We will briefly talk about recent developments on inequalities for infinite dimensional quantum symmetries.

Zoom: https://harvard.zoom.us/j/779283357

Numerous experiments have explored the phases of the cuprates with increasing doping density p from the antiferromagnetic insulator. There is now strong evidence that the small p region is a novel phase of matter, often called the pseudo gap metal, separated from conventional Fermi liquid at larger p by a quantum phase transition. Symmetry-breaking orders play a spectator role, at best, at this quantum phase transition. I will describe trial wave functions across this metal-metal transition employing hidden layers of ancilla qubits (proposed by Ya-Hui Zhang).Quantum fluctuations are described by a gauge theory of ghost fermions that carry neither spin nor charge. I will also describe a separate approach to this transition in a t-J model with random exchange interactions in the limit of large dimensions. This approach leads to a partly solvable SYK-like critical theory of holons and spinons, and a linear in temperature resistivity from time reparameterization fluctuations. Near criticality, both approaches have in common emergent fractionalized excitations, and a significantly larger entropy than naively expected.

Zoom: https://harvard.zoom.us/j/977347126

Math Table/Open Neighborhood Seminar

You’ve probably seen pictures drawn on the chalkboard of a torus (the surface of a donut) or the Klein bottle, a torus with many holes, and perhaps other surfaces.  In this talk we will take up the question of whether or not it is possible to make a list of all of the possible pictures of surfaces. The question turns out to have a surprising and beautiful answer, but it takes some real insights to make it into an actual question in mathematics.  The story here is not only one of a striking theorem, but also one about mathematicians find, ask, and refine questions.

via Zoom Video Conferencing: https://harvard.zoom.us/j/96759150216

email vcollins@math.harvard.edu or deg@math.harvard.edu for the Password

We study gapped boundaries characterized by “fermionic condensates” in 2+1 d topological order. Mathematically, each of these condensates can be described by a super commutative Frobenius algebra. We systematically obtain the species of excitations at the gapped boundary/ junctions, and study their endomorphisms (ability to trap a Majorana fermion) and fusion rules, and generalized the defect Verlinde formula to a twisted version. We illustrate these results with explicit examples. We will also comment on the connection with topological defects in spin CFTs. We will review necessary mathematical details of Frobenius algebra and their modules that we made heavy use of.

Zoom: https://harvard.zoom.us/j/977347126

Discriminating between unknown objects in a given set is a fundamental task in experimental science. Suppose you are given a quantum system which is in one of two given states with equal probability. Determining the actual state of the system amounts to doing a measurement on it which would allow you to discriminate between the two possible states. It is known that unless the two states are mutually orthogonal, perfect discrimination is possible only if you are given arbitrarily many identical copies of the state.

In this talk we consider the task of discriminating between quantum processes, instead of quantum states. In particular, we discriminate between a pair of unitary operators acting on a quantum system whose underlying Hilbert space is possibly infinite-dimensional. We prove that in contrast to state discrimination, one needs only a finite number of copies to discriminate perfectly between the two unitaries. Furthermore, no entanglement is needed in the discrimination task. The measure of discrimination is given in terms of the energy-constrained diamond norm and one of the key ingredients of the proof is a generalization of the Toeplitz-Hausdorff Theorem in convex analysis. Moreover, we employ our results to study a novel type of quantum speed limits which apply to pairs of quantum evolutions. This work was done jointly with Simon Becker (Cambridge), Ludovico Lami (Ulm) and Cambyse Rouze (Munich).

Zoom: https://harvard.zoom.us/j/779283357

This talk is an invitation to derived methods in algebraic geometry. We will use derived algebraic geometry to give a solution to the problem of descending “bases” (semiorthogonal decompositions) on the category of perfect complexes on algebraic varieties. This generalizes, and gives a uniform treatment of the work of Elagin, Shinder, Bernardara, Bergh-Schürer, Auel-Bernardara and Ballard-Duncan-McFaddin. Based on joint work with Ben Antieau in ArXiv:1912.08970.

