news

See Older News

announcements

upcoming events

«
»
Sun
Mon
Tue
Wed
Thu
Fri
Sat
September
September
September
September
1
  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Intrinsic sign problems in topological matter
    10:30 AM-12:00 PM
    October 1, 2020

    The infamous sign problem leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems. Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing efficient simulations on classical computers. Motivated by long standing open problems in many-body physics, as well as fundamental questions in quantum complexity, the possibility of intrinsic sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored. I will describe results establishing the existence of intrinsic sign problems in a broad class of topologically ordered phases in 2+1 dimensions. Within this class, these results exclude the possibility of ‘stoquastic’ Hamiltonians for bosons, and of sign-problem-free determinantal Monte Carlo algorithms for fermions. The talk is based on arxiv: 2005.05566 and 2005.05343.

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

2
3
4
5
6
  • DIFFERENTIAL GEOMETRY SEMINAR

    DIFFERENTIAL GEOMETRY SEMINAR
    The Gopakumar-Vafa invariants for local P2

    8:00 AM-9:00 AM
    October 6, 2020

    In this talk, I will introduce the Gopakumar-Vafa(GV) invariant and show one calculation on the nonreduced cycle. The GV invariant is an integral invariant predicted by physicists that counts the number of curves inside a given Calabi-Yau threefold. The definition has been conjectured by Maulik-Toda in 2016 in terms of perverse sheaf. I’ll use this definition on the total space of the canonical bundle of P2 and compute the associated invariants. This verifies a physical formula based on the work of Katz-Klemm-Vafa in 1997.

    Zoom: https://harvard.zoom.us/j/96709211410?pwd=SHJyUUc4NzU5Y1d0N2FKVzIwcmEzdz09

  • MATHEMATICAL PICTURE LANGUAGE SEMINAR

    MATHEMATICAL PICTURE LANGUAGE SEMINAR
    Reconstructing CFTs from TQFTs

    10:00 AM-11:00 AM
    October 6, 2020

    Inspired by fractional quantum Hall physics and Tannaka-Krein duality, it is conjectured that every modular tensor category (MTC) or (2+1)-topological quantum field theory (TQFT) can be realized as the representation category of a vertex operator algebra (VOA) or chiral conformal field theory (CFT).  It is obviously true for quantum group/WZW MTCs, but it is not known for MTCs appeared in subfactors such as the famous double Haagerup.  After some general discussion, I will focus on pointed MTCs or so-called abelian anyon models.  While all abelian anyon models can be realized by lattice VOAs, it is not clear whether or not they can be realized by non-lattice VOAs.  The trivial MTC is realized by the Monster moonshine module, which is a non-lattice realization.  I will provide evidence that this might be true for all abelian anyon models.  The talk is partially based on a joint work with Liang Wang: https://arxiv.org/abs/2004.12048 

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

  • CMSA EVENT: CMSA Computer Science for Mathematicians: Generation by Decomposition
    11:30 AM-12:30 PM
    October 6, 2020

    Deep learning has revolutionized our ability to generate novel images and 3D shapes. Typically neural networks are trained to map a high-dimensional latent code to full realistic samples. In this talk, I will present two recent works focusing on generation of handwritten text and 3D shapes. In these works, we take a different approach and generate image and shape samples using a more granular part-based decomposition, demonstrating that the whole is not necessarily “greater than the sum of its parts”. I will also discuss how our generation by decomposition approach allows for a semantic manipulation of 3D shapes and improved handwritten text recognition performance.

    Zoom: https://harvard.zoom.us/j/98231541450

7
  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Symmetry-enriched random critical points and topological phase transitions
    10:30 AM-12:00 PM
    October 7, 2020

    In this talk, I will describe how symmetry can enrich strong-randomness quantum critical points and phases, and lead to robust topological edge modes coexisting with critical bulk fluctuations. Our approach provides a systematic construction of strongly disordered gapless topological phases. Using real space renormalization group techniques, I will discuss the boundary and bulk critical behavior of symmetry-enriched random quantum spin chains, and argue that nonlocal observables and boundary critical behavior are controlled by new renormalization group fixed points. I will also discuss the interplay between disorder, quantum criticality and topology in higher dimensions using disordered gauge theories.

