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

TITLE: Birational geometry

ABSTRACT: About main achievements in birational geometry during the last fifty years.

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 the 90s’, Witten gave a physical derivation of an isomorphism between the Verlinde algebra of $GL(n)$ of level $l$ and the quantum cohomology ring of the Grassmannian $\text{Gr}(n,n+l)$. In the joint work arXiv:1811.01377 with Yongbin Ruan, we proposed a K-theoretic generalization of Witten’s work by relating the $\text{GL}_{n}$ Verlinde numbers to the level $l$ quantum K-invariants of the Grassmannian $\text{Gr}(n,n+l)$, and refer to it as the Verlinde/Grassmannian correspondence.

The correspondence was formulated precisely in the aforementioned paper, and we proved the rank 2 case (n=2) there. In this talk, I will discuss the proof for arbitrary rank. A new technical ingredient is the virtual nonabelian localization formula developed by Daniel Halpern-Leistner.  At the end of the talk, I will describe some applications of this correspondence.

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

TITLE: Kunihiko Kodaira and complex manifolds.

ABSTRACT: Kodaira’s motivation was to generalize the theory of Riemann surfaces in Weyl’s book to higher dimensions.  After quickly recalling the chronology of Kodaira, I will review some of Kodaira’s works in three sections on topics of harmonic analysis, deformation theory and compact complex surfaces.  Each topic corresponds to a volume of Kodaira’s collected works in three volumes, of which I will cover only tiny parts.

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 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

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

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

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

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.

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

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

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

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

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

Noisy intermediate-scale quantum (NISQ) Computers hold the key for important theoretical and experimental questions regarding quantum computers. In the lecture I will describe some questions about mathematics, statistics and computational complexity which arose in my study of NISQ systems and are related to
a) My general argument “against” quantum computers,
b) My analysis (with Yosi Rinott and Tomer Shoham) of the Google 2019 “quantum supremacy” experiment.

Relevant papers:
Yosi Rinott, Tomer Shoham and Gil Kalai, Statistical aspects of the quantum supremacy demonstration, https://gilkalai.files.wordpress.com/2019/11/stat-quantum2.pdf
Gil Kalai, The Argument against Quantum Computers, the Quantum Laws of Nature, and Google’s Supremacy Claims, https://gilkalai.files.wordpress.com/2020/08/laws-blog2.pdf
Gil Kalai, Three puzzles on mathematics, computations, and games, https://gilkalai.files.wordpress.com/2019/09/main-pr.pdf

For security reasons, you are kindly asked to show your full name while joining the meeting.

Zoom: https://harvard.zoom.us/j/95505022117

Conformal bootstrap is a powerful method to study conformal field theory (CFT) in arbitrary spacetime dimensions. Sometimes interesting CFTs such as O(N) Wilson-Fisher (WF) CFTs sit at kinks of numerical bootstrap bounds. In this talk I will first give a brief introduction to conformal bootstrap, and then discuss a new family of kinks (dubbed non-WF kinks) of numerical bootstrap bounds of O(N) symmetric CFTs. The nature of these new kinks remains mysterious, but we manage to understand few special cases, which already hint interesting physics. In 2D, the O(4) non-WF kink turns out to be the familiar SU(2)_1 Wess-Zumino-Witten model. We further consider its dimensional continuation towards the 3D SO(5) deconfined phase transition, and we find the kink disappears at fractional dimension (around D=2.7), suggesting the 3D SO(5) deconfined phase transition is pseudo-critical. At last, based on the analytical solution at infinite N limit we speculate that there exists a new family of O(N) (or SO(N)) true CFTs for N large enough, which might be a large-N generalization of SO(5) DQCP.

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

The Thouless-Anderson-Palmer (TAP) approach to the Sherrington-Kirkpatrick mean field spin glass model was proposed in one of the earliest papers on this model. Since then it has complemented subsequently elaborated methods  in theoretical physics and mathematics, such as the replica method, which are largely orthogonal to the TAP approach. The TAP approach has the advantage of being interpretable as a variational principle optimizing an energy/entropy trade-off, as commonly encountered in statistical physics and large deviations theory, and potentially allowing for a more direct characterization of the Gibbs measure and its “pure states”. In this talk I will recall the TAP approach, and present preliminary steps towards a solution of mean field spin glass models entirely within a TAP framework.

