Abstract: I will talk about the new construction of genus-zero free boundary minimal surfaces embedded in the unit ball in the Euclidean three-space which are compact and lie arbitrarily close to the boundary unit sphere with an arbitrarily large number of connected boundary components. The construction is by PDE gluing methods and the surfaces are desingularizations of unions of many catenoidal annuli and two flat discs. The talk is based on the joint walk with Professor Nicolaos Kapouleas.

https://sites.google.com/view/zoujiahua/home

Abstract: Continuous dependence on data is an important feature for solutions that model physical phenomena. Measurements of from physical experiments can never be exact, and so the goal is to provide a (narrow) range of values for the solutions. The exact dependence can be studied with analysis and partial differential equation tools and serve engineers, computational scientists, and also be important in other theoretical studies.

In this talk, we will discuss these aspects in the nonlocal framework. A main motivation for the increased interest in nonlocal models is their flexibility in handling discontinuities. We will show several results on the continuous dependence (or stability) of the solution with respect to changes in (linear and nonlinear) forcing terms, boundary or collar data, as well as with respect to the kernel of interaction of the nonlocal operator. We will close with some numerical results and extensions of the results.

https://www.wpi.edu/people/faculty/nbuczkowski

Abstract: Drawing upon methods from the field of nonlinear dispersive PDEs, we investigate well/ill-posedness for the 3D Boltzmann equation with constant collision kernel in the Sobolev spaces. We find that the threshold space is one-half derivative above the scale invariant space, and prove ill-posedness below this threshold by constructing a family of special solutions, which are neither near equilibrium nor self-similar, and exhibit a "norm deflation" behavior -- a rapid drop in the Sobolev norm that breaks the uniform continuity of the data-to-solution map. This is joint work with Xuwen Chen (University of Rochester)

https://www.math.brown.edu/jholmer/

Abstract: I will first give a brief overview of how in Cheeger-Colding theory, the Green function for the Laplacian can be used to describe the convergence of a complete Ricci-flat manifold with Euclidean volume growth to its tangent cone. When a tangent cone of the manifold has smooth cross section, Colding-Minicozzi proved a Łojasiewicz-Simon inequality which implies uniqueness of the tangent cone. As another application of their Łojasiewicz-Simon inequality, I will describe how one can identify two arbitrarily far apart scales in the manifold in a natural way. I will also discuss some Li-Yau-Hamilton estimates on Riemannian or Kähler manifolds when there is an additional assumption on sectional curvature. These inequalities are elliptic analogues of works by Hamilton and Cao-Ni for heat equation and geometric flows.

https://math.yale.edu/people/jiewon-park

Abstract: Anisotropic area, a generalization of the area functional, arises naturally in models of crystal surfaces. Due to the lack of a monotonicity formula, the regularity theory for its critical points, anisotropic minimal surfaces, is much more challenging than the area functional case. In this talk, I will discuss how one can overcome this difficulty and obtain a smooth anisotropic minimal surface for elliptic integrands in closed 3-dimensional Riemannian manifolds via min-max construction. This confirms a conjecture by Allard [Invent. Math.,1983] in dimension 3. The talk is based on joint work with Guido De Philippis and Antonio De Rosa.

https://mathematics.uchicago.edu/people/profile/yangyang-li/

Abstract: The space of all square integrable holomorphic functions on a domain in the complex plane was first systematically studied by Bergman in the 1920s. Since that time, the program to study these spaces, called Bergman spaces, has expanded to interact with many areas of complex analysis, operator theory, complex geometry and algebraic geometry. The most recent addition to the program involves studying how these Bergman spaces change when the structures defining these spaces are deformed. In this talk I will discuss my some of the major results in the area, and a new contribution I have made to it.

https://www.math.stonybrook.edu/cards/varolindror.html

Abstract: Mark Kac’s question “Can One Hear the Shape of a Drum?” asks whether one can determine the shape of a planar domain from the eigenvalues of the Laplace operator with the Dirichlet boundary condition. In this expository talk, we shall study the several complex variables analogue of Mark Kac’s question: To what extent can one determine geometric properties of a domain in several complex variables from spectral properties of the complex Neumann Laplacian? We will discuss results and open problems in both real and complex settings. We will also present relevant results on stability of the spectrum of the complex Laplacian.

https://siqifu.camden.rutgers.edu

Abstract: In this talk I will make an overview of some recent advances in applying differential and generalized geometry to exhibiting intrinsic properties of equations describing mechanical systems. Main examples are given by systems with constraints and coupled systems. From the geometric point of view, they correspond to so-called (almost) Dirac structures: those generalize simultaneously symplectic and Poisson manifolds. We will also address the question in the other direction: given a Dirac structure, how to define meaningful dynamics on it. I will conclude by mentioning some consequences of these results for construction of geometric integrators - numerical methods preserving the geometric structure of the equations and qualitative properties of respective mechanical systems.

The talk is based on:

[1] V.Salnikov, A.Hamdouni, D.Loziienko, Generalized and graded geometry for mechanics: a comprehensive introduction, Mathematics and Mechanics of Complex Systems, Vol. 9, No. 1, 2021

[2] O.Cosserat, C.Laurent-Gengoux, A.Kotov, L.Ryvkin, V.Salnikov, On Dirac structures admitting a variational approach, Mathematics and Mechanics of Complex Systems, 2023

[3] V.Salnikov, A.Hamdouni, From modelling of systems with constraints to generalized geometry and back to numerics, Z Angew Math Mech., Vol. 99, Issue 6, 2019

[4] D.Loziienko, V.Salnikov, A. Hamdouni, Construction of Pseudo-Geometric Integrators, Programming and Computing Software, Vol. 48, Issue 2, 2022

https://www.vladimir-salnikov.org

Abstract: Ricci flow has been extensively studied, and most results are either specific to dimension 3 or valid in any dimension. However, given the potential topological applications, a theory specific to the 4-dimensional situation is desirable. In this discussion, I will present tools and techniques that are unique to the dimension 4.

Together with A. Deruelle, we introduce a notion of stability for orbifold singularities. This notion helps to explain the formation of orbifold singularities along Ricci flow and lets us construct large classes of ancient Ricci flows. In collaboration with K. Naff, we utilize self-duality in dimension 4 to simplify the evolution equations of curvature. Among other things, this approach lets us to uncover intriguing links between Ricci flow and Yang-Mills flow.

https://math.mit.edu/directory/profile?pid=2272

Abstract: The classical Aviles-Giga functional is a second order energy functional that models phenomena from thin film blistering to smectic liquid crystals. In this talk we will discuss a generalized Aviles-Giga functional in 2D that was first studied by Bochard and Pegon. The zero set of the potential in this generalized functional is a strictly convex $C^1$ submanifold with certain symmetry, instead of $\mathbb{S}^1$ in the classical case. Among other things, we generalize the notion of entropies, and use this tool to establish compactness for sequences with bounded energy and regularity for finite energy states. Our regularity result is also closely related to regularity of finite entropy solutions for a class of scalar conservation laws in 1D. This is joint work with Xavier Lamy and Andrew Lorent.

https://www.wpi.edu/people/faculty/gpeng

https://www.math.columbia.edu/~johne/

https://nms.kcl.ac.uk/zoe.wyatt/