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

  • NUMBER THEORY SEMINAR
    3:00 PM-4:00 PM
    October 7, 2020

    Mazur conjectured, after Faltings’s proof of the Mordell conjecture, that the number of rational points on a curve of genus g at least 2 defined over a number field of degree d is bounded in terms of g, d and the Mordell-Weil rank. In particular the height of the curve is not involved. In this talk I will explain how to prove this conjecture and some generalizations. I will focus on how functional transcendence and unlikely intersections are applied in the proof. If time permits, I will talk about how the dependence on d can be furthermore removed if we moreover assume the relative Bogomolov conjecture. This is joint work with Vesselin Dimitrov and Philipp Habegger.

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

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

  • CMSA EVENT: CMSA New Technologies in Mathematics: Subgraph Representation Learning
    3:00 PM-4:00 PM
    October 7, 2020

    Graph representation learning has emerged as a dominant paradigm for networked data. Still, prevailing methods require abundant label information and focus on representations of nodes, edges, or entire graphs. While graph-level representations provide overarching views of graphs, they do so at the loss of finer local structure. In contrast, node-level representations preserve local topological structures, potentially to the detriment of the big picture. In this talk, I will discuss how subgraph representations are critical to advance today’s methods. First, I will outline Sub-GNN, the first subgraph neural network to learn disentangled subgraph representations. Second, I will describe G-Meta, a novel meta-learning approach for graphs. G-Meta uses subgraphs to adapt to a new task using only a handful of nodes or edges. G-Meta is theoretically justified, and remarkably, can learn in most challenging, few-shot settings that require generalization to completely new graphs and never-before-seen labels. Finally, I will discuss applications in biology and medicine. The new methods have enabled the repurposing of drugs for new diseases, including COVID-19, where predictions were experimentally verified in the wet laboratory. Further, the methods identified drug combinations safer for patients than previous treatments and provided accurate predictions that can be interpreted meaningfully.

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

8
  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Orbifold Groupoids
    10:30 AM-12:00 PM
    October 8, 2020

    Orbifolds are ubiquitous in physics, not just explicitly in CFT, but going undercover with names like Kramers-Wannier duality, Jordan-Wigner transformation, or GSO projection. All of these names describe ways to “topologically manipulate” a theory, transforming it to a new one, but leaving the local dynamics unchanged. In my talk, I will answer the question: given some (1+1)d QFT, how many new theories can we produce by topological manipulations? To do so, I will outline the relationship between these manipulations and (2+1)d Dijkgraaf-Witten TFTs, and illustrate both the conceptual and computational power of the relationship. Ideas from high-energy, condensed-matter, and pure math will show up in one form or another. Based on work with Davide Gaiotto [arxiv:2008.05960].

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

9
10
11
12
13
  • MATHEMATICAL PICTURE LANGUAGE SEMINAR
    10:00 AM-11:00 AM
    October 13, 2020

    I shall report on a new approach to study some classes of quantum circuits and exactly solvable models. Concretely, using knots gives a unified framework to characterize two famous classes of classically-simulable quantum circuits: Clifford and matchgate. We evaluate these circuits in a topological way by untying the knots. Our method is suitable for programming. The method relies on the abstraction of Ising anyons/Majorana zero modes (also known as the Z2 Quon language). It lets us partially open the black box of each small tensor in the tensor network representation. As a bonus, we find a new class of classically simulable quantum circuits. Our results have an interpretation in terms of exactly-soluble, statistical-mechanics models, and they lead to a topological extension of Kramers-Wannier duality. This point of view may also help us find new types of exactly soluble models.

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

  • HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR
    3:00 PM-4:00 PM
    October 13, 2020

    Using recent advances in the Minimal Model Program for moduli spaces of sheaves on the projective plane, we compute the cohomology of the tensor product of general semistable bundles on the projective plane.   More precisely, let V and W be two general stable bundles, and suppose the numerical invariants of W are sufficiently divisible. We fully compute the cohomology of the tensor product of V and W.  In particular, we show that if W is exceptional, then the tensor product of V and W has at most one nonzero cohomology group determined by the slope and the Euler characteristic, generalizing foundational results of Drézet, Göttsche and Hirschowitz. We also characterize when the tensor product of V and W is globally generated. Crucially, our computation is canonical given the birational geometry of the moduli space, providing a roadmap for tackling analogous problems on other surfaces.  This is joint work with Izzet Coskun and John Kopper.

