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Tame geometry originated in mathematical logic and implements strong finiteness properties by defining the notion of tame sets and functions. In part I we argued that observables in a wide class of quantum field theories are tame functions and that the tameness of a theory relies on its UV definition. The aims of this work are (1) to formalize the connection between quantum field theories and logical structures, and (2) to investigate the tameness of conformal field theories. To address the first aim, we start from a set of quantum field theories and explain how they define a logical structure that is subsequently extended to a second structure by adding physical observables. Tameness, or o-minimality, of the two structures is then a well-defined property, and sharp statements can be made by identifying these with known examples in mathematics. For the second aim we quantify our expectations on the tameness of the set of conformal field theories and effective theories that can be coupled to quantum gravity. We formulate tameness conjectures about conformal field theory observables and propose universal constraints that render spaces of conformal field theories to be tame sets. We test these conjectures in several examples and highlight first implications., Comment: 63 pages, 3 figures

Tame geometry originated in mathematical logic and implements strong finiteness properties by defining the notion of tame sets and functions. In part I we argued that observables in a wide class of quantum field theories are tame functions and that the tameness of a theory relies on its UV definition. The aims of this work are (1) to formalize the connection between quantum field theories and logical structures, and (2) to investigate the tameness of conformal field theories. To address the first aim, we start from a set of quantum field theories and explain how they define a logical structure that is subsequently extended to a second structure by adding physical observables. Tameness, or o-minimality, of the two structures is then a well-defined property, and sharp statements can be made by identifying these with known examples in mathematics. For the second aim we quantify our expectations on the tameness of the set of conformal field theories and effective theories that can be coupled to quantum gravity. We formulate tameness conjectures about conformal field theory observables and propose universal constraints that render spaces of conformal field theories to be tame sets. We test these conjectures in several examples and highlight first implications., Comment: 64 pages, 3 figures, v2: discussion on conformal blocks improved

Artificial intelligence is making spectacular progress, and one of the best examples is the development of large language models (LLMs) such as OpenAI's GPT series. In these lectures, written for readers with a background in mathematics or physics, we give a brief history and survey of the state of the art, and describe the underlying transformer architecture in detail. We then explore some current ideas on how LLMs work and how models trained to predict the next word in a text are able to perform other tasks displaying intelligence., Comment: 47 pages (v2: added references, corrected typos)

We propose a generalized finiteness principle for physical theories, in terms of the concept of tameness in mathematical logic. A tame function or space can only have a finite amount of structure, in a precise sense which we explain. Tameness generalizes the notion of an analytic function to include certain non-analytic limits, and we show that this includes many limits which are known to arise in physics. For renormalizable quantum field theories, we give a general proof that amplitudes at each order in the loop expansion are tame functions of the external momenta and the couplings. We then consider a variety of exact non-perturbative results and show that they are tame but only given constraints on the UV definition of the theory. This provides further evidence for the recent conjecture of the second author that all effective theories that can be coupled to quantum gravity are tame. We also discuss whether renormalization group flow is tame, and comment on the applicability of our results to effective theories., Comment: 50 pages

We propose a generalized finiteness principle for physical theories, in terms of the concept of tameness in mathematical logic. A tame function or space can only have a finite amount of structure, in a precise sense which we explain. Tameness generalizes the notion of an analytic function to include certain non-analytic limits, and we show that this includes many limits which are known to arise in physics. For renormalizable quantum field theories, we give a general proof that amplitudes at each order in the loop expansion are tame functions of the external momenta and the couplings. We then consider a variety of exact non-perturbative results and show that they are tame but only given constraints on the UV definition of the theory. This provides further evidence for the recent conjecture of the second author that all effective theories that can be coupled to quantum gravity are tame. We also discuss whether renormalization group flow is tame, and comment on the applicability of our results to effective theories., Comment: 50 pages

David Mumford made groundbreaking contributions in many fields, including the pure mathematics of algebraic geometry and the applied mathematics of machine learning and artificial intelligence. His work in both fields influenced my career at several key moments.

