We apply the concept of effective order to strong stability preserving (SSP) explicit Runge-Kutta methods. Relative to classical Runge-Kutta methods, methods with an effective order of accuracy are designed to satisfy a relaxed set of order conditions, but yield higher order accuracy when composed with special starting and stopping methods. We show that this allows the construction of four-stage SSP methods with effective order four (such methods cannot have classical order four). However, we also prove that effective order five methods - like classical order five methods - require the use of non-positive weights and so cannot be SSP. By numerical optimization, we construct explicit SSP Runge-Kutta methods up to effective order four and establish the optimality of many of them. Numerical experiments demonstrate the validity of these methods in practice.

These notes will give an introduction to the theory of K3 surfaces. We begin with some general results on K3 surfaces, including the construction of their moduli space and some of its properties. We then move on to focus on the theory of polarized K3 surfaces, studying their moduli, degenerations and the compactification problem. This theory is then further enhanced to a discussion of lattice polarized K3 surfaces, which provide a rich source of explicit examples, including a large class of lattice polarizations coming from elliptic fibrations. Finally, we conclude by discussing the ample and Kahler cones of K3 surfaces, and give some of their applications.

We study walk algebras and Hecke algebras for Kac-Moody root systems. Each choice of orientation for the set of real roots gives rise to a corresponding "oriented" basis for each of these algebras. We show that the notion of distinguished subexpression naturally arises when studying the transition matrix between oriented bases. We then relate these notions to the geometry of Kac-Moody flag varieties and Bott-Samelson varieties. In particular, we show that the number of points over a finite field in certain intersections of these varieties is given by change of basis coefficients between oriented bases of the Hecke algebra. Using these results we give streamlined derivations of Deodhar's formula for $R$-polynomials and point-counting formulas for specializations of nonsymmetric Macdonald polynomials $E_\lambda(\mathsf{q},t)$ at $\mathsf{q}=0,\infty$.

For every $n\ge 2$, we construct a body $U_n$ of constant width $2$ in $\mathbb{E}^n$ with small volume and symmetries of a regular $n$-simplex. $U_2$ is the Reuleaux triangle. To the best of our knowledge, $U_3$ was not previously constructed, and its volume is smaller than the volume of other three-dimensional bodies of constant width with tetrahedral symmetries. While the volume of $U_3$ is slightly larger than the volume of Meissner's bodies of width $2$, it exceeds the latter by less than $0.137\%$. For all large $n$, the volume of $U_n$ is smaller than the volume of the ball of radius $0.891$.

We study walk algebras and Hecke algebras for Kac-Moody root systems. Each choice of orientation for the set of real roots gives rise to a corresponding "oriented" basis for each of these algebras. We show that the notion of distinguished subexpression naturally arises when studying the transition matrix between oriented bases. We then relate these notions to the geometry of Kac-Moody flag varieties and Bott-Samelson varieties. In particular, we show that the number of points over a finite field in certain intersections of these varieties is given by change of basis coefficients between oriented bases of the Hecke algebra. Using these results we give streamlined derivations of Deodhar's formula for $R$-polynomials and point-counting formulas for specializations of nonsymmetric Macdonald polynomials $E_\lambda(\mathsf{q},t)$ at $\mathsf{q}=0,\infty$.

We study threefolds fibred by mirror quartic K3 surfaces. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the moduli space of mirror quartic K3 surfaces. This is then used to give a complete explicit description of all Calabi-Yau threefolds fibred by mirror quartic K3 surfaces. We conclude by studying the properties of such Calabi-Yau threefolds, including their Hodge numbers and deformation theory.

Let $K$ be an isotropic convex body in $\R^n$. Given $\eps>0$, how many independent points $X_i$ uniformly distributed on $K$ are needed for the empirical covariance matrix to approximate the identity up to $\eps$ with overwhelming probability? Our paper answers this question posed by Kannan, Lovasz and Simonovits. More precisely, let $X\in\R^n$ be a centered random vector with a log-concave distribution and with the identity as covariance matrix. An example of such a vector $X$ is a random point in an isotropic convex body. We show that for any $\eps>0$, there exists $C(\eps)>0$, such that if $N\sim C(\eps) n$ and $(X_i)_{i\le N}$ are i.i.d. copies of $X$, then $\Big\|\frac{1}{N}\sum_{i=1}^N X_i\otimes X_i - \Id\Big\| \le \epsilon,$ with probability larger than $1-\exp(-c\sqrt n)$.

