Sergei Tabachnikov

Professor of Mathematics, Penn State

tabachni@math.psu.edu


                                                                                                               Papers

Invariant differential operators in general position. (Russian) Funktsional. Anal. i Prilozhen. 16 (1982), no. 3, 86-87

Characteristic classes of homogeneous foliations. (Russian) Uspekhi Mat. Nauk 39 (1984), no. 2(236), 189-190

Characteristic classes of Grassmann foliations. (Russian) Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 83-84

Characteristic classes of parabolic foliations and symmetric functions. Serdica, 11 (1985), 86-95Characteristic classes of parabolic foliations of series B, C, and D and degrees of isotropic Grassmannians. (Russian) Funktsional. Anal. i Prilozhen. 20 (1986), no. 2, 84-85

An invariant of submanifolds transversal to a distribution. Russ. Math. Surv., 43, No 3 (1988), 193-194

Calculation of the Bennequin invariant of a Legendrian curve by the geometry of its front. Func. Anal. Appl., 22, No 3 (1988), 89-90

Characteristic classes of Lagrangian foliations. Func. Anal. Appl., 23, No 2 (1989), 90-91

Around four vertices. Russ. Math. Surv., 45, No 1 (1990), 191-192

Two remarks on the asymptotic Hopf invariant. Func. Anal. Appl., 24, No 1 (1990), 84-85

Geometry of Lagrangian and Legendrian 2-webs. Diff. Geom. Appl., 3 (1993) 265--284

Poncelet's theorem and dual billiards. L'Enseign. Math., 39 (1993), 189-194

Outer billiards. Russ. Math. Surv., 48, No 6 (1993), 75-102

Commuting dual billiard maps. Geom. Dedicata, 53 (1994), 57--68

A cone eversion. Amer. Math. Monthly, 102 (1995), 52-56

On the dual billiard problem. Adv. in Math. 115 (1995), 221-249

The four vertex theorem revisited - two variations on the old theme. Amer. Math. Monthly, 102 (1995), 912-916

Invariants of smooth triple point free plane curves. Knot Theory and Ramifications, 5 (1996), 531-552

Asymptotic dynamics of the dual billiard transformation. J. Stat. Phys., 83 (1996), 27-38

Projective connections, group Vey cocycle and deformation quantization. Int. Math. Research Notes. 1996, No 14, 705-722

Sturm theory, Ghys theorem on zeroes of the Schwarzian derivative and flattening of Legendrian curves, with V. Ovsienko. Selecta Math. (NS), 2 (1996), 297-307

On zeroes of the Schwarzian derivative. Topics in singularity theory, 229-239, AMS Transl., ser. 2, v. 180, 1997

Introducing projective billiards. Ergod. Theory and Dynam. Syst., 17 (1997), 957-976

Exact transverse line fields and projective billiards in a ball. Geom. and Funct. Anal., 7 (1997), 594-608

Parameterized curves, Minkowski caustics, Minkowski vertices and conservative line fields. L'Enseign. Math., 43 (1997), 3-26

Invariants of Legendrian and transverse knots in the standard contact space, with D. Fuchs. Topology 36 (1997), 1025-1053

Estimates for the Bennequin number of Legendrian links from state models for knot polynomials. Math. Research Lett., 4 (1997), 143--156

On functions with zero mean over a finite group. Func. Anal. Appl., 31, No 1 (1997), 93-94

Fagnano orbits of polygonal dual billiards. Geom. Dedicata, 77 (1999), 279-286

Geometry of exact transverse line fields and projective billiards. Differential and symplectic topology of knots and curves, 131-152, AMS Transl., ser. 2, v. 190, 1999

Remarks on the geometry of exact transverse line fields. Differential topology, infinite-dimensional Lie algebras and applications, 247-260, AMS Transl., ser. 2, v. 194, 1999

Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid. Comm. Math. Helv., 74 (1999), 306-321

More on paper folding, with D. Fuchs. Amer. Math. Monthly, 106 (1999), 27-35

Going in circles: variations on the Money-Coutts theorem. Geom. Dedicata, 80 (2000), 201-209