Zoom: https://harvard.zoom.us/j/91794282895?pwd=VFZxRWdDQ0VNT0hsVTllR0JCQytoZz09

We revisit ‘t Hooft anomalies in (1+1)d non-spin quantum field theory, starting from the consistency and locality conditions, and find that consistent U(1) and gravitational anomalies cannot always be canceled by properly quantized (2+1)d classical Chern-Simons actions.  On the one hand, we prove that certain exotic anomalies can only be realized by non-unitary or non-compact theories; on the other hand, without insisting on unitarity, the exotic anomalies present a small caveat to the inflow paradigm.  For the mixed U(1) gravitational anomaly, we propose an inflow mechanism involving a mixed U(1) x SO(2) classical Chern-Simons action, with a boundary condition that matches the SO(2) gauge field with the (1+1)d spin connection.  Furthermore, we show that this mixed anomaly gives rise to an isotopy anomaly of U(1) topological defect lines.  The holomorphic bc ghost system realizes all the exotic consistent anomalies.

Zoom: https://harvard.zoom.us/j/977347126

I discuss the fluctuations of individual eigenvalues of the adjacency matrix of the Erdös-Rényi graph $G(N,p)$. I show that if $N^{-1}\ll p \ll N^{-2/3}, then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. The main technical tool of the proof is a rigidity bound of accuracy $N^{-1-\varepsilon}p^{-1/2}$ for the extreme eigenvalues, which avoids the $(Np)^{-1}$-expansions from previous works. Joint work with Antti Knowles.

Zoom: https://harvard.zoom.us/j/98520388668?pwd=c1hVZk5oc3B6ZTVjUUlTN0J2dmdsQT09

I will report on a dynamical way of thinking and tell you a little about my own personal experience as a mathematician.

Zoom: https://harvard.zoom.us/j/96759150216

Certain patterns of symmetry fractionalization in (2+1)D topologically ordered phases of matter can be anomalous, which means that they possess an obstruction to being realized in purely (2+1)D. In this talk, I will explain our recent results showing how to compute the anomaly for symmetry-enriched topological (SET) states of bosons in complete generality. Given any unitary modular tensor category (UMTC) and symmetry fractionalization class for a global symmetry group G, I will show how to define a (3+1)D topologically invariant path integral in terms of a state sum for a G symmetry- protected topological (SPT) state. This also determines an exactly solvable Hamiltonian for the system which possesses a (2+1)D G symmetric surface termination that hosts deconfined anyon excitations described by the given UMTC and symmetry fractionalization class. This approach applies to general symmetry groups, including anyon-permuting and anti-unitary symmetries. In the case of unitary orientation-preserving symmetries, our results can also be viewed as providing a method to compute the H4(G,U(1)) obstruction that arises in the theory of G-crossed braided tensor categories, for which no general method has been presented to date. This is joint work with D. Bulmash, presented in arXiv:2003.11553

Zoom: https://harvard.zoom.us/j/977347126

I will describe an analogue of Saito’s theory of primitive forms for Calabi-Yau A-infinity categories. Under some conditions on the Hochschild cohomology of the category, this construction recovers the (genus zero) Gromov-Witten invariants of a symplectic manifold from its Fukaya category. This includes many compact toric manifolds, in particular projective spaces.

Zoom: https://harvard.zoom.us/j/93926041523

This talk aims to summarize a project I was involved in during the past 5 years, which links together many areas in math, CS and physics. I hope to explain our motivations and goals, summarize our understanding so far, as well as challenges and open problems. I plan to describe, through examples, many of the concepts they refer to, and the evolution of ideas leading to what we know. More details can be found at mathpicture.fas.harvard.edu/seminar. No special background is assumed.

Zoom: https://harvard.zoom.us/j/779283357

We consider a monopolist who wishes to sell n goods to a single additive buyer, where the buyer’s valuations for the goods are drawn according to independent distributions. It is well known that—unlike in the single item case—the revenue-optimal auction (a pricing scheme) may be complex, sometimes requiring a continuum of menu entries, that is, offering the buyer a choice between a continuum of lottery tickets. It is also known that simple auctions with a finite bounded number of menu entries (lotteries for the buyer to choose from) can extract a constant fraction of the optimal revenue, as well as that for the case of bounded distributions it is possible to extract an arbitrarily high fraction of the optimal revenue via a finite bounded menu size. Nonetheless, the question of the possibility of extracting an arbitrarily high fraction of the optimal revenue via a finite menu size, when the valuation distributions possibly have unbounded support (or via a finite bounded menu size when the support of the distributions is bounded by an unknown bound), remained open since the seminal paper of Hart and Nisan (2013), and so has the question of any lower-bound on the menu-size that suffices for extracting an arbitrarily high fraction of the optimal revenue when selling a fixed number of goods, even for two goods and even for i.i.d. bounded distributions.