A very popular model in machine learning is the feedforward neural network (FFN). The FFN can approximate general functions and mitigate the curse of dimensionality. Here we introduce FFNs which represent sections of holomorphic line bundles on complex manifolds, and ask some questions about their approximating power. We also explain formal similarities between the standard approach to supervised learning and the problem of finding numerical Ricci flat K\"ahler metrics, which allow carrying some ideas between the two problems., Comment: 13 pages, version to appear in PAMQ

A knowledge graph (KG) is a data structure which represents entities and relations as the vertices and edges of a directed graph with edge types. KGs are an important primitive in modern machine learning and artificial intelligence. Embedding-based models, such as the seminal TransE [Bordes et al., 2013] and the recent PairRE [Chao et al., 2020] are among the most popular and successful approaches for representing KGs and inferring missing edges (link completion). Their relative success is often credited in the literature to their ability to learn logical rules between the relations. In this work, we investigate whether learning rules between relations is indeed what drives the performance of embedding-based methods. We define motif learning and two alternative mechanisms, network learning (based only on the connectivity of the KG, ignoring the relation types), and unstructured statistical learning (ignoring the connectivity of the graph). Using experiments on synthetic KGs, we show that KG models can learn motifs and how this ability is degraded by non-motif (noise) edges. We propose tests to distinguish the contributions of the three mechanisms to performance, and apply them to popular KG benchmarks. We also discuss an issue with the standard performance testing protocol and suggest an improvement. To appear in the proceedings of Complex Networks 2021., Comment: 16 pages

A very popular model in machine learning is the feedforward neural network (FFN). The FFN can approximate general functions and mitigate the curse of dimensionality. Here we introduce FFNs which represent sections of holomorphic line bundles on complex manifolds, and ask some questions about their approximating power. We also explain formal similarities between the standard approach to supervised learning and the problem of finding numerical Ricci flat K\"ahler metrics, which allow carrying some ideas between the two problems., Comment: 13 pages, version to appear in PAMQ

The immunoreceptor tyrosine-based inhibition motif (ITIM)-containing receptor G6b-B is critical for platelet production and activation. Loss of G6b-B results in severe macrothrombocytopenia, myelofibrosis and aberrant platelet function in mice and humans. Using a combination of immunohistochemistry, affinity chromatography and proteomics, we identified the extracellular matrix heparan sulfate (HS) proteoglycan perlecan as a G6b-B binding partner. Subsequent in vitro biochemical studies and a cell-based genetic screen demonstrated that the interaction is specifically mediated by the HS chains of perlecan. Biophysical analysis revealed that heparin forms a high-affinity complex with G6b-B and mediates dimerization. Using platelets from humans and genetically modified mice, we demonstrate that binding of G6b-B to HS and multivalent heparin inhibits platelet and megakaryocyte function by inducing downstream signaling via the tyrosine phosphatases Shp1 and Shp2. Our findings provide novel insights into how G6b-B is regulated and contribute to our understanding of the interaction of megakaryocytes and platelets with glycans.

The immunoreceptor tyrosine-based inhibition motif (ITIM)-containing receptor G6b-B is critical for platelet production and activation. Loss of G6b-B results in severe macrothrombocytopenia, myelofibrosis and aberrant platelet function in mice and humans. Using a combination of immunohistochemistry, affinity chromatography and proteomics, we identified the extracellular matrix heparan sulfate (HS) proteoglycan perlecan as a G6b-B binding partner. Subsequent in vitro biochemical studies and a cell-based genetic screen demonstrated that the interaction is specifically mediated by the HS chains of perlecan. Biophysical analysis revealed that heparin forms a high-affinity complex with G6b-B and mediates dimerization. Using platelets from humans and genetically modified mice, we demonstrate that binding of G6b-B to HS and multivalent heparin inhibits platelet and megakaryocyte function by inducing downstream signaling via the tyrosine phosphatases Shp1 and Shp2. Our findings provide novel insights into how G6b-B is regulated and contribute to our understanding of the interaction of megakaryocytes and platelets with glycans.