We examine situations, where representations of a finite-dimensional $F$-algebra $A$ defined over a separable extension field $K/F$, have a unique minimal field of definition. Here the base field $F$ is assumed to be a $C_1$-field. In particular, $F$ could be a finite field or $k(t)$ or $k((t))$,where $k$ is algebraically closed. We show that a unique minimal field of definition exists if (a) $K/F$ is an algebraic extension or (b) $A$ is of finite representation type. Moreover, in these situations the minimal field of definition is a finite extension of $F$. This is not the case if $A$ is of infinite representation type or $F$ fails to be $C_1$. As a consequence, we compute the essential dimension of the functor of representations of a finite group, generalizing a theorem of N. Karpenko, J. Pevtsova and the second author.

In 2006, Ihara defined and systematically studied a generalization of the Euler-Mascheroni constant for all global fields, named the Euler-Kronecker constants. This paper examines their distribution across geometric quadratic extensions of a rational global function field, via the values of logarithmic derivatives of Dirichlet $L$-functions at 1. Using a probabilistic model, we show that the values converge to a limiting distribution with a smooth, positive density function, as the genii of quadratic fields approach infinity. We then prove a discrepancy theorem for the convergence of the frequency of these values, and obtain information about the proportion of the small values. Finally, we prove omega results on the extreme values. Our theorems imply new distribution results on the stable Taguchi heights and logarithmic Weil heights of rank 2 Drinfeld modules with CM.

We prove that the solid ergodicity property is stable with respect to taking coinduction for a fairly large class of coinduced action. More precisely, assume that \Sigma<\Gamma are countable groups such that $g\Sigma g^{-1}\cap \Sigma$ is finite for any $g\in\Gamma\setminus\Sigma$. Then any measure preserving action $\Sigma\curvearrowright X_0$ gives rise to a solidly ergodic equivalence relation if and only if the equivalence relation of the associated coinduced action $\Gamma\curvearrowright X$ is solidly ergodic. We also obtain orbit equivalence rigidity for such actions by showing that the orbit equivalence relation of a rigid or compact measure preserving action $\Sigma\curvearrowright X_0$ of a property (T) group is "remembered" by the orbit equivalence relation of $\Gamma\curvearrowright X$.

In this paper, we characterize the vanishing of twisted central $L$-values attached to newforms of square-free level in terms of so-called local polynomials and the action of finitely many Hecke operators thereon. Such polynomials are the ``local part'' of certain locally harmonic Maass forms constructed by Bringmann, Kane and Kohnen in $2015$. We offer a second perspective on this characterization for weights greater than $4$ by adapting results of Zagier to higher level. To be more precise, we establish that a twisted central $L$-value attached to a newform vanishes if and only if a certain explicitly computable polynomial is constant. We conclude by proving an identity between these constants and generalized Hurwitz class numbers, which were introduced by Pei and Wang in $2003$. We provide numerical examples in weight $4$ and levels $7$, $15$, $22$, and offer some questions for future work.

In this paper, we introduce the Fourier-restricted Euler and hypodissipative Navier--Stokes equations. These equations are analogous to the Euler and hypodissipative Navier--Stokes equations respectively, but with the Helmholtz projection replaced by a projection onto a more restrictive constraint space; the $(u\cdot\nabla)u$ nonlinearity is otherwise unchanged. The constraint space restricts the divergence-free velocity to specific Fourier modes, which have a dyadic shell structure, and are constructed iteratively using permutations. In the inviscid case -- and in the hypo-viscous case when \alpha<\frac{\log(3)}{6\log(2)} \approx .264 -- we prove finite-time blowup for a set of solutions with a discrete group of symmetries. Our blowup Ansatz is odd, permutation symmetric, and mirror symmetric about the plane $x_1+x_2+x_3=0$. The Fourier-restricted Euler and hypodissipative Navier--Stokes equations respect both the energy equality and the identity for enstrophy growth from the full Euler and hypodissipative Navier--Stokes equations respectively, which is a substantial advance over the previous literature on Euler and Navier--Stokes model equations.

For every large enough $n$, we explicitly construct a body of constant width $2$ that has volume less than $0.9^n \text{Vol}(\mathbb{B}^{n}$), where $\mathbb{B}^{n}$ is the unit ball in $\mathbb{R}^{n}$. This answers a question of O.~Schramm.

We show that any finite family of pairwise intersecting balls in $\mathbb{E}^n$ can be pierced by $(\sqrt{3/2}+o(1))^n$ points improving the previously known estimate of $(2+o(1))^n$. As a corollary, this implies that any $2$-illuminable spiky ball in $\mathbb{E}^n$ can be illuminated by $(\sqrt{3/2}+o(1))^n$ directions. For the illumination number of convex spiky balls, i.e., cap bodies, we show an upper bound in terms of the sizes of certain related spherical codes and coverings. For large dimensions, this results in an upper bound of $1.19851^n$, which can be compared with the previous $(\sqrt{2}+o(1))^n$ established only for the centrally symmetric cap bodies. We also prove the lower bounds of $(\tfrac{2}{\sqrt{3}}-o(1))^n$ for the three problems above.