In this talk, we resolve both of these questions. We first show that for every n and for every ε>0, there exists a menu-size bound C=C(n,ε) such that auctions of menu size at most C suffice for extracting a (1-ε) fraction of the optimal revenue from any valuation distributions, and give an upper bound on C(n,ε), even when the valuation distributions are unbounded and nonidentical. We then proceed to giving two lower bounds, which hold even for bounded i.i.d. distributions: one on the dependence on n when ε=1/n and n grows large, and the other on the dependence on ε when n is fixed and ε grows small. Finally, we apply these upper and lower bounds to derive results regarding the deterministic communication complexity required to run an auction that achieves such an approximation.

Based upon:
* The Menu-Size Complexity of Revenue Approximation, Moshe Babaioff, Y. A. G., and Noam Nisan, STOC 2017
* Bounding the Menu-Size of Approximately Optimal Auctions via Optimal-Transport Duality, Y. A. G., STOC 2018

Yannai Gonczarowski is a postdoctoral researcher at Microsoft Research New England. His main research interests lie in the interface between the theory of computation, economic theory, and game theory—an area commonly referred to as Algorithmic Game Theory. In particular, Yannai is interested in various aspects of complexity in mechanism design (where mechanisms are defined broadly from auctions to matching markets), including the interface between mechanism design and machine learning. Yannai received his PhD from the Departments of Math and CS, and the Center for the Study of Rationality, at the Hebrew University of Jerusalem, where he was advised by Sergiu Hart and Noam Nisan, as an Adams Fellow of the Israel Academy of Sciences and Humanities. Throughout most of his PhD studies, he was also a long-term research intern at Microsoft Research in Herzliya. He holds an M.Sc. in Math (summa cum laude) and a B.Sc. in Math and CS (summa cum laude, Valedictorian) from the Hebrew University. Yannai is also a professionally-trained opera singer, having acquired a bachelor’s degree and a master’s degree in Classical Singing at the Jerusalem Academy of Music and Dance. For his doctoral dissertation, Yannai was awarded the Hans Wiener Prize of the Hebrew University of Jerusalem, the 2018 Michael B. Maschler Prize of the Israeli Chapter of the Game Theory Society, and the ACM SIGecom Doctoral Dissertation Award for 2018. Yannai is also the recipient of the ACM SIGecom Award for Best Presentation by a Student or Postdoctoral Researcher at EC’18, and of the Best Paper Award at MATCH-UP’19. His first textbook, “Mathematical Logic through Python” (Gonczarowski and Nisan), which introduces a new approach to teaching the material of a basic Logic course to Computer Science students, tailored to the unique intuitions and strengths of this cohort of students, is forthcoming in Cambridge University Press.
Zoom: https://harvard.zoom.us/j/98684244514

I’ll review the basic construction of Randall-Sundrum
braneworlds and some of their applications to formal problems in quantum
field theory. I will highlight some recent results regarding scenarios
with mismatched brane tensions. In the last part of the talk, I’ll
review how RS branes have led to exciting new results regarding
evaporation of black holes and will put emphasis on the interesting role
the graviton mass plays in these discussions.

Zoom: https://harvard.zoom.us/j/977347126

Graph-structured data is ubiquitous throughout the natural and social sciences, from telecommunication networks to quantum chemistry. Building relational inductive biases into deep learning architectures is crucial if we want systems that can learn, reason, and generalize from this kind of data. Recent years have seen a surge in research on graph representation learning, most prominently in the development of graph neural networks (GNNs). Advances in GNNs have led to state-of-the-art results in numerous domains, including chemical synthesis, 3D-vision, recommender systems, question answering, and social network analysis. In the first part of this talk I will provide an overview and summary of recent progress in this fast-growing area, highlighting foundational methods and theoretical motivations. In the second part of this talk I will discuss fundamental limitations of the current GNN paradigm and propose open challenges for the theoretical advancement of the field.

Zoom: https://harvard.zoom.us/j/91458092166?pwd=RnFwelNhakw1b3B6dy9UMkt6T2xoQT09

The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of p-adic groups.

I will provide an overview of our understanding of the representations of p-adic groups, with an emphasis on recent progress.

I will also outline how new results about the representation theory of p-adic groups can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p, which is joint work with Sug Woo Shin. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group.

