Optimal Transport: A Semi-Discrete Approach

Abstract We prove the validity of the ε - ε β {\\varepsilon-\\varepsilon^{\\beta}} property in the isoperimetric problem with double density, generalising the known properties for the case of single density. As a consequence, we derive regularity for isoperimetric sets.

The ε - εβ Property in the Isoperimetric Problem with Double Density, and the Regularity of Isoperimetric Sets

We prove the validity of the $\varepsilon-\varepsilon^\beta$ property in the isoperimetric problem with double density, generalising the known properties for the case of single density. As a consequence, we derive regularity for isoperimetric sets.

The $\varepsilon-\varepsilon^\beta$ property in the isoperimetric problem with double density, and the regularity of isoperimetric sets

We solve a class of isoperimetric problems on $\mathbb{R}^N $ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\in [0,1]$, then among all smooth sets $\Omega$ in $\mathbb{R} ^N$ with fixed Lebesgue measure, $\int_{\partial \Omega } |x|^k \, \mathscr{H}_{N-1} (dx)$ achieves its minimum for a ball centered at the origin. Our results also imply a weighted Polya-Szego principle. In turn, we establish radiality of optimizers in some Caffarelli-Kohn-Nirenberg inequalities, and we obtain sharp bounds for eigenvalues of some nonlinear problems.

Some isoperimetric inequalities on $\mathbb{R} ^N$ with respect to weights $|x|^\alpha $

In this paper the author studies the isoperimetric problem in $\re^n$ with perimeter density $|x|^p$ and volume density $1.$ We settle completely the case $n=2,$ completing a previous work by the author: we characterize the case of equality if $0\leq p\leq 1$ and deal with the case $-\infty<p<-1$ (with the additional assumption $0\in\Omega$). In the case $n\geq 3$ we deal mainly with the case $-\infty<p<0,$ showing among others that the results in $2$ dimensions do not generalize for the range $-n+1<p<0.$

On the isoperimetric problem with perimeter density r^p

We show that the unique isoperimetric hypersurfaces in $\mathbb{R}^n$ with density $r^p$ for $n \ge 3$ and $p>0$ are spheres that pass through the origin.

Isoperimetric Regions in $\mathbb{R}^n$ with density $r^p$

The Convex Body Isoperimetric Conjecture states that the least perimeter needed to enclose a volume within a ball is greater than the least perimeter needed to enclose the same volume within any other convex body of the same volume in Rn. We focus on the conjecture in the plane and prove a new sharp lower bound for the isoperimetric profile of the disk in this case. We prove the conjecture in the case of regular polygons and show that in a general planar convex body, the conjecture holds for small areas. Acknowledgements: This paper is the work of the Williams College NSF “SMALL” 2013, 2014, and 2015 Geometry Groups. We thank our advisor Professor Frank Morgan for his support. We would like to thank the National Science Foundation, Williams College, and the MAA for supporting the “SMALL” REU and our travels to MathFest 2013, MathFest 2014, and MathFest 2015. Page 10 RHIT Undergrad. Math. J., Vol. 18, No. 2

/pdf/the-convex-body-isoperimetric-conjecture-in-the-plane-3535x2oz7v.pdf

The Convex Body Isoperimetric Conjecture in the Plane

We consider optimal transportation with constraint, as did Korman and McCann (2013, 2015), provide simplifications and generalizations of their examples and results, and provide some new examples and results.