The immunoreceptor tyrosine-based inhibition motif (ITIM)-containing receptor G6b-B is critical for platelet production and activation. Loss of G6b-B results in severe macrothrombocytopenia, myelofibrosis and aberrant platelet function in mice and humans. Using a combination of immunohistochemistry, affinity chromatography and proteomics, we identified the extracellular matrix heparan sulfate (HS) proteoglycan perlecan as a G6b-B binding partner. Subsequent in vitro biochemical studies and a cell-based genetic screen demonstrated that the interaction is specifically mediated by the HS chains of perlecan. Biophysical analysis revealed that heparin forms a high-affinity complex with G6b-B and mediates dimerization. Using platelets from humans and genetically modified mice, we demonstrate that binding of G6b-B to HS and multivalent heparin inhibits platelet and megakaryocyte function by inducing downstream signaling via the tyrosine phosphatases Shp1 and Shp2. Our findings provide novel insights into how G6b-B is regulated and contribute to our understanding of the interaction of megakaryocytes and platelets with glycans.

The immunoreceptor tyrosine-based inhibition motif (ITIM)-containing receptor G6b-B is critical for platelet production and activation. Loss of G6b-B results in severe macrothrombocytopenia, myelofibrosis and aberrant platelet function in mice and humans. Using a combination of immunohistochemistry, affinity chromatography and proteomics, we identified the extracellular matrix heparan sulfate (HS) proteoglycan perlecan as a G6b-B binding partner. Subsequent in vitro biochemical studies and a cell-based genetic screen demonstrated that the interaction is specifically mediated by the HS chains of perlecan. Biophysical analysis revealed that heparin forms a high-affinity complex with G6b-B and mediates dimerization. Using platelets from humans and genetically modified mice, we demonstrate that binding of G6b-B to HS and multivalent heparin inhibits platelet and megakaryocyte function by inducing downstream signaling via the tyrosine phosphatases Shp1 and Shp2. Our findings provide novel insights into how G6b-B is regulated and contribute to our understanding of the interaction of megakaryocytes and platelets with glycans.

Objective To determine whether the ratio of cerebrospinal fluid (CSF) immunoglobulin kappa to lambda light chains at time of multiple sclerosis (MS) diagnosis predicts disease progression and whether this was intrinsic to CSF plasmablasts. Methods CSF and peripheral blood were obtained from patients undergoing elective diagnostic lumbar puncture and included clinically isolated syndrome (CIS) (n=43), relapsing remitting MS (RRMS; n=50), primary progressive MS (PPMS; n=20) and other neurological disease controls, both inflammatory (ONID; n=23) and non-inflammatory (OND; n=114). CSF samples were assayed for free and immunoglobulin-associated light chains and on B cells and plasmablasts. Clinical follow-up data were collected during a 5-year follow-up period where available. Results There was an increased median CSF κ:λ free light chain (FLC) in all MS groups (CIS: 18.2, 95% CI 6.8 to 30.3; RRMS: 4.4, 95% CI 2.7 to 11.4; PPMS: 12.0, 95% CI 3.6 to 37.1) but not controls (OND: 1.61, 95% CI 1.4 to 1.9; ONID: 1.7, 95% CI 1.3 to 2.2; p<0.001). This ratio predicted Expanded Disability Status Scores (EDSS) progression at 5 years, with a lower median EDSS in the group with high (>10) CSF κ:λ FLC (0.0, 95% CI 0 to 2.5 vs 2.5, 95% CI 0 to 4, high vs low; p=0.049). CSF κ:λ FLC correlated with CSF IgG1 κ:λ (r=0.776; p<0.0001) and was intrinsic to CSF plasmablasts (r=0.65; p=0.026). Conclusions These data demonstrate that CSF immunoglobulin κ:λ ratios, determined at the time of diagnostic lumbar puncture, predict MS disease progression and may therefore be useful prognostic markers for early therapeutic stratification.