Zoom: https://harvard.zoom.us/j/96767001802

Password: The order of the permutation group on 9 elements.

We consider the problem of finding 3 positive integers (a,b,c) such that a/(b+c)+b/(c+a)+c/(a+b)=4. We will examine how the problem became a viral post on Facebook, solve the problem using elliptic curves, and examine some modern developments when the number 4 is replaced by N.

Zoom: https://harvard.zoom.us/j/96759150216

Quantum entanglement has played a key role in studying emergent phenomena in strongly-correlated many-body systems.  Remarkably, The entanglement properties of the ground state encodes information on the nature of excitations.  Here we introduce two new entanglement measures $g(A:B)$ and $h(A:B)$ which characterizes certain tripartite entanglement between $A$, $B$, and the environment.  The measures are based off of the entanglement of purification and the reflected entropy popular among holography.  For 1D states, the two measures are UV insensitive and yield universal quantities for symmetry-broken, symmetry preserved, and critical phases.  We conclude with a few remarks regarding applications to 2D phases.

Zoom: https://harvard.zoom.us/j/977347126

We study local and global Hamiltonian dynamical behaviors of some Lagrangian submanifolds near a Lagrangian sphere S in a symplectic manifold X. When dim S = 2, we show that there is a one-parameter family of Lagrangian tori near S, which are nondisplaceable in X. When dim S = 3, we obtain a new estimate of the displacement energy of S, by estimating the displacement energy of a one-parameter family of Lagrangian tori near S.

Zoom: https://harvard.zoom.us/j/93926041523

Fusion categories have been extensively studied by Mathematicians and have proved to have many important applications in quantum physics. A fusion category is completely determined by a set of F-symbols which satisfies the pentagon equations. In general, the fusion categories are constructed by different approaches and their F-symbols remain unknown. In this talk, we introduce the triangular prism equations for fusion categories and show that they are equivalent to the pentagon equations. Moreover, we provide a relevant way to manage the complexity by localization, and thus a possible approach to solve them for the F-symbols. As applications, we provided new criteria for categorification and a categorical approach to the neargroup construction, improving Izumi’s equations.

Zoom: https://harvard.zoom.us/j/779283357

Zoom: https://harvard.zoom.us/j/91221148687

Recently there has been tremendous progress on constructing (projective) moduli spaces of Fano varieties using K-stability. In this talk, we will show that the K-moduli space of cubic fourfolds coincide with their GIT moduli space. In particular, all smooth cubic fourfolds are K-stable as well as those with simple singularities. The key ingredients are local volume estimates in dimension 3 due to Liu-Xu, Ambro-Kawamata non-vanishing theorem for Fano 4-folds, and degeneration of K3 surfaces.

Zoom: https://harvard.zoom.us/j/91794282895?pwd=VFZxRWdDQ0VNT0hsVTllR0JCQytoZz09

A few years ago Maksym Radziwill and I showed that the average of a multiplicative function in almost all very short intervals $[x, x+h]$ is close to its average on a long interval $[x, 2x]$. This result has since been utilized in many applications.

I will talk about recent work, where Radziwill and I revisit the problem and generalise our result to functions which vanish often as well as prove a power-saving upper bound for the number of exceptional intervals (i.e. we show that there are $O(X/h^\kappa)$ exceptional $x \in [X, 2X]$).

We apply this result for instance to studying gaps between norm forms of an arbitrary number field.

Zoom: https://harvard.zoom.us/j/96767001802

Password: The order of the permutation group on 9 elements.

In this second talk, I will focus on (nearly) solvable models of metal-metal transition in random systems. The t-J model with random and all-to-all hopping and exchange can be mapped onto a quantum impurity model coupled self-consistently to an environment (the mapping also applies to a t-J model in a large dimension lattice, with random nearest-neighbor exchange). Such models will be argued to exhibit metal-metal quantum phase transitions in the universality class of the SYK model, accompanied by a linear-in-T resistivity from time reparameterization fluctuations. I will also present the results of exact diagonalization of random t-J clusters, obtained recently with Henry Shackleton, Alexander Wietek, and Antoine Georges.

