Results for "Terence Gaffney"
total 533took 0.11s
Segre Numbers and Hypersurface SingularitiesNov 01 1996We define the Segre numbers of an ideal as a generalization of the multiplicity of an ideal of finite colength. We prove generalizations of various theorems involving the multiplicity of an ideal such as a principle of specialization of integral dependence, ... More Invariants of D(q,p) singularitiesAug 01 2005Feb 20 2007The basic examples of functions defining non-isolated hypersurface singularities are the A(d) singularities and the D(q,p) singularities. The A(d) singularities, up to analytic equivalence, are the product of a Morse function and the zero map, while the ... More Nilpotents, Integral Closure and Equisingularity conditionsJul 28 2005In earlier work, the author described various stratification conditions for a complex analytic set X in terms of the theory of integral closure of modules. However, even if an analytic set has a reduced structure, often geometric operations like intersection ... More Infinitesimal bi-Lipschitz Equivalence of FunctionsJan 20 2016We introduce two different notions of infinitesimal bi-Lipschitz equivalence for functions, one related to bi-Lipschitz triviality of families of functions, one related to homeomorphisms which are bi-Lipschitz on the fibers of the functions in the family. ... More The Multiplicity-Polar TheoremMar 21 2007Given a family of pairs of modules parametrised by a smooth space Y, the Multiplicity-Polar Theorem relates the multiplicity of the pair of modules at a special point of the parameter to the multiplicity of the pair at a generic point. This theorem is ... More The Multiplicity Polar Theorem and Isolated SingularitiesSep 13 2005Using invariants from commutative algebra to count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal is used to count points of intersection of two analytic sets at points of non-transverse intersection. A problem ... More Pairs of modules and determinantal isolated singularitiesDec 31 2014Jan 01 2016We continue the development of the study of the equisingularity of isolated singularities, in the determinantal case. This version of the paper includes a substantial amount of new material (76% larger). The new material introduces the idea of the landscape ... More Weak subintegral closure of idealsAug 22 2007Sep 12 2008We describe some basic facts about the weak subintegral closure of ideals in both the algebraic and complex-analytic settings. We focus on the analogy between results on the integral closure of ideals and modules and the weak subintegral closure of an ... More Equisingularity and EIDSFeb 01 2016The study of Essentially Isolated Determinantal Singularites or EIDS was initiated by Ebeling and Gusein-Zade. They are non-smoothable as determinantal singularities, and in general have non-isolated singularities. Their singularities are generic in a ... More Specialization of integral dependence for modulesOct 03 1996May 14 1998We establish the principle of specialization of integral dependence for submodules of finite colength of free modules, as part of the general algebraic-geometric theory of the Buchsbaum--Rim multiplicity. Then we apply the principle to the study of equisingularity ... More What is good mathematics?Feb 13 2007Some personal thoughts and opinions on what ``good quality mathematics'' is, and whether one should try to define this term rigorously. As a case study, the story of Szemer\'edi's theorem is presented. Noncommutative sets of small doublingJun 11 2011Apr 03 2012A corollary of Kneser's theorem, one sees that any finite non-empty subset $A$ of an abelian group $G = (G,+)$ with $|A + A| \leq (2-\eps) |A|$ can be covered by at most $\frac{2}{\eps}-1$ translates of a finite group $H$ of cardinality at most $(2-\eps)|A|$. ... More An integration approach to the Toeplitz square peg problemNov 22 2016Jun 07 2017The "square peg problem" or "inscribed square problem" of Toeplitz asks if every simple closed curve in the plane inscribes a (non-degenerate) square, in the sense that all four vertices of that square lie on the curve. By a variety of arguments of a ... More The ergodic and combinatorial approaches to Szemerédi's theoremApr 20 2006A famous theorem of Szemer\'edi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial ... More A remark on partial sums involving the Mobius functionAug 29 2009Oct 05 2009Let $<\P > \subset \N$ be a multiplicative subsemigroup of the natural numbers $\N = \{1,2,3,...