Richard Evan Schwartz’s research while affiliated with Brown University and other places

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Publications (82)


Figure 2: The quantities used in the proof.
Figure 4: The neighborhood A of ▽ 0 .
Figure 5: Modified triangular Moebius band.
On Nearly Optimal Paper Moebius Bands
  • Preprint
  • File available

November 2024

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5 Reads

Richard Evan Schwartz

Let ϵ<1/384\epsilon<1/384 and let Ω\Omega be a smooth embedded paper Moebius band of aspect ratio less than 3+ϵ\sqrt 3 + \epsilon. We prove that Ω\Omega is within Hausdorff distance 18ϵ18 \sqrt \epsilon of an equilateral triangle of perimeter 232 \sqrt 3. This is an effective and fairly sharp version of our recent theorems in [{\bf S0\/}] about the optimal paper Moebius band.

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Pentagram Rigidity for Centrally Symmetric Octagons

April 2024

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5 Reads

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2 Citations

International Mathematics Research Notices

In this paper I will establish a special case of a conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. The special case is that of the 3-diagonal map acting on affine equivalence classes of centrally symmetric octagons. The proof involves establishing that the map is Arnold-Liouville integrable in this case, and then exploring the Lagrangian surface foliation in detail.




The Optimal Paper Moebius Band

August 2023

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1,125 Reads

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1 Citation

In this paper we prove that a smooth embedded paper Moebius band must have aspect ratio greater than 3\sqrt 3. We also prove that any sequence of smooth embedded paper Moebius bands whose aspect ratio converges to 3\sqrt 3 must converge, up to isometry, to an equilateral triangle of semi-perimeter 3\sqrt 3. These results resolve the optimal paper Moebius band conjecture of Halpern and Weaver from 1977.





Figure 4: The orbit of a hexagon projected on the (A, C)-and the (B, C)-planes, respectively.
On Projective Evolutes of Polygons

July 2022

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51 Reads

Experimental Mathematics

The evolute of a curve is the envelope of its normals. In this note we consider a projectively natural discrete analog of this construction: we define projective perpendicular bisectors of the sides of a polygon in the projective plane, and study the map that sends a polygon to the new polygon formed by the projective perpendicular bisectors of its sides. We consider this map acting on the moduli space of projective polygons. We analyze the case of pentagons; the moduli space is two-dimensional in this case. The second iteration of the map has one integral whose level curves are cubic curves, and the transformation on these level curves is conjugated to the map x↦−4x mod 1. We also present the results of an experimental study in the case of hexagons.


Citations (40)


... If the curvature, at time t = 0, along the curve is strictly increasing between the point of self-intersection and the endpoint A where the tangent is vertical, then this remains true for all t ∈ [0, T). Coiculescu and Schwartz [8] recently showed for such curves that if one re-scales the shrinking curve so that its bounding box becomes the unit square, then right before the singular time the resulting curve converges to a 'bow-tie. ' In a type-II blow-up, one zooms in on the curve at the point A of maximal curvature k max (t) and magnifies the curve by a factor k max (t). ...

Reference:

Which shapes can appear in a curve shortening flow singularity?
The affine shape of a figure 8 under the curve shortening flow
  • Citing Article
  • July 2024

Journal of Differential Geometry

... See [KS13], [MB13], [MB14], [KS16], [IK23]. The map T k has many applications and connections to other fields, such as octahedral recurrence [Sch08] [FK12], the condensation method of computing determinants [Sch08] [ Gli18], cluster algebra [Gli11] [Gek+12] [FK12], Poisson Lie groups [FM16] [Izo22a], Tsystems [KV15] [FK12], Grassmannians [FMB19], algebraically closed fields [Wei23], Poncelet polygons [Sch07] [Sch21] [Izo22b] [Sch24a], and integrable partial differential equations [Sch08] [OST10] [NS21]. ...

