Viazovska proved that the $E_8$ lattice sphere packing is the densest sphere packing in 8 dimensions. Her proof relies on two inequalities between functions defined in terms of modular and quasimodular forms. We give a direct proof of these inequalities that does not rely on computer calculations.

The oriented swap process is a natural directed random walk on the symmetric group that can be interpreted as a multi-species version of the Totally Asymmetric Simple Exclusion Process (TASEP) on a finite interval. An open problem from a 2008 paper of Angel, Holroyd, and Romik asks for the limiting distribution of the absorbing time of the process. We resolve this question by proving that this random variable satisfies GOE Tracy-Widom asymptotics. Our starting point is a distributional identity relating the behavior of the oriented swap process to last passage percolation, conjectured in a recent paper of Bisi, Cunden, Gibbons, and Romik. The main technical tool is a shift-invariance principle for multi-species TASEPs, obtained by exploiting recent results of Borodin, Gorin, and Wheeler for the stochastic colored six-vertex model.

We present new probabilistic and combinatorial identities relating three random processes: the oriented swap process (OSP) on n particles, the corner growth process, and the last passage percolation (LPP) model. We prove one of the probabilistic identities, relating a random vector of LPP times to its dual, using the duality between the Robinson–Schensted–Knuth and Burge correspondences. A second probabilistic identity, relating those two vectors to a vector of “last swap times” in the OSP, is conjectural. We give a computer‐assisted proof of this identity for n≤6 after first reformulating it as a purely combinatorial identity, and discuss its relation to the Edelman–Greene correspondence. The conjectural identity provides precise finite‐n and asymptotic predictions on the distribution of the absorbing time of the OSP, thus conditionally solving an open problem posed by Angel, Holroyd, and Romik.

In 1883, a French mathematician named Édouard Lucas came up with an intriguing scenario.There are three poles in a row, the one on the left containing a series of discs of decreasing size, with the other two, empty.The aim is to move the tower, one disc at a time, over to the right-hand pole.However, the catch is, a larger disc can never sit on top of a smaller disc.This puzzle quickly reached fame as the brainteaser now known as the Tower of Hanoi.

will appear on the journal’s web site only) Pál Turán in the 1950s proposed to use the expansion of the Riemann xi function in the Hermite polynomials as a tool to gain insight into the location of the zeros of the Riemann zeta function. In this paper we follow up and expand on Turán’s ideas in several ways by considering infinite series expansions for the Riemann xi function Ξ(t) in three specific families of orthogonal polynomials: (1) the Hermite polynomials; (2) the symmetric Meixner–Pollaczek polynomials P (3/4) n (x;π/2); and (3) the continuous Hahn polynomials pn ( x; 34 , 3 4 , 3 4 , 3 4 ) . For each of the three expansions we derive asymptotic formulas for the coefficients and prove additional results. We also apply some of the same techniques to prove a new asymptotic formula for the Taylor coefficients of the Riemann xi function, and uncover a previously unnoticed connection between the Hermite expansion of Ξ(t) and the separate program of research involving the de Bruijn–Newman constant.

We study the Taylor expansion around the point x=1 of a classical modular form, the Jacobi theta constant θ _3 . This leads naturally to a new sequence (d(n))_n=0^∞ =1,1,-1,51,849,-26199,… of integers, which arise as the Taylor coefficients in the expansion of a related “centered” version of θ _3 . We prove several results about the numbers d ( n ) and conjecture that they satisfy the congruence d(n)≡ (-1)^n-1 (mod 5) and other similar congruence relations.

We present new combinatorial and probabilistic identities relating three random processes: the oriented swap process on $n$ particles, the corner growth process, and the last passage percolation model. We prove one of the probabilistic identities, relating a random vector of last passage percolation times to its dual, using the duality between the Robinson-Schensted-Knuth and Burge correspondences. A second probabilistic identity, relating those two vectors to a vector of "last swap times" in the oriented swap process, is conjectural. We give a computer-assisted proof of this identity for $n\le 6$ after first reformulating it as a purely combinatorial identity, and discuss its relation to the Edelman-Greene correspondence.