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

  • DIFFERENTIAL GEOMETRY SEMINAR
    8:00 PM-9:00 PM
    October 13, 2020

    Quiver theory and machine learning share a common ground, namely, they both concern about linear representations of directed graphs.  The main difference arises from the crucial use of non-linearity in machine learning to approximate arbitrary functions; on the other hand, quiver theory has been focused on fiberwise-linear operations on universal bundles over the quiver moduli.
    Compared to flat spaces that have been widely used in machine learning, a quiver moduli has the advantages that it is compact, has interesting topology, and enjoys extra symmetry coming from framing.  In this talk, I will explain how fiberwise non-linearity can be naturally introduced by using Kaehler geometry of the quiver moduli.

    Zoom: https://harvard.zoom.us/j/96709211410?pwd=SHJyUUc4NzU5Y1d0N2FKVzIwcmEzdz09

14
15
16
17
18
19
20
  • MATHEMATICAL PICTURE LANGUAGE SEMINAR

    MATHEMATICAL PICTURE LANGUAGE SEMINAR
    Exploring small fusion rings and tensor categories

    10:00 AM-11:00 AM
    October 20, 2020

    I discuss some strategies for finding fusion rings of low rank (or if you prefer, fusion rules for a small number of objects) and corresponding tensor categories, or solutions to pentagon and hexagon equations. Since developing these, we have produced a large database of fusion rings by computer search, including many that we were unfamiliar with ourselves. I hope to describe the features of some of these and their generalizations at higher rank, particularly focusing on some of the less well known or studied examples, such as various rings with non-Abelian fusion. Secretly I am of course hoping that the audience will recognize some of these and share interesting information about them! This is very much work in progress. I also hope to say something about potential applications, for example to anyons on wire networks, and will briefly introduce some tools we are building to make it easy to explore and use these rings and categories.

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

  • CMSA EVENT: CMSA Computer Science for Mathematicians: Improved Lower Bounds for the Fourier Entropy/Influence Conjecture via Lexicographic Functions
    11:30 AM-12:30 PM
    October 20, 2020

    Every Boolean function can be uniquely represented as a multilinear polynomial. The entropy and the total influence are two ways to measure the concentration of its Fourier coefficients, namely the monomial coefficients in this representation: the entropy roughly measures their spread, while the total influence measures their average level. The Fourier Entropy/Influence conjecture of Friedgut and Kalai from 1996 states that the entropy to influence ratio is bounded by a universal constant C.

    Using lexicographic Boolean functions, we present three explicit asymptotic constructions that improve upon the previously best known lower bound C > 6.278944 by O’Donnell and Tan, obtained via recursive composition. The first uses their construction with the lexicographic function 𝓁⟨2/3⟩ of measure 2/3 to demonstrate that C >= 4+3 log_4 (3) > 6.377444. The second generalizes their construction to biased functions and obtains C > 6.413846 using 𝓁⟨Φ⟩, where Φ is the inverse golden ratio. The third, independent, construction gives C > 6.454784, even for monotone functions.

    Beyond modest improvements to the value of C, our constructions shed some new light on the properties sought in potential counterexamples to the conjecture.

    Additionally, we prove a Lipschitz-type condition on the total influence and spectral entropy, which may be of independent interest.

    Zoom: https://harvard.zoom.us/j/98231541450

  • HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR

    HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR
    Stringy invariants and toric Artin stacks

    3:00 PM-4:00 PM
    October 20, 2020

    Stringy Hodge numbers are certain generalizations, to the singular setting, of Hodge numbers. Unlike usual Hodge numbers, stringy Hodge numbers are not defined as dimensions of cohomology groups. Nonetheless, an open conjecture of Batyrev’s predicts that stringy Hodge numbers are nonnegative. In the special case of varieties with only quotient singularities, Yasuda proved Batyrev’s conjecture by showing that the stringy Hodge numbers are given by orbifold cohomology. For more general singularities, a similar cohomological interpretation remains elusive. I will discuss a conjectural framework, proven in the toric case, that relates stringy Hodge numbers to motivic integration for Artin stacks, and I will explain how this framework applies to the search for a cohomological interpretation for stringy Hodge numbers. This talk is based on joint work with Matthew Satriano.