A four vertex theorem for polygons. Amer. Math. Monthly, 107 (2000), 830-833

Projective geometry of polygons and discrete 4-vertex and 6-vertex theorems, with V. Ovsienko. L'Enseign. Math., 47 (2001), 3-19

Topology of cyclic configuration spaces and periodic orbits of multi-dimensional billiards, with M. Farber. Topology, 41 (2002), 553-589

Periodic trajectories in 3-dimensional convex billiards, with M. Farber. Manuscripta Math., 108, (2002), 431-437

Billiards in Finsler and Minkowski geometries, with E. Gutkin. J. Geom. and Phys., 40 (2002), 277-301

Ellipsoids, complete integrability and hyperbolic geometry. Moscow Math. J., 2 (2002), 185-198

Dual billiards in the hyperbolic plane. Nonlinearity, 15 (2002), 1051-1072

Rational billiards and flat structures, with H. Masur. Handbook of Dynamical Systems, v. 1A, North-Holland, 2002, 1015--1089

On skew loops, skew branes and quadratic hypersurfaces. Moscow Math. J., 3 (2003), 681-690

On polygonal dual billiard in the hyperbolic plane, with F. Dogru. Reg. Chaotic Dynamics, 8 (2003), 67-82

A tale of a geometric inequality. MASS Selecta, AMS, 2003, 257-262

Topological robotics: motion planning in projective spaces, with M. Farber and S. Yuzvinsky. Int. Math. Research Notes. 2003, No 34, 1853-1870

On three-periodic trajectories of multi-dimensional dual billiards. Alg. Geom. Topology, 3 (2003), 993-1004

Remarks on magnetic flows and magnetic billiards, Finsler metrics and a magnetic analog of Hilbert's fourth problem. Dynamical systems and related topics, Cambridge Univ. Press, 2004, 233-252

Dual billiards, with F. Dogru. Math. Intelligencer, 27, No 4 (2005), 18-25

Tire track geometry: variations on a theme. Israel J. Math., 151 (2006), 1-28

Non-existence of n-dimensional T-embedded discs in R^(2n), with G. Stojanovic . Comm. Math. Helv., 81 (2006), 877-882

Complexity of piecewise convex transformations in two dimensions, with applications to polygonal billiards on surfaces of constant curvature, with E. Gutkin. Moscow Math. J., 6 (2006), 673-701

On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards, with D. Genin. Journal of Modern Dynamics 1 (2007), 155-173

Book review of Arnold's Problems. Math. Intelligencer, 29, No 1 (2007), 49-52

The Poncelet grid and the billiard in an ellipse, with M. Levi. Amer. Math. Monthly, 114 (2007), 895-908

Hyperbolic Caratheodory conjecture, with V. Ovsienko. Proc. of Steklov Inst., 258 (2007), 178-193

Geodesics on an ellipsoid in Minkowski space, with D. Genin and B. Khesin. L'Enseign. Math., 53 (2007), 307-331

A proof of Culter's theorem on the existence of periodic orbits in polygonal outer billiards. Geom. Dedicata, 129 (2007), 83-87

Totally skew embeddings of manifolds, with M. Ghomi. Math. Z., 258 (2008), 499-512

On algebraically integrable outer billiards. Pacific J. Math., 235 (2008), 89-92

Converse Sturm-Hurwitz-Kellogg theorem and related results. J. Fixed Point Theory Appl., 3 (2008), 121-130

Birkhoff billiards are insecure. Discr. Cont. Dyn. Syst., 23 (2009), 1035-1040

What is ... the Schwarzian derivative? with V. Ovsienko. Notices of AMS, 56 (2009), 34-36

Quasiperiodic motion for the Pentagram Map, with V. Ovsienko and R. Schwartz. Electron. Res. Announc. Math. Sci., 16 (2009), 1-8

Spaces of pseudo-Riemannian geodesics and pseudo-Euclidean billiards, with B. Khesin. Adv. Math., 221 (2009), 1364-1396