In this paper we study gravitationally bound compact objects sourced by a string theory inspired Born-Infeld scalar field. Unlike many of their canonical scalar field counterparts, these ``boson stars'' do not have to extend out to infinity and may generate compact bodies. We analyze in detail both the junction conditions at the surface as well as the boundary conditions at the center which are required in order to have a smooth structure throughout the object and into the exterior vacuum region. These junction conditions, although involved, turn out to be relatively easy to satisfy. Analysis reveals that these compact objects have a richer structure than the canonical boson stars and some of these properties turn out to be physically peculiar: There are several branches of solutions depending on how the junction conditions are realized. Further analysis illustrates that in practice the junction conditions tend to require interior geometries reminiscent of ``bag of gold'' spacetimes, and also hide the star behind an event horizon in its exterior. The surface compactness of such objects, defined here as the ratio $2M/r$, can be made arbitrarily close to unity indicating the absence of a Buchdahl bound. Some comments on the stability of these objects is provided to find possible stable and unstable regimes. However, we argue that even in the possibly stable regime the event horizon in the vacuum region shielding the object is potentially unstable, and would cut off the star from the rest of the universe.

We study the quantization of the regularized hamiltonian, $H$, of the compactified D=11 supermembrane with non-trivial winding. By showing that $H$ is a relatively small perturbation of the bosonic hamiltonian, we construct a Dyson series for the heat kernel of $H$ and prove its convergence in the topology of the von Neumann-Schatten classes so that $e^{-Ht}$ is ensured to be of finite trace. The results provided have a natural interpretation in terms of the quantum mechanical model associated to regularizations of compactified supermembranes. In this direction, we discuss the validity of the Feynman path integral description of the heat kernel for D=11 supermembranes and obtain a matrix Feynman-Kac formula.

We find expressions for the Gateaux derivative of the matrix norms in operator spaces, and operator systems. Some applications of the results to quantum probability measures, states on C$^*$-algebras, and Birkhoff-James orthogonality are also presented.

By introducing extrinsic noise as well as intrinsic uncertainty into a network with stochastic events, this paper studies the dynamics of the resulting Markov random network and characterizes a novel phenomenon of intermittent synchronization and desynchronization that is due to an interplay of the two forms of randomness in the system. On a finite state space and in discrete time, the network allows for unperturbed (or "deterministic") randomness that represents the extrinsic noise but also for small intrinsic uncertainties modelled by a Markov perturbation. It is shown that if the deterministic random network is synchronized (resp., uniformly synchronized), then for almost all realizations of its extrinsic noise the stochastic trajectories of the perturbed network synchronize along almost all (resp., along all) time sequences after a certain time, with high probability. That is, both the probability of synchronization and the proportion of time spent in synchrony are arbitrarily close to one. Under smooth Markov perturbations, high-probability synchronization and low-probability desynchronization occur intermittently in time, which can both be precisely described via an asymptotic expansion of the invariant distribution. Existence and uniqueness of invariant distributions are established, as well as their convergence as the perturbation parameter vanishes. An explicit asymptotic expansion is derived. Ergodicity of the extrinsic noise dynamics is seen to be crucial for the characterization of (de)synchronization sets and their respective relative frequencies. An example of a smooth Markov perturbation of a synchronized probabilistic Boolean network is provided to illustrate the intermittency between high-probability synchronization and low-probability desynchronization.

We describe a notion of lattice polarization for rational elliptic surfaces and weak del Pezzo surfaces, and describe the complex moduli of the former and the Kähler cone of the latter. We then propose a version of mirror symmetry relating these two objects, which should be thought of as a form of Fano-LG correspondence. Finally, we relate this notion to other forms of mirror symmetry, including Dolgachev-Nikulin-Pinkham mirror symmetry for lattice polarized K3 surfaces and the Gross-Siebert program.

Gauge functions significantly generalize the notion of a norm, and gauge optimization, as defined by Freund (1987}, seeks the element of a convex set that is minimal with respect to a gauge function. This conceptually simple problem can be used to model a remarkable array of useful problems, including a special case of conic optimization, and related problems that arise in machine learning and signal processing. The gauge structure of these problems allows for a special kind of duality framework. This paper explores the duality framework proposed by Freund, and proposes a particular form of the problem that exposes some useful properties of the gauge optimization framework (such as the variational properties of its value function), and yet maintains most of the generality of the abstract form of gauge optimization.