Pentagram Rigidity for Centrally Symmetric Octagons
  • Citing Article
  • April 2024

International Mathematics Research Notices

... In his ICM 2022 address [Sch21], R. Schwartz stated Conjecture 6.5, which reads: "outer billiard on the regular n-gon has an aperiodic orbit if n ̸ = 3, 4, 6". This paper settles the conjecture when n is not divisible by 4. Previously, E. Gutkin and N. Simanyi [GS92] posed a more general question: "Let P be a quasi-rational but not rational polygon. ...

Survey lecture on billiards
  • Citing Chapter
  • December 2023

... When in its ground state, Reference [5] proposes that proton charge and mass are coupled and continually regenerate each other. For each proton charge arc, this coupling may be represented in the form of a virtual optimal Möbius band [5,[19][20][21][22]. Reference [5] proposes that this implies the geometry of a GSQV proton is optimal, which may explain why free or chemically bound protons do not decay. ...

The Optimal Paper Moebius Band

... Interestingly, Toeplitz's statement from 1911 that every Jordan curve admits an inscribed square is still a conjecture in the general case. Just recently, it was proved for convex or piecewise smooth curves, while extensions exist for rectangles, curves, and Klein bottles (see, e.g., [5,6]). The triangle is the simplest example of a non-smooth and piecewise linear Jordan curve; while the equilateral triangle appears to be a simple configuration, it can generate very interesting properties and applications [7]. ...

Rectangles, curves, and Klein bottles
  • Citing Article
  • September 2021

Bulletin of the American Mathematical Society

... See [CL], [HM], [Mas] for arguments which immediately work in case Ω has an open dense set of points with nonzero mean curvature. See [S0,Prop. 2.1] for the general case and more precise references. The basic idea is that each point on Ω of nonzero curvature has a unique tangent direction where the differential of the Gauss map is trivial. ...

An improved bound on the optimal paper Moebius band

Geometriae Dedicata

... Compactified configuration spaces have also been used by S.T. Vrećica and R.T.Živaljević, T. Rade [50] in their paper looking at the polygonal peg problem (inscribed affine regular hexagons in smooth Jordan curves, and inscribed parallelograms in smooth simple closed curves in R 3 ). There have also been many papers [1,3,14,15,21,22,26,31,41,40] examining quadrilaterals inscribed in curves and, more recently, making progress towards solving the rectangular-peg problem (finding rectangles of any aspect ratio inscribed in Jordan curves). ...

Inscribed rectangle coincidences
  • Citing Article
  • July 2021

Advances in Geometry

... There are not many concrete examples of the farthest point map on convex polytopes, especially in higher dimensions. Farthest point map on the regular tetrahedron is studied in [8], flat surfaces in [10], regular octahedron in [12], regular dodecahedron in [11], centrally symmetric polyhedron in [16], doubly covered simplex and convex polyhedron in [5], and doubly covered parallelotope in [13]. ...

The Farthest Point Map on the Regular Octahedron
  • Citing Article
  • April 2021

Experimental Mathematics

... The non-standard generating function for convex billiards has been already used in our paper [14], explaining conservation laws for elliptical billiards discovered recently by Dan Reznik [15,16] et al., see also [11,12,17]. Additionally, the non-standard generating function is a key ingredient in the recent proof of a part of Birkhoff conjecture for centrally symmetric billiard tables [18]. ...

Billiards in ellipses revisited

European Journal of Mathematics

... Except for results in Section 6, the present article is independent from [6]. Although it has no bearing on the methods or results of the present paper, we mention that the present paper grew out of our interest in planar arrangement theorems involving rectangles inscribed in quadrilaterals; see for example [3,4,5,8,9,10]. In the cited articles, various aspects of the flow of inscribed rectangles through complete quadrilaterals are studied. Affine transformations of these objects result in flows of inscribed parallelograms through complete quadrilaterals. ...

Four Lines and a Rectangle
  • Citing Article
  • March 2020

Experimental Mathematics