Self-dual polygons and self-dual curves, with D. Fuchs. Funct. Anal. and Other Math., 2 (2009), 203-220

On bicycle tire tracks geometry, hatchet planimeter, Menzin's conjecture and oscillation of unicycle tracks, with M. Levi. Experimental Math., 18 (2009), 173-186

Sylvester's problem (in Russian), with V. Timorin. Kvant 2009, No 5, 2-6, No 6, 6-10

Existence and non-existence of skew branes. J. Fixed Point Theory Appl., 7 (2010), 419-431

Variations on the Tait-Kneser theorem, with V. Timorin. Preprint

The Pentagram map: a discrete integrable system, with V. Ovsienko and R. Schwartz. Comm. Math. Phys., 299 (2010), 409-446

Topological aspects of the Dvoretzky Theorem, with D. Burago and S. Ivanov. J. Topology and Analysis, 2 (2010), 453-467

Contact complete integrability, with B. Khesin. Reg. Chaotic Dynamics, 15 (2010), 504-520

Elementary surprises in projective geometry, with R. Schwartz. Math. Intelligencer, 32 (2010) No 3, 31-34

Periodic trajectories in the regular pentagon, with D. Davis and D. Fuchs. Moscow Math. J., 11 (2011), 1-23

The pentagram integrals on inscribed polygons, with R. Schwartz. Electr. J. Comb., 18 (2011), P171

Variations on R. Schwartz's inequality for the Schwarzian derivative. Discr. Comp. Geom., 46 (2011), 724-742

2-frieze patterns and the cluster structure of the space of polygons, with S. Morier-Genoud and V. Ovsienko. Ann. Inst. Fourier, 62 (2012), 937-987

The (un)equal tangents problem. Amer. Math. Monthly, 119 (2012), 398-405

Higher pentagram maps, weighted directed networks, and cluster dynamics, with M. Gekhtman, M. Shapiro and V. Vainshtein. Electron. Res. Announc. Math. Sci., 19 (2012), 1-17

Discrete spherical means of directional derivatives and Veronese maps (with A. Belyaev and B. Khesin). J. Geom. Physics, 62 (2012), 124--136

Liouville-Arnold integrability of the pentagram map on closed polygons, with V. Ovsienko and R. Schwartz. Duke Math. J., 162 (2013), 2149-2196

Periodic trajectories in the regular pentagon, II, with D. Fuchs. Moscow Math. J., 13 (2013), 19-32

On fibrations with flat fibers, with V. Ovsienko. Bull. London Math. Soc., 45 (2013), 625-632

The equal tangents property, with J. Jeronimo-Castro and G. Ruiz-Hernandez. Adv. Geom., 14 (2014), 447-453

Tractrices, bicycle tire tracks, hatchet planimeters, and a 100-year-old conjecture, with R. Foote and M. Levi. Amer. Math. Monthly, 120 (2013), 199-216

Osculating curves: around the Tait-Kneser Theorem, with E. Ghys and V. Timorin. Math. Intelligencer, 35, No 1 (2013), 61-66

On the discrete bicycle transformation, with E. Tsukerman. Publ. Math. Uruguay (proc. of the Montevideo Dynamical Systems Congress 2012), 14 (2013), 201-220

Dragon curves revisited. Math. Intelligencer, 36, No 1 (2014), 13-17

Proofs (not) from The Book. Math. Intelligencer, 36, No 2 (2014), 9-14

Circumcenter of Mass and generalized Euler line, with E. Tsukerman. Discr. Comp. Geom., 51 (2014), 815-836

Linear difference equations, frieze patterns and combinatorial Gale transform, with S. Morier-Genoud, V. Ovsienko and R. Schwartz. Forum of Mathematics, Sigma, 2 (2014), e22

On curves and polygons with the equiangular chord property, with T. Aougab, X. Sun and Y. Wang. Pacific J. Math., 274 (2015), 305-324