https://msp.org/involve/2019/12-1/involve-v12-n1-p01-s.pdf

Optimal transportation with constant constraint

https://msp.org/involve/2018/11-2/involve-v11-n2-p03-s.pdf

Double bubbles in hyperbolic surfaces

: We discuss incidence estimates for neighborhoods of hyperplanes in R n , after the work of Guth, Solomon, and Wang, who proved an analogue of the Szemer´edi-Trotter theorem to estimate incidences of tubes. We use induction on scales and the high-low method of Vinh along with new geometric insights. Abstract : We will talk about singularity formation for the 3D isentropic compressible Euler and Navier-Stokes equations for ideal gases. These equations describe the motion of a compressible ideal gas, which is characterized by a parameter called adiabatic constant. Finite time singularities for generic adiabatic constants were found in the recent work of Merle, Rapha¨el, Rodnianski and Szeftel. This is done via a stability analysis around the smooth self-similar proﬁles for generic adiabatic constants. We will drop the genericity assumption and construct smooth self-similar proﬁles for all values of the adiabatic constant. In particular, we will construct the ﬁrst smooth self-similar proﬁle for a monoatomic gas. We also present a diﬀerent stability analysis around those proﬁles that allows to show singularity formation for initial data with constant density at inﬁnity. These results are joint work with Tristan Buckmaster and Javier Gomez-Serrano. Abstract : In this talk, we introduce the concept of sparse domination and discuss the work of Lacey in the sparse domination for the Spherical Maximal Function. We will then talk about some attempts of bringing sparse domination to the world of multilinear Radon type operators and a joint work with B. Foster, Y. Ou, J. Pipher, and Z. Zou, in which we proved sparse domination results for a bilinear generalization of the spherical maximal function. Namely, the maximal operator given by in any dimension d ≥ 2. Such sparse domination allows one to recover the known sharp L p × L q → L r bounds for the operator and to deduce new quanti-tative weighted norm inequalities with respect to bilinear Muckenhoupt weights. The key innovation is a group of newly developed continuity L p improving estimates for the localized version of the bilinear spherical maximal function. Andrew Abstract : I will present a joint work with Jacek Jendrej (CRNS, Sorbonne Paris Nord) on equivariant wave maps with values in the two-sphere. We prove that every ﬁnite energy solution resolves, as time passes, into a superposition of harmonic maps (solitons) and radiation, settling the soliton resolution problem for this equation. It was proved in works of Cˆote, and Jia-Kenig, that such a decomposition holds along a sequence of times. We show the resolution holds continuously-in-time via a “no-return” lemma based on the virial identity. The proof combines a modulation analysis of solutions near a multi-soliton con-ﬁguration with concentration compactness techniques. As a byproduct of our analysis we prove that there are no pure multi-solitons in equivariance class k=1 and no elastic collisions between pure multi-solitons in the higher equivariance classes. Abstract : In this talk we will sketch the proof of the unique continuation property of harmonic functions and caloric functions on any RCD ( k, 2) spaces and a counterexample for the strong unique continuation property of harmonic functions on an RCD ( k, 4) space. This characterizes one of the signiﬁcant diﬀerences between RCD spaces and smooth manifolds. We will also talk about some related open problems. The talk is based on joint works with Qin Deng. Abstract : The topic of this course is Pollicott-Ruelle resonances for Anosov ﬂows, which are strongly chaotic systems; an example of an Anosov ﬂow is the geodesic ﬂow on a compact negatively curved Riemannian manifold. Pollicott-Ruelle resonances appear in two ways: as complex characteristic frequencies in long time asymptotics of correlations and as zeros and poles of dynamical zeta functions deﬁned from the lengths of closed trajectories. I will give a ”big picture” overview of the microlocal approach to Pollicott-Ruelle resonances, focusing on general ideas rather than the details of the proofs. In particular I will discuss ergodicity, exponential mixing, and relation of dynamical zeta functions to topology. Abstract : In this talk, I will talk about the (geometric) intersection number between closed geodesics on ﬁnite volume hyperbolic surfaces. Speciﬁcally, I will discuss the optimum upper bound on the intersection number in terms of the product of hyperbolic lengths. I also talk about the equidistribution of the intersection points between closed geodesics. Abstract : Given a spin rational homology sphere Y equipped with a Z /m action preserving the spin structure, I will outline how to deﬁne equivariant reﬁnements of Manolescu’s kappa invariant. These invariants give rise to equivariant relative 10/8-ths type inequalities for Z /m -equivariant spin cobordisms between rational homology spheres. I will explain how these inequalities provide applications to knot concordance, give obstructions to extending cyclic group actions to spin ﬁllings, and via taking branched covers give us genus bounds for knots in punctured 4-manifolds. If time permits I will explain how these invariants are related to equivariant Abstract : Quiver has a rich representation theory. There are gorgeous connec-tions between quiver representations and sheaves. In this talk, we will introduce the quiver algebroid stack and an associated functor from a symplectic man-ifold, which enables us to understand these deep relations in terms of Mirror Symmetry. We will also introduce some interesting applications of this con-struction.

Sarah Tammen

Papers

Isoperimetric Regions in $\mathbb{R}^n$ with density $r^p$

The Convex Body Isoperimetric Conjecture in the Plane

Optimal transportation with constant constraint

Double bubbles in hyperbolic surfaces

The soliton resolution conjecture for equivariant wave