Let $k$ be a number field, and denote by $k^{[d]}$ the compositum of all degree $d$ extensions of $k$ in a fixed algebraic closure. We first consider the question of whether all subextensions of degree less than $d$ lie in $k^{[d]}$. We show that this occurs if and only if $d < 5$. Secondly, we consider the question of whether there exists a constant $c$ such that if $K/k$ is a finite subextension of $k^{[d]}$, then $K$ is generated over $k$ by elements of degree at most $c$. We show that such a constant exists if and only if $d < 3$. This question becomes more interesting when one restricts attention to Galois extensions $K/k$. In this setting, we show that such a constant does not exist for $d$ even or non-squarefree. If $d$ is prime, we prove that all finite Galois subextensions of $k^{[d]}$ are generated over $k$ by elements of degree at most $d$.

In this paper, we demonstrate that the complete, hyperbolic representation of various two-bridge knot and link groups enjoy a certain local rigidity property inside of the PGL(4,R) character variety. We also prove a complementary result showing that under certain rigidity hypotheses, branched covers of amphicheiral knots admit non-trivial deformations near the complete, hyperbolic representation.

In this paper, we show that a result precisely analogous to the traditional quantum no-cloning theorem holds in classical mechanics. This classical no-cloning theorem does not prohibit classical cloning, we argue, because it is based on a too-restrictive definition of cloning. Using a less popular, more inclusive definition of cloning, we give examples of classical cloning processes. We also prove that a cloning machine must be at least as complicated as the object it is supposed to clone.

In this paper, we compute the Khovanov homology for (p,-p,q) pretzel knots for odd values of p from 3 to 15 and arbitrarily large q. We provide a conjecture for the general form of the Khovanov homology of (p,-p,q) pretzel knots. These computations reveal that these knots have thin Khovanov homology. Because Greene has shown that these knots are not quasi-alternating, this provides an infinite class of non-quasi-alternating knots with thin Khovanov homology.

We study distributions of persistent homology barcodes associated to taking subsamples of a fixed size from metric measure spaces. We show that such distributions provide robust invariants of metric measure spaces, and illustrate their use in hypothesis testing and providing confidence intervals for topological data analysis.

This paper introduces the synchrosqueezed wave packet transform as a method for analyzing 2D images. This transform is a combination of wave packet transforms of a certain geometric scaling, a reallocation technique for sharpening phase space representation, and clustering algorithms for mode decomposition. For a function that is a superposition of several wave-like components satisfying certain separation conditions, we prove that the synchrosqueezed wave packet transform identies these components and estimates their instantaneous wavevectors. A discrete version of this transform is discussed in detail and numerical results are given to demonstrate properties of the proposed transform.

We exhibit an infinite family of knots with isomorphic knot Heegaard Floer homology. Each knot in this infinite family admits a nontrivial genus 2 mutant which shares the same total dimension in both knot Floer homology and Khovanov homology. Each knot is distinguished from its genus 2 mutant by both knot Floer homology and Khovanov homology as bigraded groups, as well as by the $\delta$-graded version of knot Heegaard Floer homology.

We show that any $N$-dimensional linear subspace of $L^2(\mathbb{T})$ admits an orthonormal system such that the $L^2$ norm of the square variation operator $V^2$ is as small as possible. When applied to the span of the trigonometric system, we obtain an orthonormal system of trigonometric polynomials with a $V^2$ operator that is considerably smaller than the associated operator for the trigonometric system itself.

This paper introduces a fast algorithm for computing multilinear integrals, which are defined through Fourier multipliers. The algorithm is based on generating a hierarchical decomposition of summation domain into squares, constructing a low-rank approximation for the multiplier function within each square, and applying FFT based fast convolution algorithm for the computation associated with each square. The resulting algorithm is accurate and has a linear complexity, up to logarithmic factors, with respect to the number of the unknowns in the input functions. Numerical results are presented to demonstrate the properties of this algorithm.

This paper studies the problem of continuous time expected utility maximization of consumption together with addictive habit formation in general incomplete semimartingale financial markets. Introducing an auxiliary state processes and a modified dual space, we embed our original problem into an auxiliary time-separable utility maximization problem with the shadow random endowment. We establish existence and uniqueness of the optimal solution using convex duality on the product space by defining the primal value function as depending on both the initial wealth and initial standard of living. We also provide market independent sufficient conditions on both stochastic discounting processes of the habit formation process and on the utility function for our original problem to be well posed and to modify the convex duality approach when the auxiliary dual process is not necessarily integrable.