Coxeter's frieze patterns and discretization of the Virasoro orbit, with V. Ovsienko. J. Geom. and Phys., 87 (2015), 373-381

SL 2 (Z)-tiling of the torus, Coxeter-Conway friezes and Farey triangulations, with S. Morier-Genoud and V. Ovsienko. L'Enseign. Math., 61 (2015), 71-92

A Baker's dozen of problems. Arnold Math. J., 1 (2015), 59-67

Remarks on the circumcenter of mass, with E. Tsukerman. Arnold Math. J., 1 (2015), 101-112

Introducing supersymmetric frieze patterns and linear difference operators, with S. Morier-Genoud and V. Ovsienko. Math. Zeitschrift, 281 (2015), 1061-1087

Configuration spaces of plane polygons and a sub-Riemannian approach to the equitangent problem, with J. Jeronimo-Castro. J. Dyn. Control Syst., 22 (2016), 227-250

The Six Circles Theorem revisited, with D. Ivanov. Amer. Math. Monthly, 123 (2016), 689-698

Integrable cluster dynamics of directed networks and pentagram maps, with M. Gekhtman, M. Shapiro and V. Vainshtein. Adv. in Math., 300 (2016), 390-450

On Lagrangian tangent sweeps and Lagrangian outer billiards, with D. Fuchs. Geom. Dedicata, 182 (2016), 203-213

Skewers. Arnold Math. J., 2 (2016), 171-193

Centers of mass of Poncelet polygons, 200 years after, with R. Schwartz. Math. Intelligencer, v. 38, No 2 (2016), 29-34

Hopf fibrations and Hurwitz-Radon numbers, with V. Ovsienko. Math. Intelligencer, v. 38, No 4 (2016), 11-18

Polynomials as polygons. Math. Intelligencer, v. 39, No 1 (2017), 41-43

On the bicycle transformation and the filament equation: results and conjectures. J. Geom. and Phys., 115 (2017), 116-123

Billiard transformations of parallel flows: a periscope theorem, with A. Plakhov and D. Treschev. J. Geom. and Phys., 115 (2017), 157-166

Iterating evolutes and involutes, with M. Arnold, D. Fuchs, I. Izmestiev, and E. Tsukerman. Discr. Comp. Geom., 58 (2017), 80-143

Ivory's Theorem revisited, with I. Izmestiev. J. Integrable Systems, 2 (2017), 1-36

Iterating evolutes of spatial polygons and of spatial curves, with D. Fuchs. Moscow Math. J., 17 (2017), 667-689

Vladimir Igorevich Arnold, with B. Khesin. Biogr. Mems Fell. Royal Soc., publ. online 08/30/17

Dual numbers, weighted quivers, and extended Somos and Gale-Robinson sequences, with V. Ovsienko. Algebras Represent. Theory, 21 (2018), 1119-1132

Book review of "A singular mathematical promenade" by E. Ghys. Math. Intelligencer, v. 40, No 2 (2018), 85-88

Introducing symplectic billiards, with P. Albers. Advances in Math., 333 (2018), 822-867         

A four vertex theorem for frieze patterns? Math. Intelligencer, v. 40, No 4 (2018), 14-18

Open problems, questions, and challenges in finite-dimensional integrable systems, with A. Bolsinov, V. Matveev, and E. Miranda. Philosoph. Trans. Royal Society A, 376 (2018), 20170430 (40 pp)

On centro-affine curves and Backlund transformations of the KdV equation. Arnold Math. J., 4 (2018), 445-458

Projective configuration theorems: old wine into new wineskins. in Geometry in History, Springer, 2019, 401--434

Kasner meets Poncelet. Math. Intelligencer, v. 41, No 4 (2019), 56-59

Fun problems in geometry and beyond, with B. Khesin. SIGMA 15 (2019), 097, 21 pages

Tire tracks and integrable curve evolution, with G. Bor, M. Levi, and R. Perline. Int. Math. Res. Notes, 2020, no. 9, 2698-2768

Descartes circle theorem, Steiner porism, and spherical designs, with R. Schwartz. Amer. Math. Monthly, v. 127 (2020), 238-248

Book review of "The Scottish Book. Mathematics from the Scottish Cafe" . Math. Intelligencer, v. 42, No1 (2020), 71-74

Counting periodic trajectories of Finsler billiards, with P. Blagojevic, M. Harrison, and G. Ziegler. SIGMA 16 (2020), 022, 33 pages

Wire billiards, the first steps, with M. Bialy and A. Mironov. Advances in Math., 368 (2020), 107154, 27 pp.