Objective To determine whether the ratio of cerebrospinal fluid (CSF) immunoglobulin kappa to lambda light chains at time of multiple sclerosis (MS) diagnosis predicts disease progression and whether this was intrinsic to CSF plasmablasts. Methods CSF and peripheral blood were obtained from patients undergoing elective diagnostic lumbar puncture and included clinically isolated syndrome (CIS) (n=43), relapsing remitting MS (RRMS; n=50), primary progressive MS (PPMS; n=20) and other neurological disease controls, both inflammatory (ONID; n=23) and non-inflammatory (OND; n=114). CSF samples were assayed for free and immunoglobulin-associated light chains and on B cells and plasmablasts. Clinical follow-up data were collected during a 5-year follow-up period where available. Results There was an increased median CSF κ:λ free light chain (FLC) in all MS groups (CIS: 18.2, 95% CI 6.8 to 30.3; RRMS: 4.4, 95% CI 2.7 to 11.4; PPMS: 12.0, 95% CI 3.6 to 37.1) but not controls (OND: 1.61, 95% CI 1.4 to 1.9; ONID: 1.7, 95% CI 1.3 to 2.2; p<0.001). This ratio predicted Expanded Disability Status Scores (EDSS) progression at 5 years, with a lower median EDSS in the group with high (>10) CSF κ:λ FLC (0.0, 95% CI 0 to 2.5 vs 2.5, 95% CI 0 to 4, high vs low; p=0.049). CSF κ:λ FLC correlated with CSF IgG1 κ:λ (r=0.776; p<0.0001) and was intrinsic to CSF plasmablasts (r=0.65; p=0.026). Conclusions These data demonstrate that CSF immunoglobulin κ:λ ratios, determined at the time of diagnostic lumbar puncture, predict MS disease progression and may therefore be useful prognostic markers for early therapeutic stratification.

We propose a new approach for multiverse analysis based on computational complexity, which leads to a new family of "computational" measure factors. By defining a cosmology as a space-time containing a vacuum with specified properties (for example small cosmological constant) together with rules for how time evolution will produce the vacuum, we can associate global time in a multiverse with clock time on a supercomputer which simulates it. We argue for a principle of "limited computational complexity" governing early universe dynamics as simulated by this supercomputer, which translates to a global measure for regulating the infinities of eternal inflation. The rules for time evolution can be thought of as a search algorithm, whose details should be constrained by a stronger principle of "minimal computational complexity." Unlike previously studied global measures, ours avoids standard equilibrium considerations and the well-known problems of Boltzmann Brains and the youngness paradox. We also give various definitions of the computational complexity of a cosmology, and argue that there are only a few natural complexity classes. (v2: version submitted for publication: clarified section 5.3; added references) (v3: added discussion of marginally hospitable vacua. Version to appear in Annals of Physics), Comment: 50 pages, 6 figures

Yau proved an existence theorem for Ricci-flat K\"ahler metrics in the 1970's, but we still have no closed form expressions for them. Nevertheless there are several ways to get approximate expressions, both numerical and analytical. We survey some of this work and explain how it can be used to obtain physical predictions from superstring theory., Comment: 10 pages. Version to appear in Nuclear Physics B special issue: 60 Years of Calabi Conjecture

It is well understood --- through string dualities --- that there are 20 massless vector fields in the spectrum of eight-dimensional F-theory compactifications on smooth elliptically fibered K3 surfaces at a generic point in the K3 moduli space. Such F-theory vacua, which do not have any enhanced gauge symmetries, can be thought of as supersymmetric type IIB compactifications on P1 with 24 (p,q) seven-branes. Naively, one might expect there to be 24 massless vector fields in the eight-dimensional effective theory coming from world-volume gauge fields of the 24 branes. In this paper, we show how the vector field spectrum of the eight-dimensional effective theory can be obtained from the point of view of type IIB supergravity coupled to the world-volume theory of the seven-branes. In particular, we first show that the two-forms of the type IIB theory absorb the seven-brane world-volume gauge fields via the Cremmer-Scherk mechanism. We then proceed to show that the massless vector fields of the eight-dimensional theory come from KK-reducing the SL(2,Z) doublet two-forms of type IIB theory along SL(2,Z) doublet one-forms on the P1. We also discuss the relation between these vector fields and the "eaten" world-volume vector fields of the seven-branes., Comment: 33 pages + appendices, 5 figures; v2: minor corrections, reference added

We show that the entropy of certain conformal interfaces between $N=(2,2)$ sigma models that belong to the same moduli space, has a natural geometric interpretation in the large volume limit as Calabi's diastasis function. This is an extension of the well-known relation between the quantum K\"ahler potential and the overlap of canonical Ramond-Ramond ground states in $N=(2,2)$ models., Comment: 18 pages, 1 figure