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

  • DIFFERENTIAL GEOMETRY SEMINAR

    DIFFERENTIAL GEOMETRY SEMINAR
    Self-duality in quantum K-theory

    8:00 PM-9:00 PM
    October 20, 2020

    When we upgrade from equivariant cohomology to equivariant
    K-theory, many important algebraic/geometric tools such as dimensional vanishing become inapplicable in general. I will explain some nice conditions we can impose on K-theory classes to restore some of these tools. These conditions hold for many types of curve-counting theories (e.g. quasimaps) and are crucial for the development of those flavors of quantum K-theory, but they notably are not present in Gromov-Witten theory. I will describe an attempt to twist GW theory to fulfill these conditions.

    Zoom: https://harvard.zoom.us/j/96709211410?pwd=SHJyUUc4NzU5Y1d0N2FKVzIwcmEzdz09

21
  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Multipole Insulators and Higher-Form symmetries
    10:30 AM-12:00 PM
    October 21, 2020

    The most basic characteristic of an electrically insulating system is the absence of charged currents. This property alone guarantees the conservation of the overall dipole moment (i.e., the first multipole moment) in the low-energy sector. It is then natural to inquire about the fate of the transport properties of higher electric multipole moments, such as the quadrupole and octupole moments, and ask what properties of the insulating system can guarantee their conservation. In this talk I will present a suitable refinement of the notion of an insulator by investigating a class of systems that conserve both the total charge and the total dipole moment. In particular, I will consider microscopic models for systems that conserve dipole moments exactly and show that one can divide charge insulators into two new classes: (i) a dipole metal, which is a charge-insulating system that supports dipole-moment currents, or (ii) a dipole insulator which is a charge-insulating system that does not allow dipole currents and thus, conserves an overall quadrupole moment. In the second part of my talk I will discuss a more mathematical description of dipole-conserving systems where I show that a conservation of the overall dipole moment can be naturally attributed to a global 1-form electric U(1) symmetry, which is in direct analogy to how the electric charge conservation is guaranteed by the global U(1) phase-rotation symmetry for electrically charged particles. Finally, this new approach will allow me to construct a topological response action which is especially useful for characterizing Higher-Order Topological phases carrying quantized quadrupole moments.

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

  • NUMBER THEORY SEMINAR
    3:00 PM-4:00 PM
    October 21, 2020

    In this talk, we will discuss a geometric construction of p-adic analogues of Maass–Shimura differential operators on automorphic forms on Shimura varieties of PEL type A or C (that is, unitary or symplectic), at p an unramified prime. Maass–Shimura operators are smooth weight raising differential operators used in the study of special values of L-functions, and in the arithmetic setting for the construction of p-adic L-functions.  In this talk, we will focus in particular on the case of unitary groups of arbitrary signature, when new phenomena arise for p  non split.  We will also discuss an application to the study of modular mod p Galois representations. This talk is based on joint work with Ellen Eischen (in the unitary case for p non split), and with Eischen, Flanders, Ghitza, and Mc Andrew (in the other cases).

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

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

  • NUMBER THEORY SEMINAR
    3:00 PM-4:00 PM
    October 21, 2020

    In this talk, we will discuss a geometric construction of p-adic analogues of Maass–Shimura differential operators on automorphic forms on Shimura varieties of PEL type A or C (that is, unitary or symplectic), at p an unramified prime. Maass–Shimura operators are smooth weight raising differential operators used in the study of special values of L-functions, and in the arithmetic setting for the construction of p-adic L-functions.  In this talk, we will focus in particular on the case of unitary groups of arbitrary signature, when new phenomena arise for p  non split.  We will also discuss an application to the study of modular mod p Galois representations. This talk is based on joint work with Ellen Eischen (in the unitary case for p non split), and with Eischen, Flanders, Ghitza, and Mc Andrew (in the other cases).

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

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

  • NUMBER THEORY SEMINAR
    3:00 PM-4:00 PM
    October 21, 2020

    In this talk, we will discuss a geometric construction of p-adic analogues of Maass–Shimura differential operators on automorphic forms on Shimura varieties of PEL type A or C (that is, unitary or symplectic), at p an unramified prime. Maass–Shimura operators are smooth weight raising differential operators used in the study of special values of L-functions, and in the arithmetic setting for the construction of p-adic L-functions.  In this talk, we will focus in particular on the case of unitary groups of arbitrary signature, when new phenomena arise for p  non split.  We will also discuss an application to the study of modular mod p Galois representations. This talk is based on joint work with Ellen Eischen (in the unitary case for p non split), and with Eischen, Flanders, Ghitza, and Mc Andrew (in the other cases).