We consider alternative orders of summation for the conditionally convergent series defining the weight-2 Eisenstein series G(2) and the Weierstrass p-function. The resulting sums differ from the standard ones by a residual term that can be thought of as a function of the shapes with respect to which we sum. We compute this residual function explicitly and give some examples. The results generalize the well-known quasimodularity relationship between G(2) and its series summed in the reverse order.

We study infinite series expansions for the Riemann xi function Ξ(t) in three specific families of orthogonal polynomials: (1) the Hermite polynomials; (2) the symmetric Meixner-Pollaczek polynomials P_n^(3/4)(x;π/2); and (3) the continuous Hahn polynomials p_n(x; 3/4,3/4,3/4,3/4). The first expansion was discussed in earlier work by Turán, and the other two expansions are new. For each of the three expansions, we derive formulas for the coefficients, show that they appear with alternating signs, derive formulas for their asymptotic behavior, and derive additional interesting properties and relationships. We also apply some of the same techniques to prove a new asymptotic formula for the Taylor coefficients of the Riemann xi function. Our results continue and expand the program of research initiated in the 1950s by Turán, who proposed using the Hermite expansion of the Riemann xi function as a tool to gain insight into the location of the Riemann zeta zeros. We also uncover a connection between Turán's ideas and the separate program of research involving the so-called De Bruijn-Newman constant. Most significantly, the phenomena associated with the new expansions in the Meixner-Pollaczek and continuous Hahn polynomial families suggest that those expansions may be even more natural tools than the Hermite expansion for approaching the Riemann hypothesis and related questions.

Pál Turán in the 1950s proposed to use the expansion of the Riemann xi function in the Hermite polynomials as a tool to gain insight into the location of the zeros of the Riemann zeta function. In this paper we follow up and expand on Turán’s ideas in sev

The moving sofa problem, posed by L. Moser in 1966, asks for the planar shape of maximal area that can move around a right-angled corner in a hallway of unit width. It is known that a maximal area shape exists, and that its area is at least 2.2195…—the area of an explicit construction found by Gerver in 1992—and at most 22≈2.82, with the lower bound being conjectured as the true value. We prove a new and improved upper bound of 2.37. The method involves a computer-assisted proof scheme that can be used to rigorously derive further improved upper bounds that converge to the correct value.

The moving sofa problem, posed by L. Moser in 1966, asks for the planar shape of maximal area that can move around a right-angled corner in a hallway of unit width, and is conjectured to have as its solution a complicated shape derived by Gerver in 1992. We extend Gerver's techniques by deriving a family of six differential equations arising from the area-maximization property. We then use this result to derive a new shape that we propose as a possible solution to the "ambidextrous moving sofa problem," a variant of the problem previously studied by Conway and others in which the shape is required to be able to negotiate a right-angle turn both to the left and to the right. Unlike Gerver's construction, our new shape can be expressed in closed form, and its boundary is a piecewise algebraic curve. Its area is equal to $X+\arctan Y$, where $X$ and $Y$ are solutions to the cubic equations $x^2(x+3)=8$ and $x(4x^2+3)=1$, respectively.

We derive new results about properties of the Witten zeta function associated with the group SU(3), and use them to prove an asymptotic formula for the number of n-dimensional representations of SU(3) counted up to equivalence. Our analysis also relates the Witten zeta function of SU(3) to a summation identity for Bernoulli numbers discovered in 2008 by Agoh and Dilcher. We give a new proof of that identity and show that it is a special case of a stronger identity involving the Eisenstein series.

