(Princeton) 
Stable shock wave formation for the isentropic compressible Euler equations.  I will talk about recent work with Steve Shkoller, and Vlad Vicol, regarding shock wave formation for the isentropic compressible Euler equations.  
(Harvard) 
The Penrose inequality and the hoop conjecture  The Penrose Inequality is a conjecture from 1972 that aims to refine the Positive Mass Theorem, which was proven independently by SchoenYau and Witten in 197981 and which is a fundamental result in the study of scalar curvature. A special case of the inequality ('the Riemannian case') was proven, independently, by HuiskenIlmanen and Bray in 2001 by developing, respectively, a weak inverse mean curvature flow and a novel conformal flow of metrics. The Hoop Conjecture is a (deliberately) vaguely formulated attempt by physicists to capture the physical idea that sufficiently dense concentrations of curvature herald the presence of black holes. The Penrose Inequality and Hoop Conjecture turn out to be related, and in this talk I will present the most recent results in both directions. Both results make fundamental use of the weak inverse mean curvature flow of HuiskenIlmanen, and the WangYau quasilocal mass (which is related to the problem of isometric embeddings into Euclidean space). The work is joint with S.T.Yau and A.Alaee.  
(Duke) 
Small scale creation in solutions to modified SQG equations  The modified SQG equations have similarities both to 2D and 3D Euler equations. The question of global regularity vs finite time blow up remains open for smooth initial data, for the entire range of BiotSavart law parameter 0< alpha<1. For patch solutions in halfplane, an example of finite time singularity has been constructed by Ryzhik, Yao, Zlatos and myself in the case of small alpha. However, for smooth initial data there have been no examples of any infinite in time growth. In this talk I will review the history of the problem and describe a recent construction by Siming He and myself yielding solutions with exponential in time growth of derivatives.  
(Univ. Bicocca, Milan) 
Liouville type theorems and local regularity for degenerate or singular problems  We consider a class of equations in divergence form with a weight which is singular or degenerate on a hyperplane, the characteristic manifold. We study local regularity of solutions, showing how boundary conditions on the characteristic manifold do affect such regularity. Our analysis relies in uniform bounds in Holder spaces for solutions to a class of uniformly elliptic approximating problems as the parameter epsilon of regularization tends to zero. Our method is based upon blowup and appropriate Liouville type theorems. This is a joint project with Y. Sire and S. Terracini.  
(UNC) 
Random Waves and the Spectral Function on Manifolds without Conjugate Points  In this talk, we discuss offdiagonal Weyl asymptotics on a compact manifold M, with the goal of understanding the statistical properties of monochromatic random waves. These waves can be thought of as randomized "approximate eigenfunctions," and their statistics are completely determined by an associated covariance kernel which coincides exactly with a rescaled version of the spectral function of the LaplaceBeltrami operator. We prove that in the geometric setting of manifolds without conjugate points, one can obtain a logarithmic improvement in the twopoint Weyl law for this spectral function, provided one restricts to a small enough neighborhood of the diagonal in M x M. This then implies that the covariance kernel of a monochromatic random wave locally converges to a universal scaling limit at a logarithmic rate as we take the frequency parameter to infinity. This result generalizes the work of Berard, who obtained the logarithmic improvement in the ondiagonal case for manifolds with nonpositive curvature.  
(Princeton) 
A Bound on the Cohomology of Quasiregularly Elliptic Manifolds  A classical result gives that if there exists a holomorphic mapping f: C > M, then M is homeomorphic to S^2 or S^1 x S^1, where M is a compact Riemann surface. I will discuss a generalization of this problem to higher dimensions. I will show that if M is an ndimensional, closed, connected, orientable Riemannian manifold that admits a quasiregular mapping from R^n, then the dimension of the degree l de Rham cohomology of M is bounded above by n choose I. This is a sharp upper bound that proves a conjecture by Bonk and Heinonen. A corollary of this theorem answers an open problem posed by Gromov. He asked whether there exists a simply connected manifold that does not admit a quasiregular map from R^n. The result gives an affirmative answer to this question.  
(Helsinki) 
Imaginary multiplicative chaos, Onsager inequalities, and Sobolevregularity  We recall the notion of Gaussian multiplicative chaos, especially the imaginary one. Our aim is to present some basic regularity properties for the chaos, and its uniqueness, which leads to revisiting Onsager type inequalities. Join work with Janne Junnila (EPFL) and Christian Webb (Aalto University).  


(NYU) 
TBA  TBA  
(Toronto) 
Scalar curvature and harmonic functions  We'll discuss a new technique for relating scalar curvature bounds to the global structure of 3dimensional manifolds, exploiting a relationship between scalar curvature and the topology of level sets of harmonic functions. We will describe several geometric applications in both the compact and asymptotically flat settings, including a simple and effective new proof (joint with Bray, Kazaras, and Khuri) of the threedimensional Riemannian positive mass theorem.  
(LSU) 

(Texas A&M) 

4pm eastern itme, Meeting ID: 865 257 351, Password: 761520 
(UConn) 
Entire spacelike constant $\sigma_{n1}$ curvature in Minkowski space  We prove that, in the Minkowski space, if a spacelike, (n ā 1)convex hypersurface M with constant $\sigma_{nā1}$ curvature has bounded principal curvatures, then M is convex. Moreover, if M is not strictly convex, after an R^{n,1} rigid motion, M splits as a product $M^{nā1}\times R.$ We also construct nontrivial examples of strictly convex, spacelike hypersurface M with constant $\sigma_{nā1}$ curvature and bounded principal curvatures. This is a joint work with Changyu Ren and Zhizhang Wang. 
(Chicago) 

Meeting ID: 92836386241 
(OSU) 
Fully nonlinear elliptic equations for conformal deformation of ChernRicci curvatures  There are several ways to define ChernRicci curvatures for the Chern connection on a nonKahler Hermitian manifold. We introduce a notion of mixedChernRicci forms, which naturally occur in geometric problems and seem interesting to study, and consider fully nonlinear elliptic equations for their conformal deformation. We establish a priori estimates and prove existence results for these equations under some general structure conditions. Our work is motivated by the close connections of these equations to problems in nonKahler complex geometry, and the fact that there have been increasing interests in fully nonlinear pde's beyond the complex MongeAmpere equation from geometry and complex analysis. This talk is based in part on work with Chunhui Qiu and Rirong Yuan. 
(TBD) 

(University of Edinburgh) 
TBA  TBA 
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