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

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

  • OPEN NEIGHBORHOOD SEMINAR
22
  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: The uses of lattice topological defects
    10:30 AM-12:00 PM
    October 22, 2020

    I will give an overview of my work with Aasen and Mong on using fusion categories to find and analyse topological defects in two-dimensional classical lattice models and quantum chains.

    These defects possess a variety of remarkable properties. Not only is the partition function independent of deformations of their path, but they can branch and fuse in a topologically invariant fashion.  One use is to extend Kramers-Wannier duality to a large class of models, explaining exact degeneracies between non-symmetry-related ground states as well as in the low-energy spectrum. The universal behaviour under Dehn twists gives exact results for scaling dimensions, while gluing a topological defect to a boundary allows universal ratios of the boundary g-factor to be computed exactly on the lattice.  I also will describe how terminating defect lines allows the construction of fractional-spin conserved currents, giving a linear method for Baxterization, I.e. constructing integrable models from a braided tensor category.

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

  • HARVARD-MIT-BU-BRANDEIS-NORTHEASTERN COLLOQUIUM

    HARVARD-MIT-BU-BRANDEIS-NORTHEASTERN COLLOQUIUM
    Galois symmetries of the stable homology of integer symplectic groups

    4:30 PM-5:30 PM
    October 22, 2020

    There are many natural sequences of moduli spaces in algebraic geometry whose homology approaches a “limit”, despite the fact that the spaces themselves have growing dimension.  If these moduli spaces are defined over a field K,  this limiting homology carries an extra structure — an action of the Galois group of K —  which is arithmetically interesting.

    In joint work with Feng and Galatius, we compute this action (or rather a slight variant) in the case of the moduli space of abelian varieties. I will explain the answer and why I find it interesting. No familiarity with abelian varieties will be assumed — I will emphasize topology over algebraic geometry.

    Zoom: https://mit.zoom.us/j/98577860372

23
24
25
26
27
  • DIFFERENTIAL GEOMETRY SEMINAR
    8:00 AM-9:00 AM
    October 27, 2020

    The derived category of a Fano threefold Y of Picard rank 1 and index 2 admits a semiorthogonal decomposition. This defines a non-trivial subcategory Ku(Y) called the Kuznetsov component, which encodes most of the geometry of Y. I will present joint work with M. Altavilla and M. Petkovic, in which we describe certain moduli spaces of Bridgeland-stable objects in Ku(Y), via the stability conditions constructed by Bayer, Macrì, Lahoz and Stellari. Furthermore, in our work we study the behavior of the Abel-Jacobi map on these moduli space. As an application in the case of degree d = 2, we prove a strengthening of a categorical Torelli Theorem by Bernardara and Tabuada.

    Zoom: https://harvard.zoom.us/j/96709211410?pwd=SHJyUUc4NzU5Y1d0N2FKVzIwcmEzdz09

  • MATHEMATICAL PICTURE LANGUAGE SEMINAR

    MATHEMATICAL PICTURE LANGUAGE SEMINAR
    The quest of a finite purely quantum group

    10:00 AM-11:00 AM
    October 27, 2020

    An important open problem is whether there exists a finite quantum group which cannot be cooked up from (classical) finite groups. A finite purely quantum group would be a finite dimensional Hopf C*-algebra (Kac algebra) K such that the unitary integral fusion category Rep(K) is not weakly group-theoretical, and admits no such fusion subcategory other than the trivial one. This talk will expose the first results in the quest of such an object, through joint works with Zhengwei Liu, Yunxiang Ren and Jinsong Wu, involving subfactor planar algebras, quantum Fourier analysis and fusion categories.

