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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., 46 pages

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., 64 pages, 3 figures, v2: discussion on conformal blocks improved

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

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

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., 16 pages

We study four-dimensional fractional quantum Hall states on CP2 geometry from microscopic approaches. While in 2d the standard Laughlin wave function, given by a power of Vandermonde determinant, admits a product representation in terms of the Jastrow factor, this is no longer true in higher dimensions. In 4d we can define two different types of Laughlin wavefunctions, the Determinant-Laughlin (Det-Laughlin) and Jastrow-Laughlin (Jas-Laughlin) states. We find that they are exactly annihilated by, respectively, two-particle and three-particle short ranged interacting Hamiltonians. We then mainly focus on the ground state, low energy excitations and the quasi-hole degeneracy of Det-Laughlin state. The quasi-hole degeneracy exhibits an anomalous counting, indicating the existence of multiple forms of quasi-hole wavefunctions. We argue that these are captured by the mathematical framework of the "commutative algebra of N-points in the plane". We also generalize the pseudopotential formalism to dimensions higher than two, by considering coherent state wavefunction of bound states. The microscopic wavefunctions and Hamiltonians studied in this work pave the way for systematic study of high dimensional topological phase of matter that is potentially realizable in cold atom and optical experiments.

We propose machine learning inspired methods for computing numerical Calabi-Yau (Ricci flat K\"ahler) metrics, and implement them using Tensorflow/Keras. We compare them with previous work, and find that they are far more accurate for manifolds with little or no symmetry. We also discuss issues such as overparameterization and choice of optimization methods., Comment: Version accepted by MSML 2021

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; p10) 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

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

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

Current pharmacological and surgical treatments for Parkinson's disease offer symptomatic improvements to those suffering from this incurable degenerative neurological disorder, but none of these has convincingly shown effects on disease progression. Novel approaches based on gene therapy have several potential advantages over conventional treatment modalities. These could be used to provide more consistent dopamine supplementation, potentially providing superior symptomatic relief with fewer side effects. More radically, gene therapy could be used to correct the imbalances in basal ganglia circuitry associated with the symptoms of Parkinson's disease, or to preserve or restore dopaminergic neurons lost during the disease process itself. The latter neuroprotective approach is the most exciting, as it could theoretically be disease modifying rather than simply symptom alleviating. Gene therapy agents using these approaches are currently making the transition from the laboratory to the bedside. This paper summarises the theoretical approaches to gene therapy for Parkinson's disease and the findings of clinical trials in this rapidly changing field.

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 consider a probe in a BPS black hole in type II strings compactified on Calabi-Yau manifolds, and conjecture that its moduli space metric is the balanced metric., Comment: 15 pages

We present evidence that the number of string/$M$ theory vacua consistent with experiments is a finite number. We do this both by explicit analysis of infinite sequences of vacua and by applying various mathematical finiteness theorems.

We discuss conjectures following from the attractor mechanism in type II string theory about the possible Chern classes of stable holomorphic vector bundles on Calabi-Yau threefolds. In particular, we give sufficient conditions for Chern classes to correspond to stable bundles., 32 pages, Latex

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., 8 pages, latex, JHEP

This article will appear in the Encyclopedia of Mathematical Physics (Elsevier, 2006) and follows its referencing guidelines., Comment: 21 pages, Latex

This is a submission to the Encyclopedia of Mathematical Physics (Elsevier, 2006) and conforms to its referencing guidelines., Comment: 16 pages, Latex

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 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.)