We conjecture that a non-flat $D$-real-dimensional compact Calabi-Yau manifold, such as a quintic hypersurface with D=6, or a K3 manifold with D=4, has locally length minimizing closed geodesics, and that the number of these with length less than L grows asymptotically as L^{D}. We also outline the physical arguments behind this conjecture, which involve the claim that all states in a nonlinear sigma model can be identified as "momentum" and "winding" states in the large volume limit., Comment: minor corrections, 43 pages, to appear in frontiers in mathematical physics. Frontiers in Physics, Dec 16, 2013

The connection of maximally supersymmetric Yang-Mills theory to the (2,0) theory in six dimensions has raised the possibility that it might be perturbatively ultraviolet finite in five dimensions. We test this hypothesis by computing the coefficient of the first potential ultraviolet divergence of planar (large N_c) maximally supersymmetric Yang-Mills theory in D = 5, which occurs at six loops. We show that the coefficient is nonvanishing. Furthermore, the numerical value of the divergence falls very close to an approximate exponential formula based on the coefficients of the divergences through five loops. This formula predicts the approximate values of the ultraviolet divergence at loop orders L > 6 in the critical dimension D = 4 + 6/L. To obtain the six-loop divergence we first construct the planar six-loop four-point amplitude integrand using generalized unitarity. The ultraviolet divergence follows from a set of vacuum integrals, which are obtained by expanding the integrand in the external momenta. The vacuum integrals are integrated via sector decomposition, using a modified version of the FIESTA program., Comment: 31 pages, revtex, 12 figures

We briefly survey our present understanding of the string landscape, and use it to discuss the chances that we will see low energy supersymmetry at the LHC., Comment: 27 pp, ws-rv9x6.cls (included) v2: improved section 6

We study effective potentials coming from compactifications of string theory. We show that, under mild assumptions, such potentials are bounded from below in four dimensions, giving an affirmative answer to a conjecture proposed by the second author in arXiv:0911.3378v4 [hep-th]. We also derive some sufficient conditions for the existence of critical points. All proofs and mathematical hypotheses are discussed in the context of their relevance to the physics of the problem.

We discuss how D=5 maximally supersymmetric Yang-Mills theory (MSYM) might be used to study or even to define the (2,0) theory in six dimensions. It is known that the compactification of (2,0) theory on a circle leads to D=5 MSYM. A variety of arguments suggest that the relation can be reversed, and that all of the degrees of freedom of (2,0) theory are already present in D=5 MSYM. If so, this relation should have consequences for D=5 SYM perturbation theory. We explore whether it might imply all orders finiteness, or else an unusual relation between the cutoff and the gauge coupling. S-duality of the reduction to D=4 may provide nonperturbative constraints or tests of these options., Comment: 17 pages (v2: added references)

We derive a holographic dual description of free quantum field theory in arbitrary dimensions, by reinterpreting the exact renormalization group, to obtain a higher spin gravity theory of the general type which had been proposed and studied as a dual theory. We show that the dual theory reproduces all correlation functions., Comment: 5 pages, 1 figure; v2: various improvements

We discuss the four dimensional effective action for type IIB flux compactifications, and obtain the quadratic terms taking warp effects into account. The analysis includes both the 4-d zero modes and their KK excitations, which become light at large warping. We identify an 'axial' type gauge for the supergravity fluctuations, which makes the four dimensional degrees of freedom manifest. The other key ingredient is the existence of constraints coming from the ten dimensional equations of motion. Applying these conditions leads to considerable simplifications, enabling us to obtain the low energy lagrangian explicitly. In particular, the warped Kahler potential for metric moduli is computed and it is shown that there are no mixings with the KK fluctuations and the result differs from previous proposals. The four dimensional potential contains a generalization of the Gukov-Vafa-Witten term, plus usual mass terms for KK modes.

We construct the effective theory of the universal Kaehler modulus in warped compactifications using the Hamiltonian formulation of general relativity. The spacetime dependent 10d solution is constructed at the linear level for both the volume modulus and its axionic partner, and nontrivial cancellations of warping effects are found in the dimensional reduction. Our main result is that the Kaehler potential is not corrected by warping, up to an overall shift in the background value of the volume modulus. We extend the analysis beyond the linearized approximation by computing the fully backreacted 10d metric corresponding to a finite volume modulus fluctuation. Also, we discuss the behavior of the modulus in strongly warped regions and show that there are no mixings with light Kaluza-Klein modes. These results are important for the phenomenology and cosmology of flux compactifications., Comment: 28 pages, 1 figure; v2. corrected typos, added refs & minor clarifications