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

  • CMSA EVENT: CMSA Computer Science for Mathematicians: Depth-Width Trade-offs for Neural Networks through the lens of Dynamical Systems
    11:30 AM-12:30 PM
    October 27, 2020

    How can we use the theory of dynamical systems in analyzing the capabilities of neural networks? Understanding the representational power of Deep Neural Networks (DNNs) and how their structural properties (e.g., depth, width, type of activation unit) affect the functions they can compute, has been an important yet challenging question in deep learning and approximation theory. In a seminal paper, Telgarsky reveals the limitations of shallow neural networks by exploiting the oscillatory behavior of a simple triangle function and states as a tantalizing open question to characterize those functions that cannot be well-approximated by small depths.
    In this work, we point to a new connection between DNNs expressivity and dynamical systems, enabling us to get trade-offs for representing functions based on the presence of a generalized notion of fixed points, called periodic points that have played a major role in chaos theory (Li-Yorke chaos and Sharkovskii’s theorem). Our main results are general lower bounds for the width needed to represent periodic functions as a function of the depth, generalizing previous constructions relying on specific functions.

    Based on two recent works:
    with Ioannis Panageas, Sai Ganesh Nagarajan, Xiao Wang from ICLR’20 (spotlight):  https://arxiv.org/abs/1912.04378
    with Ioannis Panageas, Sai Ganesh Nagarajan from ICML’20: https://arxiv.org/abs/2003.00777

    Zoom: https://harvard.zoom.us/j/98231541450

  • HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR
    3:00 PM-4:00 PM
    October 27, 2020

    The generic vanishing theorem of Green-Lazarsfeld says that for general elements in the Picard variety of a projective manifold, their cohomology groups vanish in all degrees. Moreover, the cohomological jumping locus, that is, the locus where generic vanishing fails, is a union of torsion translated abelian subvarieties. If one replaces the Picard variety by the character variety of rank 1 local systems, then one can study a similar phenomenon, which are works by Simpson and Budur-Wang topologically and Esnault-Kerz arithmetically.  In this talk, I will focus on the same phenomenon but from algebraic perspectives by using D-modules. More precisely, I will discuss zero loci of Bernstein-Sato ideals and explain why the zero loci can be treated as the algebraic analogue of topological jumping loci by using relative D-modules. Then I will prove a conjecture of Budur that zero loci of Bernstein-Sato ideals are related to the topological jumping loci in the sense of Riemann-Hilbert Correspondence. This is based on joint work with Nero Budur, Robin van der Veer and Peng Zhou.

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

28
  • CMSA EVENT: CMSA Strongly Correlated Quantum Materials and High-Temperature Superconductors Series: The not-so-normal normal state of underdoped Cuprate
    10:30 AM-12:00 PM
    October 28, 2020

    The underdoped Cuprate exhibits a rich variety of unusual properties that have been exposed after years of experimental investigations. They include a pseudo-gap near the anti-nodal points and “Fermi arcs” of gapless excitations, together with a variety of order such as charge order, nematicity and possibly loop currents and time reversal and inversion breaking. I shall argue that by making a single assumption of strong pair fluctuations at finite momentum (Pair density wave), a unified description of this phenomenology is possible. As an example, I will focus on a description of the ground state that emerges when superconductivity is suppressed by a magnetic field, which supports small electron pockets. [Dai, Senthil, Lee, Phys Rev B101, 064502 (2020)] There is some support for the pair density wave hypothesis from STM data that found charge order at double the usual wave-vector in the vicinity of vortices, as well as evidence for a fragile form of superconductivity persisting to fields much above Hc2. I shall suggest a more direct experimental probe of the proposed fluctuating pair density wave.

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

  • RANDOM MATRIX SEMINAR
    2:00 PM-3:00 PM
    October 28, 2020

    From the viewpoint of spin glass theory, restricted Boltzmann machines represent a veritable challenge, as to the lack of convexity prevents us to use Guerra’s bounds. Therefore even the replica symmetric approximation for the free energy presents some challenges. I will present old and new results around the topic along with some open problems.

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

    Password: rmtpt2020

  • CMSA EVENT: CMSA New Technologies in Mathematics: Generalization bounds for rational self-supervised learning algorithms
    3:00 PM-4:00 PM
    October 28, 2020

    The generalization gap of a learning algorithm is the expected difference between its performance on the training data and its performance on fresh unseen test samples. Modern deep learning algorithms typically have large generalization gaps, as they use more parameters than the size of their training set. Moreover the best known rigorous bounds on their generalization gap are often vacuous.

    In this talk we will see a new upper bound on the generalization gap of classifiers that are obtained by first using self-supervision to learn a complex representation of the (label free) training data, and then fitting a simple (e.g., linear) classifier to the labels. Such classifiers have become increasingly popular in recent years, as they offer several practical advantages and have been shown to approach state-of-art results.