Motivated by the vacuum selection problem of string/M theory, we study a new geometric invariant of a positive Hermitian line bundle $(L, h)\to M$ over a compact K\"ahler manifold: the expected distribution of critical points of a Gaussian random holomorphic section $s \in H^0(M, L)$ with respect to the Chern connection $\nabla_h$. It is a measure on $M$ whose total mass is the average number $\mathcal{N}^{crit}_h$ of critical points of a random holomorphic section. We are interested in the metric dependence of $\mathcal{N}^{crit}_h$, especially metrics $h$ which minimize $\mathcal{N}^{crit}_h$. We concentrate on the asymptotic minimization problem for the sequence of tensor powers $(L^N, h^N)\to M$ of the line bundle and their critical point densities $\mathcal{K}^{crit}_{N,h}(z)$. We prove that $\mathcal{K}^{crit}_{N,h}(z)$ has a complete asymptotic expansion in $N$ whose coefficients are curvature invariants of $h$. The first two terms in the expansion of $\mathcal{N}^{crit}_{N,h}$ are topological invariants of $(L, M)$. The third term is a topological invariant plus a constant $\beta_2(m)$ (depending only on the dimension $m$ of $M$) times the Calabi functional $\int_M \rho^2 dV_h$, where $\rho$ is the scalar curvature of the K\"ahler metric $\omega_h:=\frac i2 \Theta_h$. We give an integral formula for $\beta_2(m)$ and show, by a computer assisted calculation, that $\beta_2(m)>0$ for $m\leq 5$, hence that $\mathcal{N}^{crit}_{N,h}$ is asymptotically minimized by the Calabi extremal metric (when one exists). We conjecture that $\beta_2(m)>0$ in all dimensions, i.e. the Calabi extremal metric is always the asymptotic minimizer., Comment: 35 pages; 2 Maple worksheets. Final version for Journal of Differential Geometry. This is the second paper in a three-part series including math.CV/0402326 and math-ph/0506015. This revision contains additional conjectures and numerical results, as well as stylistic changes and minor corrections

A popular way to study N=1 supersymmetric gauge theories is to realize them geometrically in string theory, as suspended brane constructions, D-branes wrapping cycles in Calabi-Yau manifolds, orbifolds, and otherwise. Among the applications of this idea are simple derivations and generalizations of Seiberg duality for the theories which can be so realized. We abstract from these arguments the idea that Seiberg duality arises because a configuration of gauge theory can be realized as a bound state of a collection of branes in more than one way, and we show that different brane world-volume theories obtained this way have matching moduli spaces, the primary test of Seiberg duality. Furthermore, we do this by defining ``brane'' and all the other ingredients of such arguments purely algebraically, for a very large class of N=1 quiver supersymmetric gauge theories, making physical intuitions about brane-antibrane systems and tachyon condensation precise using the tools of homological algebra. These techniques allow us to compute the spectrum and superpotential of the dual theory from first principles, and to make contact with geometry and topological string theory when this is appropriate, but in general provide a more abstract notion of ``noncommutative geometry'' which is better suited to these problems. This makes contact with mathematical results in the representation theory of algebras; in this language, Seiberg duality is a tilting equivalence between the derived categories of the quiver algebras of the dual theories., 52 pages, 6 figures, Latex in JHEP format

We discuss some mathematical conjectures which have come out of the Dirichlet branes in superstring theory, focusing on the case of supersymmetric branes in Calabi-Yau compactification. This has led to the formulation of a notion of stability for objects in a derived category, contact with Kontsevich's homological mirror symmetry conjecture, and "physics proofs" for many of the subsequent conjectures based on it, such as the representation of Calabi-Yau monodromy by autoequivalences of the derived category., Comment: 15 pp., amstex using special .sty (included). To appear in the 2002 ICM proceedings

We discuss the recent proposal that BPS D-branes in Calabi-Yau compactification of type II string theory are Pi-stable objects in the derived category of coherent sheaves., Comment: amslatex, 17 pp. To appear in the proceedings of Strings 2001, Mumbai, India

We give an overview of recent work on Dirichlet branes on Calabi-Yau threefolds which makes contact with Kontsevich's homological mirror symmetry proposal, proposes a new definition of stability which is appropriate in string theory, and provides concrete quiver categories equivalent to certain categories of branes on CY. To appear in the proceedings of the 3rd European Congress of Mathematics., Comment: latex (birkart.cls), 18pp