Zoom: https://harvard.zoom.us/j/977347126

Modern machine learning methods –most noticeably multi-layer neural networks– require to fit highly non-linear models comprising tens of thousands to millions of parameters. Despite this, little attention is paid to the regularization mechanism to control model’s complexity. Indeed, the resulting models are often so complex as to achieve vanishing training error: they interpolate the data. Despite this, these models generalize well to unseen data: they have small test error. I will discuss several examples of this phenomenon, beginning with a simple linear regression model, and ending with two-layers neural networks in the so-called lazy regime. For these examples precise asymptotics could be determined mathematically, using tools from random matrix theory. I will try to extract a unifying picture.
A common feature is the fact that a complex unregularized nonlinear model becomes essentially equivalent to a simpler model, which is however regularized in a non-trivial way.
[Based on joint papers with: Behrooz Ghorbani, Song Mei, Theodor Misiakiewicz, Feng Ruan, Youngtak Sohn, Jun Yan, Yiqiao Zhong]

Zoom: https://harvard.zoom.us/j/91458092166

We’ll start with a type of doodle most of you have done since you were little, and start wondering about it. We’ll be led through a number of questions I’ve heard starting from long ago, wandering through a number of different ideas, and hopefully ending in the vicinity of some of the groundbreaking ideas of Maryam Mirzakhani. This is what mathematics, and basic science, is really about — observing something out there “in nature”, wondering about it, explaining it, and realizing that it connects to other deeper questions, and then repeating the process.

Zoom: https://harvard.zoom.us/j/96759150216

The cuprate superconductors exhibit the highest ambient-pressure superconducting transition temperatures (T c ), and after more than three decades of extraordinary research activity, continue to pose formidable scientific challenges. A major experimental obstacle has been to distinguish universal phenomena from materials- or technique-dependent ones. Angle-resolved photoemission spectroscopy (ARPES) measures momentum-dependent single-particle electronic excitations and has been invaluable in the endeavor to determine the anisotropic momentum-space properties of the cuprates. HgBa 2 CuO 4+d (Hg1201) is a single-layer cuprate with a particularly high optimal T c and a simple crystal structure; yet there exists little information from ARPES about the electronic properties of this model system. I will present recent ARPES studies of doping-, temperature-, and momentum-dependent systematics of near- nodal dispersion anomalies in Hg1201. The data reveal a hierarchy of three distinct energy scales which establish several universal phenomena, both in terms of connecting multiple experimental techniques for a single material, and in terms of connecting comparable spectral features in multiple structurally similar cuprates.

Zoom: https://harvard.zoom.us/j/977347126

TITLE: Area-minimizing integral currents and their regularity

ABSTRACT: Caccioppoli sets and integral currents (their generalization in higher codimension) were introduced in the late fifties and early sixties to give a general geometric approach to the existence of area-minimizing oriented surfaces spanning a given contour. These concepts started a whole new subject which has had tremendous impacts in several areas of mathematics: superficially through direct applications of the main theorems, but more deeply because of the techniques which have been invented to deal with related analytical and geometrical challenges. In this lecture I will review the basic concepts, the related existence theory of solutions of the Plateau problem, and what is known about their regularity. I will also touch upon several fundamental open problems which still defy our understanding.

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

For more information, please visit the event page.

Register here to attend.

TITLE: The origins of Langlands’ conjectures

ABSTRACT: Langlands has made many contributions to number theory, but the principal one is probably his discovery in 1966–67, followed by work in subsequent years, of the role of the dual group in the theories of automorphic forms and L-functions.  In order to try to understand what this amounted to, I will trace the origins of this development through work of Ramanujan, Hecke, Siegel, Maass, Selberg, and other mathematicians of the twentieth century.

Talk chair:  Wilfried Schmid

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

For more information, please visit the event page.

Register here to attend.

TITLE: From Deep Learning to Deep Understanding

ABSTRACT: In this talk I will discuss a couple of research directions for robust AI beyond deep neural networks. The first is the need to understand what we are learning, by shifting the focus from targeting effects to understanding causes. The second is the need for a hybrid neural/symbolic approach that leverages both commonsense knowledge and massive amount of data. Specifically, as an example, I will present some latest work at Microsoft Research on building a pre-trained grounded text generator for task-oriented dialog. It is a hybrid architecture that employs a large-scale Transformer-based deep learning model,  and symbol manipulation modules such as business databases, knowledge graphs and commonsense rules. Unlike GPT or similar language models learnt from data, it is a multi-turn decision making system which takes user input, updates the belief state, retrieved from the database via symbolic reasoning, and decides how to complete the task with grounded response.

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

For more information, please visit the event page.

Register here to attend.

