We will consider the following problem: if (C,x_1,…,x_n) is a fixed general pointed curve, and X is a fixed target variety with general points y_1,…,y_n, then how many maps f:C -> X in a given homology class are there, such that f(x_i)=y_i? When considered virtually in Gromov-Witten theory, the answer may be expressed in terms of the quantum cohomology of X, leading to explicit formulas in some cases (Buch-Pandharipande). The geometric question is more subtle, though in the presence of sufficient positivity, it is expected that the virtual answers are enumerative. I will give an overview of recent progress on various aspects of this problem, including joint work with Farkas, Pandharipande, and Cela, as well as work of other authors.

https://harvard.zoom.us/j/97335783449?pwd=S3U0eVdyODFEdzNaRXVEUTF3R3NwZz09

loci, equidistribution of Hecke translates of a curve in the moduli space of abelian varieties and equidistribution of some families of CM points in Shimura varieties. The results of this talk are joint work with Nicolas Tholozan.

Consider the set of Galois extensions L of Q whose Galois group is the quaternion group. For large X, Klüners counted extensions with |disc(L)| <= X. We discuss asymptotics when bounding invariants other than the discriminant.

One way that mathematics grows is by finding new questions to study beyond the standard topics of serious mathematical research. We give three examples, ranging from the recreational (how many digits in 6561101970383!, and how did I find this curious factorial? What’s the unusual feature of the “elementary” identity 12 + 21 + 27 + 38 + 44 + 53 = 2 + 5 + 11 + 22 + 33 + 48 + 74?) to a collaboration with faculty in the School of Public Health.

I will introduce a class of non-invertible topological defects in (3 + 1)d gauge theories whose fusion rules are the higher-dimensional analogs of those of the Kramers-Wannier defect in the (1 + 1)d critical Ising model. As in the lower-dimensional case, the presence of such non-invertible defects implies self-duality under a particular gauging of their discrete (higher-form) symmetries. Examples of theories with such a defect include SO(3) Yang-Mills (YM) at θ = π, N = 1 SO(3) super YM, and N = 4 SU(2) super YM at τ = i. I will also explain an analogous construction in (2+1)d, and give a number of examples in Chern-Simons-matter theories. This talk is based on https://arxiv.org/abs/2111.01141.

https://harvard.zoom.us/j/977347126
Password: cmsa

Slice a triply periodic wooden sculpture along an irrational plane. If you ink the cut surface and press it against a page, the pattern you print will be quasiperiodic. Patterns like these help physicists see how metals conduct electricity in strong magnetic fields. I’ll show you some block prints that imitate the printing process described above, and I’ll point out the visual features that reveal conductivity properties.

Zoom ID: 950 2372 5230 (Password: cmsa)

Interactive slides: https://www.ihes.fr/~fenyes/seeing/slices/

The (tree) amplituhedron was introduced in 2013 by Arkani-Hamed and Trnka in their study of N=4 SYM scattering amplitudes. A central conjecture in the field was to prove that the m=4 amplituhedron is triangulated by the images of certain positroid cells, called the BCFW cells. In this talk I will describe a resolution of this conjecture. The seminar is based on a recent joint work with Chaim Even-Zohar and Tsviqa Lakrec.

https://harvard.zoom.us/j/91799784675?pwd=MS9LV25DWk9RcmJoRVM0K3RGWkFRdz09

Password: 1251442

In metals, orbital motions of conduction electrons are quantized in magnetic fields, which is manifested by quantum oscillations in electrical resistivity. This Landau quantization is generally absent in insulators, in which all the electrons are localized. Here we report a notable exception in an insulator — ytterbium dodecaboride (YbB12). The resistivity of YbB12, despite much larger than that of usual metals, exhibits profound quantum oscillations under intense magnetic fields. This unconventional oscillation is shown to arise from the insulating bulk instead of conducting surface states. The large effective masses indicate strong correlation effects between electrons. Our result is the first discovery of quantum oscillations in the electrical resistivity of a strongly correlated insulator and will bring crucial insight into understanding the ground state in gapped Kondo systems.

https://harvard.zoom.us/j/977347126
Password: cmsa

I will talk about work to uncover connections between invariant theory and maximum likelihood estimation. I will describe how norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We will see the role played by polytopes and discuss connections to scaling algorithms. Based on joint work with Carlos Améndola, Kathlén Kohn, and Philipp Reichenbach.

https://harvard.zoom.us/j/91799784675?pwd=MS9LV25DWk9RcmJoRVM0K3RGWkFRdz09

Password: 1251442

Strominger-Yau-Zaslow conjecture is one of the guiding principles in mirror symmetry, which not only predicts the geometric structures of Calabi-Yau manifolds but also provides a recipe for mirror construction. Besides mirror symmetry, the SYZ conjecture itself is the holy grail in geometrical analysis and closely related to the behavior of the Ricci-flat metrics. In this talk, we will explain how SYZ fibrations on log Calabi-Yau surfaces detect the non-standard semi-flat metric which generalized the semi-flat metrics of Greene-Shapere-Vafa-Yau. Furthermore, we will use the SYZ fibration on log Calabi-Yau surfaces to prove the Torelli theorem of gravitational instantons of type ALH^*. This is based on the joint works with T. Collins and A. Jacob.

https://harvard.zoom.us/j/97335783449?pwd=S3U0eVdyODFEdzNaRXVEUTF3R3NwZz09

Theoretical computer science has given rise to new, exciting questions in algebraic geometry and representation theory. In this talkI will focus on the problem of matrix multiplication. It is generally conjectured by computer scientists that as n grows very large, it becomes almost as easy to multiply nxn matrices as it is to add them! After giving a brief history of the problem I will focus on algebraic geometry and representation theory relevant for the problem and conclude by discussing  recent work with A. Conner, A. Harper, and H. Huang.