We discuss the four dimensional effective action for type IIB flux compactifications, and obtain the quadratic terms taking warp effects into account. The analysis includes both the 4-d zero modes and their KK excitations, which become light at large warping. We identify an 'axial' type gauge for the supergravity fluctuations, which makes the four dimensional degrees of freedom manifest. The other key ingredient is the existence of constraints coming from the ten dimensional equations of motion. Applying these conditions leads to considerable simplifications, enabling us to obtain the low energy lagrangian explicitly. In particular, the warped Kahler potential for metric moduli is computed and it is shown that there are no mixings with the KK fluctuations and the result differs from previous proposals. The four dimensional potential contains a generalization of the Gukov-Vafa-Witten term, plus usual mass terms for KK modes.

We construct the effective theory of the universal Kaehler modulus in warped compactifications using the Hamiltonian formulation of general relativity. The spacetime dependent 10d solution is constructed at the linear level for both the volume modulus and its axionic partner, and nontrivial cancellations of warping effects are found in the dimensional reduction. Our main result is that the Kaehler potential is not corrected by warping, up to an overall shift in the background value of the volume modulus. We extend the analysis beyond the linearized approximation by computing the fully backreacted 10d metric corresponding to a finite volume modulus fluctuation. Also, we discuss the behavior of the modulus in strongly warped regions and show that there are no mixings with light Kaluza-Klein modes. These results are important for the phenomenology and cosmology of flux compactifications., Comment: 28 pages, 1 figure; v2. corrected typos, added refs & minor clarifications

We discuss the four dimensional effective action for type IIB flux compactifications, and obtain the quadratic terms taking warp effects into account. The analysis includes both the 4-d zero modes and their KK excitations, which become light at large warping. We identify an 'axial' type gauge for the supergravity fluctuations, which makes the four dimensional degrees of freedom manifest. The other key ingredient is the existence of constraints coming from the ten dimensional equations of motion. Applying these conditions leads to considerable simplifications, enabling us to obtain the low energy lagrangian explicitly. In particular, the warped Kahler potential for metric moduli is computed and it is shown that there are no mixings with the KK fluctuations and the result differs from previous proposals. The four dimensional potential contains a generalization of the Gukov-Vafa-Witten term, plus usual mass terms for KK modes.

We review recent work in which compactifications of string and M theory are constructed in which all scalar fields (moduli) are massive, and supersymmetry is broken with a small positive cosmological constant, features needed to reproduce real world physics. We explain how this work implies that there is a ``landscape'' of string/M theory vacua, perhaps containing many candidates for describing real world physics, and present the arguments for and against this idea. We discuss statistical surveys of the landscape, and the prospects for testable consequences of this picture, such as observable effects of moduli, constraints on early cosmology, and predictions for the scale of supersymmetry breaking., Comment: 66 pages, 3 figures, Latex with revtex4 macros. v3: version to appear in RMP

We study the computational complexity of the physical problem of finding vacua of string theory which agree with data, such as the cosmological constant, and show that such problems are typically NP hard. In particular, we prove that in the Bousso-Polchinski model, the problem is NP complete. We discuss the issues this raises and the possibility that, even if we were to find compelling evidence that some vacuum of string theory describes our universe, we might never be able to find that vacuum explicitly. In a companion paper, we apply this point of view to the question of how early cosmology might select a vacuum., Comment: JHEP3 Latex, 53 pp, 2 .eps figures

We give a ``physics proof'' of a conjecture made by the first author at Strings 2005, that the moduli spaces of certain conformal field theories are finite volume in the Zamolodchikov metric, using an RG flow argument., Comment: 8 pages, latex, JHEP

We give a ``physics proof'' of a conjecture made by the first author at Strings 2005, that the moduli spaces of certain conformal field theories are finite volume in the Zamolodchikov metric, using an RG flow argument., Comment: 8 pages, latex, JHEP

We give results for the distribution and number of flux vacua of various types, supersymmetric and nonsupersymmetric, in IIB string theory compactified on Calabi-Yau manifolds. We compare this with related problems such as counting attractor points.

We find IIb compactifications on Calabi-Yau orientifolds in which all Kahler moduli are stabilized, along lines suggested by Kachru, Kallosh, Linde and Trivedi.