In this talk, I will discuss a reformulation of the Wilson loop in large N gauge theories in terms of matrix product states. The construction is motivated by the analysis of supersymmetric Wilson loops in the maximally super Yang–Mills theory in four dimensions, but can be applied to any other large N gauge theories and matrix models, although less effective. For the maximally super Yang–Mills theory, one can further perform the computation exactly as a function of ‘t Hooft coupling by combining our formulation with the relation to integrable spin chains.

Zoom: https://harvard.zoom.us/j/93926041523

TITLE: A personal story of the 4D Poincare conjecture.

ABSTRACT: The proof of PC4 involved the convergence of several historical streams.  To get started: high dimensional manifold topology (Smale), a new idea on how to study 4-manifolds (Casson), wild “Texas” topology (Bing). Once inside the proof: there are three submodules: Casson towers come to life (in the sense of reproduction), a very intricate explicit shrinking argument (provided by Edwards), and the “blind fold” shrinking argument (which in retrospect is in the linage of Brown’s proof of the Schoenflies theorem). Beyond those mentioned: Kirby, Cannon, Ancel, Quinn, and Starbird helped me understand my proof. I will discuss the main points and how they fit together.

Talk Chair: Peter Kronheimer

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

For more information, please visit the event page.

Register here to attend.

Subfactors and K-theory are useful mechanisms for understanding modular tensor categories and conformal field theories CFT. As part of this, one issue is to try and construct or reconstruct a conformal field theory as the representation theory of a conformal net of algebras, or as a vertex operator algebra from a given abstractly presented modular tensor category. Freed, Hopkins and Teleman realized the chiral Verlinde rings of WZW models as twisted equivariant K-theory. I will describe work which has led to represent the full CFT and modular invariant partition function K-theoretically and descriptions of Verlinde rings as Hilbert modules over an operator algebra, and higher equivariant twists with bundles beyond compact operators. Orbifold models play an important role and orbifolds of Tambara-Yamagami systems are relevant to understanding the double of the Haagerup as a conformal field theory. This is joint work with Andreas Aaserud, Terry Gannon and Ulrich Pennig.

Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

Given an n x n matrix A with bounded entries, we study the problem of testing whether A is positive semi-definite (PSD) via only a small number of queries to the entries of A. While in general one must read the entire matrix A to determine if it is PSD, we demonstrate that testing whether A is PSD or “far” from PSD (under some norm) is indeed possible with query complexity which only depends on the distance to the PSD cone. This setup is commonly referred to as the property testing framework. We consider two natural norms of n x n matrices: the spectral norm and the Frobenius (Euclidean) norm. We give a query-optimal algorithm for the former, and present new upper and lower bounds for the latter.

Both of these algorithms have a very simple structure: they randomly sample a collection of principal submatrices and check whether these submatrices are PSD.  Thus, our problem can phrased purely as a question in random matrix theory: given a (entry-wise bounded) matrix A which is at distance D (in some norm) from the PSD cone, what is the probability that a random k x k principal submatrix is not PSD? For the spectral norm, this problem can be tightly answered by classical arguments (e.g. scalar valued concentration), however the case of the Euclidean norm appears to be more involved, and will require matrix concentration based arguments.

If a complex, integral, projective curve C has only planar singularities, then its Jacobian admits a natural compactification with interesting topology. Work of Oblomkov, Shende, and others suggests the existence of a variety, stratified by algebraic tori and defined solely in terms of the topology of C, that retracts transcendentally onto this compactified Jacobian. We expect this retraction to be a new type of nonabelian Hodge correspondence: in particular, at the level of cohomology, it should map a (halved) weight filtration onto a filtration defined via perverse sheaves. I will construct a candidate for the larger variety using the combinatorics of braids and flag varieties, related to but ultimately different from a construction of Shende-Treumann-Zaslow. I will present evidence that the entire story is the SL_n case of a recipe that works for any semisimple group G, and that in a precise sense, these retractions should respect the isomorphism between the unipotent locus of G and the nilpotent locus of Lie(G). The key ingredient is a map from elements of the loop Lie algebra to conjugacy classes in a generalized braid group. The latter are the “algebraic braids” of the title.

