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We derive an effective recursion for Witten's r-spin intersection numbers, using Witten's conjecture relating r-spin numbers to the Gel'fand-Dikii hierarchy. Consequences include closed-form descriptions of the intersection numbers (for example, in terms of gamma functions). We use these closed-form descriptions to prove Harer-Zagier's formula for the Euler characteristic of Mg,1. Finally, we extend Witten's series expansion formula for the Landau-Ginzburg potential to study r-spin numbers in the small phase space in genus zero. Our key tool is the calculus of formal pseudodifferential operators, and is partially motivated by work of Brézin and Hikami. [ABSTRACT FROM AUTHOR]

The article discusses the mathematical aspects of doodle which involves looking for some shapes and drawing curves around the shape. It says that the idea behind the circular shape of a doodle points to a fundamental mathematical fact which is the triangle inequality. It mentions the geometry of a doodle including the number of circular arcs, areas, and perimeters. The relation of a doodle to differential geometry, Hilbert problem, and hyperbolic geometry is also discussed.

The authors offer a mathematical solution on finding the value of π in a two-dimensional space. They mention that π is defined as the ratio of the circumference of the circle to two times its radius. They discuss the transformation of the equation of the unit circle, where its center is the origin and radius is one, to indicate the value of π.

The article provides an answer to a question of the point whose distances from the corners of the unit square are rational.

We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ℤ (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over $\mathbb{F}_{p}$ that lifts to ℤ/ p 7 but not ℤ/ p 8. (Of course the results hold in the holomorphic category as well.) It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. The essential starting point is Mnëv’s universality theorem. [ABSTRACT FROM AUTHOR]

Focuses on a collection of cartographical miscellania that appeals mathematically from person to person in the U.S. community. Description of a cartogram; Means of interpolating the colors on a country-by-county level; Presentation of a cartographic and photographic views of an interchange highway.

Focuses on the two-dimensional rotor-router shape. Definition of the two-dimensional rotor-router walk by mathematician Jim Propp; Simulations in two dimensions indicating the shape generated by the rotor-router walk is significantly rounder than that of internal diffusion limited aggregation (IDLA); Information on the difference between the inradius and outradius of the IDLA shape during simulations up to a million particles.

Presents discussions on the number four and twenty-four. Details on number factorizations; Functions of computers in naming numbers; Discussions on the fortuitous coincidence of numbers.

Given an object over ℚ, there is often no reason for invariants of the corresponding holomorphic object to be preserved by the absolute Galois group Gal(ℚ/ℚ), and in general this is not true, although it is sometimes surprising to observe in practice. The case of covers of the projective line branched only over the points 0, 1, and ∞, through Belyi's theorem [9], leads to Grothendieck's dessins d'enfants program for understanding the absolute Galois group through its faithful action on such covers. This note is motivated by Catanese's question about a higher-dimensional analogue: does the absolute Galois group act faithfully on the deformation equivalence classes of smooth surfaces? (These equivalence classes are of course by definition the strongest deformation invariants.) We give a short proof of a weaker result: the absolute Galois group acts faithfully on the irreducible components of the moduli space of smooth surfaces (of general type, canonically polarized). Bauer, Catanese, and Grunewald have recently answered Catanese's original question using a different construction (using surfaces isogenous to a product) [7]. [ABSTRACT FROM PUBLISHER]

The answer to a mathematical puzzle, published in the volume 27, number 3, of the journal "Mathematical Entertainments," is provided.

Presents a crossword puzzle in mathematics.

Presents the mathematical problem of the keg index and a theory of drunkenness. Probabilities used; Equations expressing the expected number of swigs before the passage of the keg.