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

Classical learning theory suggests that the optimal generalization performance of a machine learning model should occur at an intermediate model complexity, striking a balance between simpler models that exhibit high bias and more complex models that exhibit high variance of the predictive function. However, such a simple trade-off does not adequately describe the behavior of many modern deep learning models, which simultaneously attain low bias and low variance in the heavily overparameterized regime. Recent efforts to explain this phenomenon theoretically have focused on simple settings, such as linear regression or kernel regression with unstructured random features, which are too coarse to reveal important nuances of actual neural networks. In this talk, I will describe a precise high-dimensional asymptotic analysis of Neural Tangent Kernel regression that reveals some of these nuances, including non-monotonic behavior deep in the overparameterized regime. I will also present a novel bias-variance decomposition that unambiguously attributes these surprising observations to particular sources of randomness in the training procedure.

Zoom: https://harvard.zoom.us/j/93890093197?pwd=QWtXUkJycU5scmh6QVhBdVp0UkhlUT09

The Bloch–Kato conjecture predicts that the dimension of the Selmer group of a global Galois representation is equal to the order of vanishing of its L-function. In this talk, I will focus on the 4-dimensional Galois representations arising from cohomological automorphic representations of GSp(4) (i.e. from genus two Siegel modular forms). I will show that if the L-function is non-vanishing at some critical value, then the corresponding Selmer group is zero, under a long list of technical hypotheses. The proof of this theorem relies on an Euler system, a p-adic L-function, and a reciprocity law connecting those together. I will also survey work in progress aiming to extend this result to some other classes of automorphic representations.

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

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

Proving a result that’s more general than intended can often be a cause to celebrate! But it can also be a red flag, indicating that a particular technique is not incisive enough to tackle more advanced problems. In this talk, we’ll see one such situation, the notion of relativization in the theory of computational complexity. We’ll define what it means for a theorem to relativize — hold true even if computers are given superpowers — and describe the famous P vs. NP problem and why it cannot be resolved by relativizing techniques. If we have time, we’ll sketch a proof of the time hierarchy theorem, a nice example of a relativizing theorem.

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

The pseudogap phase of cuprate superconductors is arguably the most enigmatic phase of quantum matter. We aim to shed new light on this phase by investigating the non- superconducting ground state of several cuprate materials at low temperature across a wide doping range, suppressing superconductivity with a magnetic field. Hall effect measurements across the pseudogap critical doping p* reveal a sharp drop in carrier density n from n = 1 + p above p* to n = p below p*, signaling a major transformation of the Fermi surface. Angle-dependent magneto-resistance (ADMR) directly reveals a change in Fermi surface topology across p*. From specific heat measurements, we observe the classic thermodynamic signatures of quantum criticality: the electronic specific heat C el shows a sharp peak at p*, where it varies in temperature as C el ~ – T logT. At p* and just above, the electrical resistivity is linear in T at low T, with an inelastic scattering rate that obeys the Planckian limit. Finally, the pseudogap phase is found to have a large negative thermal Hall conductivity, which extends to zero doping. We show that the pseudogap phase makes phonons become chiral.
Understanding the mechanisms responsible for these various new signatures will help elucidate the nature of the pseudogap phase.

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

3d mirror symmetry is a proposed duality relating a pair of 3-dimensional supersymmetric gauge theories. Various consequences of this duality have been heavily explored by representation theorists in recent years, under the name of “symplectic duality”. In joint work in progress with Justin Hilburn, for the case of abelian gauge groups, we provide a fully mathematical explanation of this duality in the form of an equivalence of 2-categories of boundary conditions for topological twists of these theories. We will also discuss some applications to homological mirror symmetry and geometric Langlands duality.

Zoom: https://harvard.zoom.us/j/91780604388?pwd=d3BqazFwbDZLQng0cEREclFqWkN4UT09

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

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

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

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

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

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.

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.

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.

The lecture follows the title as a guiding thread.

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

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

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

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

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

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

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

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

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

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

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.

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

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