Zoom: https://harvard.zoom.us/j/91794282895?pwd=VFZxRWdDQ0VNT0hsVTllR0JCQytoZz09

I will discuss some aspects of SYZ mirror symmetry for pairs (X,D) where X is a del Pezzo surface or a rational elliptic surface and D is an anti-canonical divisor which is either smooth or a wheel of rational curves. In particular I will explain the existence of special Lagrangian fibrations and mirror symmetry for (suitably interpreted) Hodge numbers. If time permits, I will describe a proof of SYZ mirror symmetry for del Pezzo surfaces. This is joint work with A. Jacob and Y.-S. Lin.
Zoom: https://harvard.zoom.us/j/96709211410?pwd=SHJyUUc4NzU5Y1d0N2FKVzIwcmEzdz09

TITLE: Hodge structures and the topology of algebraic varieties

ABSTRACT: We review the major progress made since the 50’s in our understanding of the topology of complex algebraic varieties. Most of the results  we will discuss  rely on Hodge theory, which  has some analytic aspects giving the Hodge and Lefschetz decompositions, and the Hodge-Riemann relations. We will see that a crucial ingredient, the existence of a polarization,  is missing in the general Kaehler context.
We will also discuss some results and problems related to algebraic cycles and motives.

Talk chair: Joe Harris

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

For more information, please visit the event page.

Register here to attend.

We discover a wide range of new nonperturbative effects in quantum gravity, namely moduli spaces of constrained instantons of the Einstein-Hilbert action.  We find these instantons in all spacetime dimensions, for AdS and dS.  Many can be written in closed form and are quadratically stable.  In 3D AdS, where the full gravitational path integral is more tractable, we study constrained instantons corresponding to Euclidean wormholes.  We show that they encode the energy level statistics of microstates of BTZ black holes, which precisely agrees with a quantitative prediction from random matrix theory.

Zoom: https://harvard.zoom.us/j/977347126

TITLE: Immersions of manifolds and homotopy theory

ABSTRACT: The interface between the study of the topology of differentiable manifolds and algebraic topology has been one of the richest areas of work in topology since the 1950’s. In this talk I will focus on one aspect of that interface: the problem of studying embeddings and immersions of manifolds using homotopy theoretic techniques. I will discuss the history of this problem, going back to the pioneering work of Whitney, Thom, Pontrjagin, Wu, Smale, Hirsch, and others. I will discuss the historical applications of this homotopy theoretic perspective, going back to Smale’s eversion of the 2-sphere in 3-space. I will then focus on the problems of finding the smallest dimension Euclidean space into which every n-manifold embeds or immerses. The embedding question is still very much unsolved, and the immersion question was solved in the 1980’s. I will discuss the homotopy theoretic techniques involved in the solution of this problem, and contributions in the 60’s, 70’s and 80’s of Massey, Brown, Peterson, and myself. I will also discuss questions regarding the best embedding and immersion dimensions of specific manifolds, such has projective spaces. Finally, I will end by discussing more modern approaches to studying spaces of embeddings due to Goodwillie, Weiss, and others. This talk will be geared toward a general mathematical audience.

Talk chair: Michael Hopkins

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

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$\ell$-torsion conjecture states that $\ell$-torsion of the class group $|\text{Cl}_K[\ell]|$ for every number field $K$ is bounded by $\text{Disc}(K)^{\epsilon}$. It follows from a classical result of Brauer-Siegel, or even earlier result of Minkowski that the class number $|\text{Cl}_K|$ of a number field $K$ are always bounded by $\text{Disc}(K)^{1/2+\epsilon}$, therefore we obtain a trivial bound $\text{Disc}(K)^{1/2+\epsilon}$ on $|\text{Cl}_K[\ell]|$. We will talk about results on this conjecture, and recent works on breaking the trivial bound for $\ell$-torsion of class groups in some cases based on a work of Ellenberg-Venkatesh.

Zoom: https://harvard.zoom.us/j/96767001802

Password: The order of the permutation group on 9 elements.

The geometry of rhombic tilings and tessellations like the Penrose tiling have captivated mathematicians and artists alike.  Hidden in the geometry of certain rhombic tilings of certain polygons, though, is an unexpected combinatorial structure that not only lends itself to some combinatorial objects, but also is often rather useful for their enumeration.  In this talk, we will highlight the connection between one type of rhombic tiling and the world of plane partitions, monotone discrete functions, and stacks of cubes; and another type of rhombic tiling to the world of permutations, Coxeter groups, and reduced words.

Come learn some neat mathematics connecting permutations, polygons, geometry, and groups.

Zoom link is posted here:
https://calendar.college.harvard.edu/event/math_table