Two variations on the periscope theorem. Reg. Chaotic Dynamics, 25 (2020), 11-17

Book review of "You Failed Your Math Test, Comrade Einstein". Math. Intelligencer, v. 42, No 3 (2020), 83-87

Four equivalent properties of integrable billiards, with A. Glutsyuk and I. Izmestiev. Israel J. Math., 241 (2021), 693-719

Billiards in ellipses revisited, with A. Akopyan and R. Schwartz. European J. Math., 8 (2022), 1313-1327

Dan Reznik's identities and more, with M. Bialy. European J. Math., 8 (2022), 1341-1354

Remarks on Joachimsthal integral and Poritsky property, with M. Arnold. Arnold Math. J., 7 (2021), 483-491

Polar bear or penguin? Musings on Earth cartography and Chebyshev nets, with B. Khesin. Math. Intelligencer, v. 43, No 1 (2021), 20-24

Cross-ratio dynamics on ideal polygons, with M. Arnold, D. Fuchs, and I. Izmestiev. Int. Math. Res. Notes, 2022, no. 9, 6770-6853

Symplectically convex and symplectically star-shaped curves - a variational problem, with P. Albers. J. Fixed Point Theory Appl. (Viterbo Festschrift), 24 (2022), no. 2, Paper No. 27, 24 pp.

Variations on the Tait-Kneser theorem, with G. Bor and C. Jackman. Math. Intelligencer, v. 43, No 3 (2021), 8-14

Loewner's "forgotten" theorem, with P. Albers. Math. Intelligencer, v. 44, No 1 (2022), 7-11

Remarks on rigidity properties of conics. Reg. Chaotic Dynamics, 27 (2022), 18-23

Open problems on billiards and geometric optics, with M. Bialy, C, Fierobe, A. Glutsyuk, M. Levi, and A. Plakhov. Arnold Math. J., 8 (2022), 411-422

On cusps of caustics by reflection: a billiard variation on Jacobi's Last Geometric Statement, with G. Bor. Amer. Math. Monthly, v. 130 (2023), 454-467

Self-Backlund curves in centroaffine geometry and Lame's equation, with M. Bialy and G. Bor. Communication of AMS, 2 (2022), 232-282

A family of integrable transformations of centroaffine polygons: geometrical aspects, with M. Arnold and D. Fuchs. Ann. Inst. Fourier, to appear

On projective evolutes of polygons, with M. Arnold and R. Schwartz. Experimental Math., online first

On conformal points of area preserving maps and related topics, with P. Albers. J. Geom. Physics, 180, Oct. 2022, 104644

Bicycling geodesics are Kirchhoff rods, with G. Bor and C. Jackman. Nonlinearity, 36 (2023), 3572-3602

Polygonal symplectic billiards, with P. Albers, G. Banhatti, F. Sadlo, and R. Schwartz. J. Experimental Math, to appear

Differential geometry of space curves: forgotten chapters, with D. Fuchs, I. Izmestiev, M. Raffaelli, and G. Szewieczek. Math Intelligencer,  v. 46, No 1 (2024), 9-21

Iterating skew evolutes and skew involutes: a linear analog of the bicycle kinematics. Moscow Math. J. (Ilyashenko Festschrift),  23 (2023), 625-640

Monotone twist maps and Dowker-type theorems (with P. Albers). Preprint

Geometry of integrable linkages (with R. Perline).  Preprint

A family of maps and a vector field on plane polygons (with A. Arnold and L. Costa). Preprint



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