We develop the Littlewood-Richardson homotopy algorithm, which uses numerical continuation to compute solutions to Schubert problems on Grassmannians and is based on the geometric Littlewood-Richardson rule. One key ingredient of this algorithm is our new optimal formulation of Schubert problems in local Stiefel coordinates as systems of equations. Our implementation can solve problem instances with tens of thousands of solutions. [ABSTRACT FROM AUTHOR]

This paper deals with the geometry of the space [Geometric Invariant Theory (GIT) quotient] M8 of eight points in ℙ1, and the Gale-quotient N′8 of the GIT quotient of eight points in ℙ3.The space M8 comes with a natural embedding in ℙ13, or more precisely, the projectivization of the 8-representation V4, 4. There is a single 8-skew cubic  in ℙ13. The fact that M8 lies on the skew cubic  is a consequence of Thomae's formula for hyperelliptic curves, but more is true: M8 is the singular locus of . These constructions yield the free resolution of M8 and are used in the determination of the ‘single’ equation cutting out the GIT quotient of n points in ℙ1 in general [20].The space N′8 comes with a natural embedding in ℙ13, or more precisely, ℙ V2, 2, 2, 2. There is a single skew quintic  containing N′8 and N′8 is the singular locus of the skew quintic .The skew cubic  and skew quintic  are projectively dual. (In particular, they are surprisingly singular, in the sense of having a dual of remarkably low degree.) The divisor on the skew cubic blown down by the dual map is the secant variety Sec(M8) and the contraction Sec(M8) → N′8 factors through N8 via the space of eight points on a quadric surface. We conjecture (Conjecture 1.1) that the divisor on the skew quintic blown down by the dual map is the quadrisecant variety of N′8 (the closure of the union of quadrisecant lines), and that the quintic  is the trisecant variety. The resulting picture extends the classical duality in the 6-point case between the Segre cubic threefold and the Igusa quartic threefold.We note that there are a number of geometrically natural varieties that are (related to) the singular loci of remarkably singular cubic hypersurfaces (see for example [3, 5]). [ABSTRACT FROM PUBLISHER]

Consider the projective coordinate ring of the Geometric Invariant Theory (GIT) quotient (ℙ1) n //SL(2), with the usual linearization, where n is even. In 1894, Kempe proved that this ring is generated in degree one. In [2] we showed that, over ℚ, the relations between degree one invariants are generated by a class of quadratic relations—the simplest binomial relations—with the exception of n = 6, where there is a single cubic relation. The purpose of this article is to show that these results hold over , and to suggest why they may be true over . [ABSTRACT FROM PUBLISHER]

The ring of projective invariants of n ordered points on the projective line is one of the most basic and earliest studied examples in Geometric Invariant Theory. It is a remarkable fact and the point of this paper that, unlike its close relative the ring of invariants of n unordered points, this ring can be completely and simply described. In 1894 Kempe found generators for this ring, thereby proving the First Main Theorem for it (in the terminology introduced by Weyl). In this paper we compute the relations among Kempe's invariants, thereby proving the SecondMain Theorem (again in the terminology of Weyl), and completing the description of the ring 115 years later. This paper introduces a number of new tools to the problem, and uses the graphical algebra formalismto intermediate between representation-theoretic arguments (for symmetric and Lie groups), and the symmetry-breaking of the Speyer-Sturmfels degeneration. [ABSTRACT FROM AUTHOR]

Abstract: We consider the ring of invariants of n points on the projective line. The space is perhaps the first nontrivial example of a GIT quotient. The construction depends on the weighting of the n points. Kempe found generators (in the unit weight case) in 1894. We describe the full ideal of relations for all weightings. In some sense, there is only one equation, which is quadratic except for the classical case of the Segre cubic primal, for and weight 16. The cases of up to 6 points are long known to relate to beautiful familiar geometry. The case of 8 points turns out to be richer still. To cite this article: B. Howard et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

Abstract: We use a simple description of the outer automorphism of to cleanly describe the invariant theory of six points in , , and . [Copyright &y& Elsevier]

Presents a mathematical problem involving a light bulb and 100 prisoners. Solutions proposed for the problem; Mathematical proofs of the solution; Variations from the solutions proposed; Mathematical formulae used to solve the problem.

Focuses on the card game set, a fast-paced game that has a rich mathematical structure. Origin of the game; Techniques to play the game; Description of the fourier transform method, a tool for analyzing problems associated with symmetry groups.