\}$ generated by an arbitrary set $\P$ of primes (finite or infinite). We given an elementary proof that the partial sums $\sum_{n \in < \P >: n \leq x} \frac{\mu(n)}{n}$ ... More Global regularity of wave maps II. Small energy in two dimensionsNov 22 2000Jan 10 2001We show that wave maps from Minkowski space $\R^{1+n}$ to a sphere $S^{m-1}$ are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$, in all dimensions $n \geq 2$. This generalizes the results ... More Small time asymptotics for stochastic evolution equationsDec 03 2010We obtain a large deviation principle describing the small time asymptotics of the solution of a stochastic evolution equation with multiplicative noise. Our assumptions are a condition on the linear drift operator that is satisfied by generators of analytic ... More An uncertainty principle for cyclic groups of prime orderAug 29 2003Jul 22 2004Let $G$ be a finite abelian group, and let $f: G \to \C$ be a complex function on $G$. The uncertainty principle asserts that the support $\supp(f) := \{x \in G: f(x) \neq 0\}$ is related to the support of the Fourier transform $\hat f: G \to \C$ by the ... More Mixing for progressions in non-abelian groupsDec 11 2012May 31 2013We study the mixing properties of progressions $(x,xg,xg^2)$, $(x,xg,xg^2,xg^3)$ of length three and four in a model class of finite non-abelian groups, namely the special linear groups $SL_d(F)$ over a finite field $F$, with $d$ bounded. For length three ... More A counterexample to an endpoint bilinear Strichartz inequalitySep 29 2006The endpoint Strichartz estimate $\| e^{it\Delta} f \|_{L^2_t L^\infty_x(\R \times \R^2)} \lesssim \|f\|_{L^2_x(\R^2)}$ is known to be false by the work of Montgomery-Smith, despite being only ``logarithmically far'' from being true in some sense. In ... More Geometric renormalization of large energy wave mapsNov 16 2004There has been much progress in recent years in understanding the existence problem for wave maps with small critical Sobolev norm (in particular for two-dimensional wave maps with small energy); a key aspect in that theory has been a renormalization ... More Freiman's theorem for solvable groupsJun 18 2009Feb 19 2010Freiman's theorem asserts, roughly speaking, if that a finite set in a torsion-free abelian group has small doubling, then it can be efficiently contained in (or controlled by) a generalised arithmetic progression. This was generalised by Green and Ruzsa ... More The Erdos discrepancy problemSep 17 2015Jan 13 2017We show that for any sequence $f: {\bf N} \to \{-1,+1\}$ taking values in $\{-1,+1\}$, the discrepancy $$ \sup_{n,d \in {\bf N}} \left|\sum_{j=1}^n f(jd)\right| $$ of $f$ is infinite. This answers a question of Erd\H{o}s. In fact the argument also applies ... More Global well-posedness of the Benjamin-Ono equation in H^1(R)Jul 22 2003Jun 26 2004We show that the Benjamin-Ono equation is globally well-posed in $H^s(\R)$ for $s \geq 1$. This is despite the presence of the derivative in the non-linearity, which causes the solution map to not be uniformly continuous in $H^s$ for any $s$. The main ... More Structure and randomness in combinatoricsJul 29 2007Aug 03 2007Combinatorics, like computer science, often has to deal with large objects of unspecified (or unusable) structure. One powerful way to deal with such an arbitrary object is to decompose it into more usable components. In particular, it has proven profitable ... More Inverse theorems for sets and measures of polynomial growthJul 05 2015Oct 01 2015We give a structural description of the finite subsets $A$ of an arbitrary group $G$ which obey the polynomial growth condition $|A^n| \leq n^d |A|$ for some bounded $d$ and sufficiently large $n$, showing that such sets are controlled by (a bounded number ... More Upper semicontinuity of the lamination hullSep 30 2016Let $K \subseteq \mathbb{R}^{2 \times 2}$ be a compact set, let $K^{rc}$ be its rank-one convex hull, and let $L(K)$ be its lamination convex hull. It is shown that the mapping $K \to \overline{L(K)}$ is not upper semicontinuous on the diagonal matrices ... More An integration approach to the Toeplitz square peg problemNov 22 2016The "square peg problem" or "inscribed square problem" of Toeplitz asks if every simple closed curve in the plane inscribes a (non-degenerate) square, in the sense that all four vertices of that square lie on the curve. By a variety of arguments of a ... More The sum-product phenomenon in arbitrary ringsJun 16 2008Feb 23 2009The \emph{sum-product phenomenon} predicts that a finite set $A$ in a ring $R$ should have either a large sumset $A+A$ or large product set $A \cdot A$ unless it is in some sense "close" to a finite subring of $R$. This phenomenon has been analysed intensively ... More The weak-type (1,1) of L \log L homogeneous convolution operatorsMar 29 1999Sep 06 1999We show that a homogeneous convolution kernel on an arbitrary homogeneous group which is