    We show that (under the assumptions described below) the generalization gap of such classifiers tends to zero as long as the complexity of the simple classifier is asymptotically smaller than the number of training samples. We stress that our bound is independent of the complexity of the representation that can use an arbitrarily large number of parameters.
    Our bound assuming that the learning algorithm satisfies certain noise-robustness (adding small amount of label noise causes small degradation in performance) and rationality (getting the wrong label is not better than getting no label at all) conditions that widely (and sometimes provably) hold across many standard architectures.
    We complement this result with an empirical study, demonstrating that our bound is non-vacuous for many popular representation-learning based classifiers on CIFAR-10 and ImageNet, including SimCLR, AMDIM and BigBiGAN.

    The talk will not assume any specific background in machine learning, and should be accessible to a general mathematical audience. Joint work with Gal Kaplun.

    Zoom: https://harvard.zoom.us/j/96047767096?pwd=M2djQW5wck9pY25TYmZ1T1RSVk5MZz09

  • NUMBER THEORY SEMINAR
    3:00 PM-4:00 PM
    October 28, 2020

    The local Langlands conjectures predict that (packets of) irreducible complex representations of p-adic reductive groups (such as GL_n(Q_p), GSp_2n(Q_p), etc.) should be parametrized by certain representations of the Weil-Deligne group.  A special role in this hypothetical correspondence is held by the supercuspidal representations, which generically are expected to correspond to irreducible objects on the Galois side, and which serve as building blocks for all irreducible representations.  Motivated by recent advances in the mod-p local Langlands program (i.e., with mod-p coefficients instead of complex coefficients), I will give an overview of what is known about supersingular representations of p-adic reductive groups, which are the “mod-p coefficients” analogs of supercuspidal representations.  This is joint work with Florian Herzig and Marie-France Vigneras.

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

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

  • MATH TABLE
    4:30 PM-5:30 PM
    October 28, 2020

    The purest forms of functional programming use monads to define computations that happen within contexts. For instance, the IO monad, which is a standard object in Haskell as well as languages inspired by Haskell, is used to handle processes that require interaction with the outside world. Monads are the dread of many fledgling programmers learning functional programming for the first time, but they are actually familiar constructions from category theory. This talk will discuss the definitions of monad in functional programming and category theory and describe how they are manifested in the context of a Haskell program that reads in and prints an integer.

    Zoom: https://harvard.zoom.us/j/96759150216?pwd=Tk1kZlZ3ZGJOVWdTU3JjN2g4MjdrZz09

29
  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Symmetry, Insulating States and Excitations of Twisted Bilayer Graphene with Coulomb Interaction
    10:30 AM-12:00 PM
    October 29, 2020

    The twisted bilayer graphene (TBG) near the magic angle around 1 degree hosts topological flat moiré electron bands, and exhibits a rich tunable strongly interacting physics. Correlated insulators and Chern insulators have been observed at integer fillings nu=0,+-1,+-2,+-3 (number of electrons per moiré unit cell). I will first talk about the enhanced U(4) or U(4)xU(4) symmetries of the projected TBG Hamiltonian with Coulomb interaction in various combinations of the flat band limit and two chiral limits. The symmetries in the first chiral and/or flat limits allow us to identify exact or approximate ground/low-energy (Chern) insulator states at all the integer fillings nu under a weak assumption, and to exactly compute charge +-1, +-2 and neutral excitations. In the realistic case away from the first chiral and flat band limits, we find perturbatively that the ground state at integer fillings nu has Chern number +-mod(nu,2), which is intervalley coherent if nu=0,+-1,+-2, and is valley polarized if nu=+-3. We further show that at nu=+-1 and +-2, a first order phase transition to a Chern number 4-|nu| state occurs in an out-of-plane magnetic field. Our calculation of excitations also rules out the Cooper pairing at integer fillings nu from Coulomb interaction in the flat band limit, suggesting other superconductivity mechanisms. These analytical results at nonzero fillings are further verified by a full Hilbert space exact diagonalization (ED) calculation. Furthermore, our ED calculation for nu=-3 implies a phase transition to possible translationally breaking or metallic phases at large deviation from the first chiral limit.

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

30
31