We prove variational forms of the Barban-Davenport-Halberstam Theorem and the large sieve inequality. We apply our result to prove an estimate for the sum of the squares of prime differences, averaged over arithmetic progressions.

We consider a model of optimal investment and consumption with both habit-formation and partial observations in incomplete Ito processes markets. The individual investor develops addictive consumption habits gradually while he can only observe the market stock prices but not the instantaneous rates of return, which follow Ornstein-Uhlenbeck processes. Applying the Kalman-Bucy filtering theorem and Dynamic Programming arguments, we solve the associated HJB equation fully explicitly for this path dependent stochastic control problem in the case of power utility preferences. We will provide optimal investment and consumption policies in explicit feedback forms using rigorous verification arguments.

We investigate the square variation operator $V^2$ (which majorizes the partial sum maximal operator) on general orthonormal systems (ONS) of size $N$. We prove that the $L^2$ norm of the $V^2$ operator is bounded by $O(\ln(N))$ on any ONS. This result is sharp and refines the classical Rademacher-Menchov theorem. We show that this can be improved to $O(\sqrt{\ln(N)})$ for the trigonometric system, which is also sharp. We show that for any choice of coefficients, this truncation of the trigonometric system can be rearranged so that the $L^2$ norm of the associated $V^2$ operator is $O(\sqrt{\ln\ln(N)})$. We also show that for $p>2$, a bounded ONS of size $N$ can be rearranged so that the $L^2$ norm of the $V^p$ operator is at most $O_p(\ln \ln (N))$ uniformly for all choices of coefficients. This refines Bourgain's work on Garsia's conjecture, which is equivalent to the $V^{\infty}$ case. Several other results on operators of this form are also obtained. The proofs rely on combinatorial and probabilistic methods.

We prove certain endpoint restriction estimates for the paraboloid over finite fields in three and higher dimensions. Working in the bilinear setting, we are able to pass from estimates for characteristic functions to estimates for general functions while avoiding the extra logarithmic power of the field size which is introduced by the dyadic pigeonhole approach. This allows us to remove logarithmic factors from the estimates obtained by Mockenhaupt and Tao in three dimensions and those obtained by Iosevich and Koh in higher dimensions.

We have developed a treecode-based O(N logN) algorithm for the generalized Born (GB) implicit solvation model. Our treecode-based GB (tGB) is based on the GBr6 (J. Phys. Chem. B 111, 3055 (2007)), an analytical GB method with a pairwise descreening approximation for the R6 volume integral expression. The algorithm is composed of a cutoff scheme for the effective Born radii calculation, and a treecode implementation of the GB chargecharge pair interactions. Test results demonstrate that the tGB algorithm can reproduce the Poisson solvation energy with an average relative error less than 0.6% while provide an almost linear-scaling calculation for a representative set of 25 proteins with different sizes (from 2815 atoms to 65456 atoms). For a typical system of 10k atoms, the tGB calculation is 3 times faster than the direct summation as implemented in the original GBr6 model. Thus, our tGB method provides an efficient way for performing implicit solvent GB simulations of larger biomolecular systems at longer time scales.

Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is essential to the numerical stability of the method. We also show how to define interpolation functions and discrete Hodge stars on dual meshes which can be used to create previously unconsidered mixed methods. Examples from magnetostatics and Darcy flow are examined in detail.

We establish an exact asymptotic formula for the square variation of certain partial sum processes. Let $\{X_{i}\}$ be a sequence of independent, identically distributed mean zero random variables with finite variance $\sigma$ and satisfying a moment condition $\mathbb{E}[|X_{i}|^{2+\delta} ] < \infty$ for some $\delta > 0$. If we let $\mathcal{P}_{N}$ denote the set of all possible partitions of the interval $[N]$ into subintervals, then we have that $\max_{\pi \in \mathcal{P}_{N}} \sum_{I \in \pi} | \sum_{i\in I} X_{i}|^2 \sim 2 \sigma^2N \ln \ln(N)$ holds almost surely. This can be viewed as a variational strengthening of the law of the iterated logarithm and refines results of J. Qian on partial sum and empirical processes. When $\delta = 0$, we obtain a weaker `in probability' version of the result.