This is a write-up of lectures given at the 1998 Spring School at the Abdus Salam ICTP. We give a conceptual introduction to D-geometry, the study of geometry as seen by D-branes in string theory, and to noncommutative geometry as it has appeared in D-brane and Matrix theory physics., LaTeX, 26 pp. Added reference

This is a write-up of two lectures on AdS/CFT correspondance given by the authors at the 1998 Spring School at the Abdus Salam ICTP, Comment: Uses Sprocl.sty - supplied

We show that discrete torsion is implemented in a D-brane world-volume theory by using a projective representation of the orbifold point group. We study the example of C^3/Z_2 x Z_2 and show that the resolution of singularities agrees with that proposed by Vafa and Witten. A new type of fractional brane appears., 10 pp., harvmac

We consider (3 + 1)-dimensional N=4 super Yang-Mills theory with a nonvanishing scalar Higgs vacuum expectation value, and compare this theory to AdS supergravity with branes in the bulk. We show that the one-loop effective potential for excitations of the Yang-Mills field agrees with the classical linearized potential for brane waves in the AdS picture in the limit of long wavelengths. This supports the idea that the AdS/CFT correspondance fits into string theory as expected from previous work., latex, 8 pp. (v2) Added references and a comment on other base spaces. (v3) fixed sign error

We identify a non Fermi Liquid (NFL) class of fixed points describing the infrared behaviour of interacting chiral fermions in one dimension. The thermodynamic properties and asymptotic correlation functions are characterized by universal exponents, which we determine by means of conformal and Bethe-Ansatz techniques. The mechanism leading to the breakdown of Fermi-Liquid theory is quite general and can be expected to be realized in systems with broken T-invariance. As an example we study the edge states of interacting QHE systems. We calculate the universal frequency dependence of the spin conductance in these systems as well as the NMR response., Revtex, 8 pages, 10 figures

We explore the extent to which a local string theory dynamics in anti-de Sitter space can be determined from its proposed Conformal Field Theory (CFT) description. Free fields in the bulk are constructed from the CFT operators, but difficulties are encountered when one attempts to incorporate interactions. We also discuss general features of black hole dynamics as seen from the CFT perspective. In particular, we argue that the singularity of AdS_3 black holes is resolved in the CFT description., Comment: harvmac, 26 pp

We study toroidal compactification of Matrix theory, using ideas and results of non-commutative geometry. We generalize this to compactification on the noncommutative torus, explain the classification of these backgrounds, and argue that they correspond in supergravity to tori with constant background three-form tensor field. The paper includes an introduction for mathematicians to the IKKT formulation of Matrix theory and its relation to the BFSS Matrix theory., Comment: harvmac, 41 pp. A slightly simpler BPS mass formula is proposed

We give a broad overview of superstring duality, Dirichlet branes, and some implications of both for questions about the structure of space-time at short distances., To appear in the proceedings of the LXIV Les Houches session on `Quantum Symmetries', August 1995. latex, 29 pp

We develop and study a D-brane realization of 4D N=2 super Yang-Mills theory. It is a type IIB string theory compactified on R^6\times K3 and containing parallel 7-branes. It can also be regarded as a subsector of Vafa's F-theory compactified on K3\times K3 and is thus dual to the heterotic string on K3\times T^2. We show that the one-loop prepotential in this gauge theory is exactly equal to the interaction produced by classical closed string exchange. A monopole configuration corresponds to an open Dirichlet 5-brane wrapping around K3 with ends attached to two 7-branes., Comment: harvmac, 13 pp. Revised discussion of (p,q) 7-branes

Effective field theories in type I and II superstring theories for D-branes located at points in the orbifold C^2/Z_n are supersymmetric gauge theories whose field content is conveniently summarized by a `quiver diagram,' and whose Lagrangian includes non-metric couplings to the orbifold moduli: in particular, twisted sector moduli couple as Fayet-Iliopoulos terms in the gauge theory. These theories describe D-branes on resolved ALE spaces. Their spaces of vacua are moduli spaces of smooth ALE metrics and Yang-Mills instantons, whose metrics are explicitly computable. For U(N) instantons, the construction exactly reproduces results of Kronheimer and Nakajima., Comment: 50 pages, 11 figures, harvmac/epsf

