on af fine invariance.\nOn the one hand\, this allows to reprove the $C^{1\,\\a lpha}$-regularity result (Figalli-Kim\, De Philippis-Figalli) bypassing Ca ffarelli’s celebrated theory. This also extends to boundary regularity ( Chen-Figalli)\, which is joint work with T. Miura.\nOn the other hand\, du e to its robustness\, it can be used as a large-scale regularity theory fo r the problem of matching the Lebesgue measure to the Poisson measure in t he thermodynamic limit. This is joint work with M. Goldman and M. Huesmann .\n LOCATION:https://stable.researchseminars.org/talk/IASanalysis/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Steven Zelditch (Northwestern University) DTSTART:20200428T150000Z DTEND:20200428T160000Z DTSTAMP:20260404T094310Z UID:IASanalysis/3 DESCRIPTION:Title: Ellipses of small eccentricity are determined by their Dirichle t (or\, Neumann) spectra\nby Steven Zelditch (Northwestern University) as part of IAS Analysis Seminar\n\n\nAbstract\nIn 1965\, M. Kac proved th at discs were uniquely determined by their Dirichlet (or\, Neumann) spectr a. Until recently\, disks were the only smooth plane domains known to be determined by their eigenvalues. Recently\, H. Hezari and I proved that el lipses of small eccentricity are also determined uniquely by their Dirichl et (or\, Neumann) spectra. The proof uses recent results of Avila\, de Sim oi\, and Kaloshin\, proving that nearly circular plane domains with ratio nally integrable billiards must be ellipses. It also uses a ``bounce deco mposition'' for the wave trace\, representing the wave trace as a sum of q -bounce oscillatory integrals. It is shown that for nearly circular domain s\, each is a spectral invariant and that the ellipse is uniquely determin ed by its q-bounce invariants.\n LOCATION:https://stable.researchseminars.org/talk/IASanalysis/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Zhiyuan Zhang (Université Paris 13) DTSTART:20200504T150000Z DTEND:20200504T160000Z DTSTAMP:20260404T094310Z UID:IASanalysis/4 DESCRIPTION:Title: Exponential mixing of 3D Anosov flows\nby Zhiyuan Zhang (Un iversité 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 mea sure with Hölder potential. This is a joint work with Masato Tsujii.\n LOCATION:https://stable.researchseminars.org/talk/IASanalysis/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Guy C. David (Ball State University) DTSTART:20200512T150000Z DTEND:20200512T160000Z DTSTAMP:20260404T094310Z 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\nAbstr act\nGiven a Lipschitz map\, it is often useful to chop the domain into pi eces on which the map has simple behavior. For example\, depending on the dimensions 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 man y issues\, it is even more useful if this decomposition is quantitative\, i.e.\, with bounds independent of the particular map or spaces involved. A fter surveying the question of bi-Lipschitz decomposition\, we will discus s the more complicated case in which dimension decreases\, e.g.\, for maps from $\\mathbb{R}^3$ to $\\mathbb{R}^2$. This is recent joint work with R aanan Schul\, improving a previous result of Azzam-Schul.\n LOCATION:https://stable.researchseminars.org/talk/IASanalysis/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Hong Wang (Institute for Advanced Study) DTSTART:20200518T150000Z DTEND:20200518T160000Z DTSTAMP:20260404T094310Z UID:IASanalysis/6 DESCRIPTION:Title: Square function estimate for the cone in R^3\nby Hong Wang (Institute for Advanced Study) as part of IAS Analysis Seminar\n\n\nAbstra ct\nWe prove a sharp square function estimate for the cone in R^3 and cons equently the local smoothing conjecture for the wave equation in 2+1 dimen sions. The proof uses induction on scales and an incidence estimate for po ints and tubes.\n\nThis is joint work with Larry Guth and Ruixiang Zhang.\ n LOCATION:https://stable.researchseminars.org/talk/IASanalysis/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Max Engelstein (University of Minnesota) DTSTART:20200601T150000Z DTEND:20200601T160000Z DTSTAMP:20260404T094310Z UID:IASanalysis/7 DESCRIPTION:Title: Winding for Wave Maps\nby Max Engelstein (University of Min nesota) as part of IAS Analysis Seminar\n\n\nAbstract\nWave maps are harmo nic maps from a Lorentzian domain to a Riemannian target. Like solutions t o many energy critical PDE\, wave maps can develop singularities where the energy concentrates on arbitrary small scales but the norm stays bounded. Zooming in on these singularities yields a harmonic map (called a soliton or bubble) in the weak limit. One fundamental question is whether this we ak limit 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 uniqueness may not hold if the target manifold is not analytic. Our construction is heavily inspired by Peter Topping's analogous example of a ``winding" bubble in harmonic map heat flow. However\, the Hamiltonian na ture of the wave maps will occasionally necessitate different arguments. This is joint work with Dana Mendelson (U Chicago).\n LOCATION:https://stable.researchseminars.org/talk/IASanalysis/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Alik Mazel (AMC Health) DTSTART:20200525T150000Z DTEND:20200525T160000Z DTSTAMP:20260404T094310Z 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 Analysis Seminar\n\n\nAbstract\nThe construction of pure phases from grou nd states is performed for $ u > u_*(d)$ for all values of $d$ except for 39 special ones. For values $d$ with a single equivalence class all period ic ground states generate the corresponding pure phase which provides a co mplete description of extreme Gibbs measures (complete phase diagram). For a general $d$ we prove that at least one class of ground states generates pure phases and propose an algorithm that decides\, after finitely many i terations\, which classes of ground states generate pure phases. We conjec ture that in case of several classes only one of them generates pure phase s which is confirmed by (numerical) application of our algorithm to severa l (relatively small) values of $d$.\n LOCATION:https://stable.researchseminars.org/talk/IASanalysis/8/ END:VEVENT END:VCALENDAR