We investigate the structure of finite sets $A \subseteq \Z$ where $|A+A|$ is large. We present a combinatorial construction that serves as a counterexample to natural conjectures in the pursuit of an "anti-Freiman" theory in additive combinatorics. In particular, we answer a question along these lines posed by O'Bryant. Our construction also answers several questions about the nature of finite unions of $B_2[G]$ and $B^\circ_2[G]$ sets, and enables us to construct a $\Lambda(4)$ set which does not contain large $B_2[G]$ or $B^\circ_2[G]$ sets.

We prove the optimal convergence estimate for first order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic approach defines the functions as the solution of a PDE. We show that given certain conditions on the geometry of the polygon, each of these constructions can obtain the optimal convergence estimate. In particular, we show that the well-known maximum interior angle condition required for interpolants over triangles is still required for Wachspress functions but not for Sibson functions.

The classical isoperimetric inequality relates the lengths of curves to the areas that they bound. More specifically, we have that for a smooth, simple closed curve of length L bounding area A on a surface of constant curvature c,L2 � 4�A � cA2with equality holding only if the curve is a geodesic circle. We prove generalizations of the isoperimetric inequality for both spherical and hyperbolic wave fronts (i.e. piecewise smooth curves which may have cusps). We then discuss �bicycle curves� using the generalized isoperimetric inequalities. The euclidean model of a bicycle is a unit segment AB that can move so that it remains tangent to the trajectory of point A (the rear wheel is fixed on the bicycle frame), as discussed in [Finn, The College Mathematics Journal 33, 2002], [Tabachnikov, Israel J. Math. 151: 1�28, 2006], and [Levi, Tabachnikov, Experiment. Math. 18: 173�186, 2009]. We extend this definition to a general Riemannian manifold, and concern ourselves in particular with bicycle curves in the hyperbolic plane H2 and on the sphere S2. We prove results along the lines of those in [Levi, Tabachnikov, Experiment. Math. 18: 173�186, 2009] and resolve both spherical and hyperbolic versions of Menzin's conjecture, which relates the area bounded by a curve to its associated monodromy map.

In the continual memory leakage model, security against attackers who can repeatedly obtain leakage is achieved by periodically updating the secret key. This is an appealing model which captures a wide class of side-channel attacks, but all previous constructions in this model provide only a very minimal amount of leakage tolerance \emph{during secret key updates}. Since key updates may happen frequently, improving security guarantees against attackers who obtain leakage during these updates is an important problem. In this work, we present the first cryptographic primitives which are secure against a super-logarithmic amount of leakage during secret key updates. We present signature and public key encryption schemes in the standard model which can tolerate a constant fraction of the secret key to be leaked between updates as well as \emph{a constant fraction of the secret key and update randomness} to be leaked during updates. Our signature scheme also allows us to leak a constant fraction of the entire secret state during signing. Before this work, it was unknown how to tolerate super-logarithmic leakage during updates even in the random oracle model. We rely on subgroup decision assumptions in composite order bilinear groups.

We prove local-in-time existence and uniqueness of an inviscid Boussinesq-type system. We assume the density equation contains nonzero diffusion and that our initial vorticity and density belong to a space of borderline Besov type.

We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we prove that flat non-local almost minimal boundaries are smooth. This can be viewed as a non-local version of the Almgren-De Giorgi-Tamanini regularity theory. The main result has several applications, among these $C^{1,\alpha}$ regularity for sets with prescribed non-local mean curvature in $L^p$ and regularity of solutions to non-local obstacle problems.

In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x,y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recent developments in the regularity theory for Monge-Ampere type equations. An application to microeconomics will also be described, which amounts to finding the equilibrium price distribution for a monopolist marketing a multidimensional line of products to a population of anonymous agents whose preferences are known only statistically.

In this work we provide an Aleksandrov-Bakelman-Pucci type estimate for a certain class of fully nonlinear elliptic integro-differential equations and generalizations of both the Monge-Amp\`ere operator and the convex envelope to a nonlocal, fractional-order setting. This particular elliptic family under consideration is large enough to capture the second order theory as the order of the integro-differential equations tends to 2. Moreover, our estimate is uniform in the order of the equations, resulting in a genuine generalization of the existing ABP estimate. This result also gives a new comparison theorem for viscosity solutions of such equations which only depends on the $L^\infty$ and $L^n$ norms of the right hand side, in contrast to previous comparison results which utilize the continuity of the right hand side for their conclusions. These results appear to be new even for the linear case of the relevant equations.