We study models of chiral interacting fermions by means of conformal and Bethe-Ansatz techniques, and determine their thermodynamic properties and asymptotic correlation functions. We identify a class of fixed points characterizing the infrared behavior of the models. They display {\it chirally stabilized} non Fermi-Liquid behavior characterized by universal exponents. A realization of these fluids may be found in the edge states of bilayered QHE systems. We calculate the universal temperature dependence of the staggered conductance in these systems., revtex, no figures

We discuss a set of universal couplings between superstring Ramond-Ramond gauge fields and the gauge fields internal to D-branes, with emphasis on their topological consequences, and argue that instanton solutions in these internal theories are equivalent to D-branes. A particular example is the Dirichlet 5-brane in type I theory, which Witten recently showed is the zero size limit of an SO(32) instanton. Its effective world-volume theory is an Sp(1) gauge theory, unbroken in the zero size limit. We show that the zero size limit of an instanton in this theory is a 1-brane, which can be described as a bound state of the Dirichlet 1-brane with the 5-brane. Considering several 1 and 5-branes provides a description of moduli spaces of Sp(N) instantons, and a type II generalization is given which should describe U(N) instantons., Comment: harvmac, 10 pp. (References added)

We reinterpret U(N) Chern-Simons-Witten theory quantized on a torus as a free fermion system. Its Hilbert space and some observables are simply related to those of group quantum mechanics, even at finite N and k. Its large N limit can be described using techniques developed for matrix quantum mechanics and two-dimensional Yang-Mills theory. We discuss the bosonization of this theory, which for YM_2 gave a precise interpretation of Wilson loop operators in terms of string creation and annihilation operators, and examine its consequences for a string interpretation here. The formalism seems entirely adequate for the leading large N results and in a sense can be thought of as a `classical string field theory'. In considering subleading orders in 1/N, we identify some major differences between CSW and YM_2, which must be dealt with to find a CSW gauge string interpretation. Although these particular differences are probably not relevant for `QCD string,' they do illustrate some of the issues there, and we comment on this. We also propose an approach to dealing with large N transitions., RU-94-29, 30 pp, 1 figure, harvmac

We try to draw lessons for higher dimensions from the string representations recently derived for large $N$ Yang-Mills theory by Gross and Taylor, Kostov, and others, and call attention to three characteristics that should be expected of a string theory precisely equivalent to a higher dimensional gauge theory: continuous world-sheets; strong coupling at short distances; and negative weights. To appear in the proceedings of the Strings '93 Berkeley conference., RU-94-9. (harvmac, 9 pp.)

We show how to use quantum mechanics on the group manifold U(N) as a tool for problems in U(N) representation theory. The quantum mechanics reduces to free fermions on the circle, which in the large N limit become relativistic. The theory can be bosonized giving the Das-Jevicki-Sakita collective field theory. The formalism is particularly suited to problems involving tensor product multiplicity (Littlewood-Richardson) coefficients. As examples, we discuss the partition function of two-dimensional Yang-Mills theory on the sphere, and the zero magnetic field limit of D-dimensional Eguchi-Kawai Yang-Mills theory. We give the leading O(N^0) solution of the latter theory, using a method which allows computing corrections. Largely (but not completely) superseded by hep-th/9311130., RU-93-13, NSF-ITP-93-39, 13 pp

Following some motivating comments on large N two-dimensional Yang-Mills theory, we discuss techniques for large N group representation theory, using quantum mechanics on the group manifold U(N), its equivalence to a quasirelativistic two-dimensional free fermion theory, and bosonization. As applications, we compute the free energy for two-dimensional Yang-Mills theory on the torus to O(1/N^2), and an interesting approximation to the leading answer for the sphere. We discuss the question of whether the free energy for the torus has R -> 1/R invariance. A substantially revised version of hep-th/9303159 with many new results., 18 pp. (LaTeX)