L \log L on the unit annulus is bounded on L^p for 1 < p < \infty and is of weak-type (1,1), generalizing the result of Seeger. The proof is in a similar spirit to ... More The Nonequilibrium Dynamics of Driven Line LiquidsJun 24 1992We study the nonequilibrium dynamics of line liquids as realized in the nonlinear motion of flux lines of a superconductor driven by an applied electric current. Our analysis suggests a transition in the dynamics of the lines from a smooth, laminar phase ... More A remark on primality testing and decimal expansionsFeb 22 2008Apr 18 2010We show that for any fixed base $a$, a positive proportion of primes have the property that they become composite after altering any one of their digits in the base $a$ expansion; the case $a=2$ was already established by Cohen-Selfridge and Sun, using ... More Product set estimates for non-commutative groupsJan 18 2006Oct 26 2011We develop the Pl\"unnecke-Ruzsa and Balog-Szemer\'edi-Gowers theory of sum set estimates in the non-commutative setting, with discrete, continuous, and metric entropy formulations of these estimates. We also develop a Freiman-type inverse theorem for ... More Cancellation for the multilinear Hilbert transformMay 24 2015May 29 2015For any natural number $k$, consider the $k$-linear Hilbert transform $$ H_k( f_1,\dots,f_k )(x) := \operatorname{p.v.} \int_{\bf R} f_1(x+t) \dots f_k(x+kt)\ \frac{dt}{t}$$ for test functions $f_1,\dots,f_k: {\bf R} \to {\bf C}$. It is conjectured that ... More Two remarks on the generalised Korteweg de-Vries equationJun 09 2006Jan 20 2009We make two observations concerning the generalised Korteweg de Vries equation $u_t + u_{xxx} = \mu (|u|^{p-1} u)_x$. Firstly we give a scaling argument that shows, roughly speaking, that any quantitative scattering result for $L^2$-critical equation ... More Low regularity semi-linear wave equationsSep 19 1997We prove local well-posedness results for the semi-linear wave equation for data in $H^\gamma$, $0 < \gamma < \frac{n-3}{2(n-1)}$, extending the previously known results for this problem. The improvement comes from an introduction of a two-scale Lebesgue ... More An inverse theorem for an inequality of KneserNov 12 2017Jun 30 2018Let $G = (G,+)$ be a compact connected abelian group, and let $\mu_G$ denote its probability Haar measure. A theorem of Kneser (generalising previous results of Macbeath and Raikov) establishes the bound $$ \mu_G(A + B) \geq \min( \mu_G(A)+\mu_G(B), 1 ... More Some remarks on the lonely runner conjectureJan 09 2017Nov 02 2017The lonely runner conjecture of Wills and Cusick, in its most popular formulation, asserts that if $n$ runners with distinct constant speeds run around a unit circle ${\bf R}/{\bf Z}$ starting at a common time and place, then each runner will at some ... More Sumset and inverse sumset theorems for Shannon entropyJun 24 2009Oct 26 2009Let $G = (G,+)$ be an additive group. The sumset theory of Pl\"unnecke and Ruzsa gives several relations between the size of sumsets $A+B$ of finite sets $A, B$, and related objects such as iterated sumsets $kA$ and difference sets $A-B$, while the inverse ... More A remark on circular means of Fourier transforms of measuresJan 06 2000Jul 11 2000In a recent paper of Wolff the optimal decay of circular L^p means of compactly supported measures of finite energy was given for p>=2, with application to Falconer's distance problem. The question was then raised in that paper as to whether any non-trivial ... More Every odd number greater than 1 is the sum of at most five primesJan 31 2012Jul 03 2012We prove that every odd number $N$ greater than 1 can be expressed as the sum of at most five primes, improving the result of Ramar\'e that every even natural number can be expressed as the sum of at most six primes. We follow the circle method of Hardy-Littlewood ... More The Gaussian primes contain arbitrarily shaped constellationsJan 20 2005Dec 31 2011We show that the Gaussian primes $P[i] \subseteq \Z[i]$ contain infinitely constellations of any prescribed shape and orientation. More precisely, given any distinct Gaussian integers $v_0,...,v_{k-1}$, we show that there are infinitely many sets $\{a+rv_0,...,a+rv_{k-1}\}$, ... More Szemerédi's regularity lemma revisitedApr 22 2005Nov 16 2005Szemer\'edi's regularity lemma is a basic tool in graph theory, and also plays an important role in additive combinatorics, most notably in proving Szemer\'edi's theorem on arithmetic progressions . In this note we revisit this lemma from the perspective ... More Why are solitons stable?Feb 18 2008Feb 20 2008The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once nonlinear effects are taken into account, \emph{solitary wave} ... More