We prove a limit shape theorem describing the asymptotic shape of bumping routes when the Robinson–Schensted algorithm is applied to a finite sequence of independent, identically distributed random variables with the uniform distribution U[0,1] on the unit interval, followed by an insertion of a deterministic number α. The bumping route converges after scaling, in the limit as the length of the sequence tends to infinity, to an explicit, deterministic curve depending only on α. This extends our previous result on the asymptotic determinism of Robinson–Schensted insertion, and answers a question posed by Moore in 2006. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 171–182, 2016

We study an infinite version of the "jeu de taquin" sliding game, which can be thought of as a natural measure-preserving transformation on the set of infinite Young tableaux equipped with the Plancherel probability measure. We use methods from representation theory to show that the Robinson Schensted Knuth (RSK) algorithm gives an isomorphism between this measure-preserving dynamical system and the one-sided shift dynamics on a sequence of independent and identically distributed random variables distributed uniformly on the unit interval. We also show that the jeu de taquin paths induced by the transformation are asymptotically straight lines emanating from the origin in a random direction whose distribution is computed explicitly, and show that this result can be interpreted as a statement on the limiting speed of a second-class particle in the Plancherel-TASEP particle system (a variant of the Totally Asymmetric Simple Exclusion Process associated with Plancherel growth), in analogy with earlier results for second class particles in the ordinary TASEP.

The optimal condition of the cone volume measure of a pair of antipodal points is proved and analyzed.

The dense O(1) loop model is a statistical physics model with connections to the quantum XXZ spin chain, alternating sign matrices, the six-vertex model and critical bond percolation on the square lattice. When cylindrical boundary conditions are imposed, the model possesses a commuting family of transfer matrices. The original proof of the commutation property is algebraic and is based on the Yang-Baxter equation. In this paper we give a new proof of this fact using a direct combinatorial bijection.

Loop percolation, also known as the dense O(1) loop model, is a variant of critical bond percolation in the square lattice Z^2 whose graph structure consists of a disjoint union of cycles. We study its connectivity pattern, which is a random noncrossing matching associated with a loop percolation configuration. These connectivity patterns exhibit a striking rationality property whereby probabilities of naturally-occurring events are dyadic rational numbers or rational functions of a size parameter n, but the reasons for this are not completely understood. We prove the rationality phenomenon in a few cases and prove an explicit formula expressing the probabilities in the "cylindrical geometry" as coefficients in certain multivariate polynomials. This reduces the rationality problem in the general case to that of proving a family of conjectural constant term identities generalizing an identity due to Di Francesco and Zinn-Justin. Our results make use of, and extend, algebraic techniques related to the quantum Knizhnik-Zamolodchikov equation.

In a surprising sequence of developments, the longest increasing subsequence problem, originally mentioned as merely a curious example in a 1961 paper, has proven to have deep connections to many seemingly unrelated branches of mathematics, such as random permutations, random matrices, Young tableaux, and the corner growth model. The detailed and playful study of these connections makes this book suitable as a starting point for a wider exploration of elegant mathematical ideas that are of interest to every mathematician and to many computer scientists, physicists and statisticians. The specific topics covered are the Vershik-Kerov–Logan-Shepp limit shape theorem, the Baik–Deift–Johansson theorem, the Tracy–Widom distribution, and the corner growth process. This exciting body of work, encompassing important advances in probability and combinatorics over the last forty years, is made accessible to a general graduate-level audience for the first time in a highly polished presentation.

The refined enumeration of alternating sign matrices (ASMs) of given order having prescribed behavior near one or more of their boundary edges has been the subject of extensive study, starting with the Refined Alternating Sign Matrix Conjecture of Mills-Robbins-Rumsey, its proof by Zeilberger, and more recent work on doubly-refined and triply-refined enumeration by several authors. In this paper we extend the previously known results on this problem by deriving explicit enumeration formulas for the "top-left-bottom" (triply-refined) and "top-left-bottom-right" (quadruply-refined) enumerations. The latter case solves the problem of computing the full boundary correlation function for ASMs. The enumeration formulas are proved by deriving new representations, which are of independent interest, for the partition function of the square ice model with domain wall boundary conditions at the "combinatorial point" 2{\pi}/3.