We prove under general assumptions that solutions of the thin obstacle or Signorini problem in any space dimension achieve the optimal regularity C^{1,1/2}. This improves the known optimal regularity results by allowing the thin obstacle to be de�ned in an arbitrary C^{1,beta} hypersurface, beta > 1/2, additionally, our proof covers any linear elliptic operator in divergence form with smooth coef�cients. The main ingredients of the proof are a version of Almgren�s monotonicity formula and the optimal regularity of global solutions

In this paper, we introduce two new invariants that are closely related to Milnor�s curvature-torsion invariant. The first, a particularly natural invariant called the spiral index of a knot, captures the number of local maxima in a knot projection that is free of inflection points. This invariant is sandwiched between the bridge and braid index of a knot, and captures more subtle properties. The second invariant, the projective superbridge index, provides a method of counting the greatest number of local maxima that occur in a given projection. In addition to investigating the relationships among these invariants, we use them to classify all those knots for which Milnor�s curvature-torsion invariant is 6�.

For an isotrivial semi-abelian varieties over a global field of positive characteristic, we show that on the set of adelic points of a large class of its subvarieties, the adelic closure of a finitely generated subgroup of the group of its global points cuts out exactly the global points of those subvarieties which lies in this subgroup.

We construct locally defined symplectic torus actions on ribbon graph complexes. Symplectic reduction techniques allow for a recursive formula for the symplectic volumes of these spaces. Taking the Laplace transform results in the Eynard-Orantin recursion formulas for the Airy curve x = y^2 / 2.

We determine decompositions of the space of algebraic numbers modulo torsion by Galois field and degree which are orthogonal with respect to the natural inner product associated to the $L^2$ Weil height recently introduced by Allcock and Vaaler. Using these decompositions, we then introduce vector space norms associated to the Mahler measure. We formulate $L^p$ Lehmer conjectures involving lower bounds on these norms and prove that these new conjectures are equivalent to their classical counterparts, specifically, the classical Lehmer conjecture in the $p=1$ case and the Schinzel-Zassenhaus conjecture in the $p=\infty$ case.

In a previous paper, the authors introduced several vector space norms on the space of algebraic numbers modulo torsion which corresponded to the Mahler measure on a certain class of numbers and allowed the authors to formulate $L^p$ Lehmer conjectures which were equivalent to their classical counterparts. In this paper, we introduce and study several analogous norms which are constructed in order to satisfy an extremal property with respect to the Mahler measure. These norms are a natural generalization of the metric Mahler measure introduced by Dubickas and Smyth. We evaluate these norms on certain classes of algebraic numbers and prove that the infimum in the construction is achieved in a certain finite dimensional space.

In this paper we determine topologically the canonical component of the $SL_2(\mathbb{C})$ character variety of the Whitehead link complement.

This paper exhibits an infinite family of hyperbolic knot complements that have three knot complements in their respective commensurability classes.

We show that the optimal constant in Erd\"{o}s' sum-free subset theorem cannot be larger than $11/28 \approx .393$.

We present a classification of the so-called �additive symmetric 2-cocycles� of arbitrary degree and dimension over F_p, along with a partial result and some conjectures for m-cocycles over F_p, m>2. This expands greatly on a result originally due to Lazard and more recently investigated by Ando, Hopkins and Strickland, and together with their work this culminates in a complete classification of 2-cocycles over an arbitrary commutative ring. The ring classifying these polynomials finds application in algebraic topology, to be fully explored in a sequel.

Given a computer model of a physical object, it is often quite difficult to visualize and quantify any global effects on the shape representation caused by local uncertainty and local errors in the data. This problem is further amplified when dealing with hierarchical representations containing varying levels of detail and / or shapes undergoing dynamic deformations. In this paper, we compute, quantify and visualize the complementary topological and geometrical features of 3D shape models, namely, the tunnels, pockets and internal voids of the object. We find that this approach sheds a unique light on how a model is affected by local uncertainty, errors or modifications and show how the presence or absence of complementary shape features can be essential to an object's structural form and function.

The selection of appropriate level sets for the quantitative visualization of three dimensional imaging or simulation data is a problem that is both fundamental and essential. The selected level set needs to satisfy several topological and geometric constraints to be useful for subsequent quantitative processing and visualization. For an initial selection of an isosurface, guided by contour tree data structures, we detect the topological features by computing stable and unstable manifolds of the critical points of the distance function induced by the isosurface. We further enhance the description of these features by associating geometric attributes with them. We then rank the attributed features and provide a handle to them for curation of the topological anomalies.