It is known that string theory compactifications leading to low energy effective theories with different chiral matter content (e.g. different numbers of standard model generations) are connected through phase transitions, described by non-trivial quantum fixed point theories. We point out that such compactifications are also connected on a purely classical level, through transitions that can be described using standard effective field theory. We illustrate this with examples, including some in which the transition proceeds entirely through supersymmetric configurations.

We give an introduction to the statistical approach to studying vacua of string/M theory, and discuss recent results of Ashok and Douglas on counting supersymmetric flux vacua in type IIb Calabi-Yau compactification. To appear in the proceedings of the 2003 String Phenomenology workshop in Durham, UK., Comment: 14 pages, latex, documentclass ws-p8-50x6-00

We find IIb compactifications on Calabi-Yau orientifolds in which all Kahler moduli are stabilized, along lines suggested by Kachru, Kallosh, Linde and Trivedi.

We give results for the distribution and number of flux vacua of various types, supersymmetric and nonsupersymmetric, in IIB string theory compactified on Calabi-Yau manifolds. We compare this with related problems such as counting attractor points.

It is known that string theory compactifications leading to low energy effective theories with different chiral matter content (e.g. different numbers of standard model generations) are connected through phase transitions, described by non-trivial quantum fixed point theories. We point out that such compactifications are also connected on a purely classical level, through transitions that can be described using standard effective field theory. We illustrate this with examples, including some in which the transition proceeds entirely through supersymmetric configurations.

We discuss the question of what type and scale of supersymmetry breaking might be statistically favored among vacua of string/M theory, building on comments in Denef and Douglas, hep-th/0404116., Comment: 8 pp. Latex (v4: we identify a serious error in the original argument, and attempt to address it.)

We give an introduction to the statistical approach to studying vacua of string/M theory, and discuss recent results of Ashok and Douglas on counting supersymmetric flux vacua in type IIb Calabi-Yau compactification. To appear in the proceedings of the 2003 String Phenomenology workshop in Durham, UK., Comment: 14 pages, latex, documentclass ws-p8-50x6-00

We used a special purpose computer to integrate the orbits of the outer five planets for 100 Myr into the future and 100 Myr into the past. The strongest features in the Fourier transforms of the orbital elements of the Jovian planets can be indentified with the frequencies predicted by linear secular theory. Many of the weaker features in the Fourier spectra are identified as linear combinations of the basic frequencies. We note serious differences between our measurements and the predictions of Bretagnon (1974). The amplitude of the 3.796 Myr period libration of Pluto's longitude of perihelion is modulated with a period of 34 Myr. Very long periods, on the order of 137 million years, are also seen.

We used a special purpose computer to integrate the orbits of the outer five planets for 100 Myr into the future and 100 Myr into the past. The strongest features in the Fourier transforms of the orbital elements of the Jovian planets can be indentified with the frequencies predicted by linear secular theory. Many of the weaker features in the Fourier spectra are identified as linear combinations of the basic frequencies. We note serious differences between our measurements and the predictions of Bretagnon (1974). The amplitude of the 3.796 Myr period libration of Pluto's longitude of perihelion is modulated with a period of 34 Myr. Very long periods, on the order of 137 million years, are also seen.

We used a special purpose computer to integrate the orbits of the outer five planets for 100 Myr into the future and 100 Myr into the past. The strongest features in the Fourier transforms of the orbital elements of the Jovian planets can be indentified with the frequencies predicted by linear secular theory. Many of the weaker features in the Fourier spectra are identified as linear combinations of the basic frequencies. We note serious differences between our measurements and the predictions of Bretagnon (1974). The amplitude of the 3.796 Myr period libration of Pluto's longitude of perihelion is modulated with a period of 34 Myr. Very long periods, on the order of 137 million years, are also seen.

We discuss systematic approaches to the classification of string/M theory vacua, and physical questions this might help us resolve. To this end, we initiate the study of ensembles of effective Lagrangians, which can be used to precisely study the predictive power of string theory, and in simple examples can lead to universality results. Using these ideas, we outline an approach to estimating the number of vacua of string/M theory which can realize the Standard Model., Comment: harvmac, 72pp (v4: fixed error in discussion of quiver ensembles)