There are several properties of closed geodesics which are proven using its Hamiltonian formulation, which has no analogue for minimal surfaces. I will talk about some recent progress in proving some of these properties for minimal surfaces.

Zoom Link: https://cuhk.zoom.us/j/94766047171

(Meeting ID: 947 6604 7171; Passcode: 20220209 )

Over the past several years, attention mechanisms (primarily in the form of the Transformer architecture) have revolutionized deep learning, leading to advances in natural language processing, computer vision, code synthesis, protein structure prediction, and beyond. Attention has a remarkable ability to enable the learning of long-range dependencies in diverse modalities of data. And yet, there is at present limited principled understanding of the reasons for its success. In this talk, I’ll explain how attention mechanisms and Transformers work, and then I’ll share the results of a preliminary investigation into whythey work so well. In particular, I’ll discuss an inductive bias of attention that we call sparse variable creation: bounded-norm Transformer layers are capable of representing sparse Boolean functions, with statistical generalization guarantees akin to sparse regression.

https://harvard.zoom.us/j/99651364593?pwd=Q1R0RTMrZ2NZQjg1U1ZOaUYzSE02QT09

Grothendieck’s Section Conjecture posits that the set of rational points on a smooth projective curve Y of genus at least two should be equal to a certain “section set” defined purely in terms of the etale fundamental group of Y. In this talk, I will preview some upcoming work with Jakob Stix in which we prove a partial finiteness result for this section set, thereby giving an unconditional verification of a prediction of the Section
Conjecture for a general curve Y. We do this by adapting the recent p-adic proof of the Mordell Conjecture due to Brian Lawrence and Akshay Venkatesh.

Superstring theory as we know it started from the discovery by Green and Schwarz in 1984 that the perturbative anomalies of heterotic strings miraculously cancel. But the cancellation of global anomalies of heterotic strings remained an open problem for a long time.

In this talk, I would like to report how this issue was finally resolved last year, by combining two developments outside of string theory. Namely, on one hand, the study of topological phases in condensed matter theory has led to our vastly improved understanding of the general form of global anomalies. On the other hand, the study of topological modular forms in algebraic topology allows us to constrain the data of heterotic worldsheet theories greatly, as far as their contributions to the anomalies are concerned. Putting them together, it is possible to show that global anomalies of heterotic strings are always absent.

The talk is based on https://arxiv.org/abs/2103.12211 and https://arxiv.org/abs/2108.13542 , in collaboration with Mayuko Yamashita.

https://harvard.zoom.us/j/977347126
Password: cmsa

A real algebraic variety is the set of points in real Euclidean space that satisfy a system of polynomial equations.  Metric algebraic geometry is the study of properties of real algebraic varieties that depend on a distance metric. In this talk, we introduce metric algebraic geometry through a discussion of Voronoi cells, bottlenecks, and the reach of an algebraic variety. We also show applications to the computational study of the geometry of data with nonlinear models.

Zoom ID: 950 2372 5230 (Password: cmsa)

It is well-established that the Standard Model (SM) of particle physics is based on su(3)Xsu(2)Xu(1) Lie-algebra. What is less appreciated, however, is that SM accommodates a Z_6 1-form global symmetry.  Gauging this symmetry, or a subgroup of it, changes the global structure of the SM gauge group and amounts to summing over sectors of instantons with fractional topological charges. After a brief review of the concept of higher-form symmetries, I will explain the origin of the Z_6 1-form symmetry and construct the explicit fractional-instanton solutions on compact manifolds. The new instantons mediate baryon-number and lepton-number violating processes, which can win over the weak BPST-instanton processes, provided that SM accommodates extra hyper-charged particles above the TeV scale. I will also comment on the cosmological aspects of the new solutions.

https://harvard.zoom.us/j/977347126
Password: cmsa

My lab is interested in the active and adaptive materials that underlie control of cell shape.  This has centered around understanding force transmission and sensing within the actin cytoskeleton.  I will first review our current understanding of the types of active matter that can be constructed by actin polymers.  I will then turn to our recent experiments to understand how Cell shape changes in epithelial tissue.  I will describe the two sources of active stresses within these tissues, one driven by the cell cycle and controlling cell-cell stresses and the other controlled by cell-matrix signaling controlling motility.  I will then briefly describe how we are using optogenetics to locally control active stresses to reveal adaptive and force-sensitive mechanics of the cytoskeletal machinery. Hopefully, I will convince you that recent experimental and theoretical advances make this a very promising time to study this quite complicated form of active matter!

How many rational preperiodic points can a degree d polynomial in Q[x] have? Conjecturally, there is some uniform bound B_d(Q) on the number of such preperiodic points — but how big is B_d(Q)? Using interpolation we can easily find examples with d + 1 rational preperiodic points, but every point beyond that has to be fought for. In this talk I will share some recent work, joint with John R. Doyle, in which we prove that B_d(Q) is at least d + c*log(d) for some constant c and for all sufficiently large d. As a bonus, I’ll show how our construction also produces examples of pairs of degree d polynomials f(x) and g(x) with more than d^2 common complex preperiodic points.

http://people.math.harvard.edu/~demarco/AlgebraicDynamics/ for more information

In this talk, I will introduce a virtual variant of the quantized Coulomb branch constructed by Braverman-Finkelberg-Nakajima, where the convolution product is modified by a virtual intersection. The resulting virtual Coulomb branch acts on the moduli space of quasimaps into the holomorphic symplectic quotient T^*N//G. When G is abelian, over the torus fixed points, this representation is a Verma module. The vertex function, a K-theoretic enumerative invariant introduced by A. Okounkov, can be expressed as a Whittaker function of the algebra. The construction also provides a description of the quantum q-difference module. As an application, this gives a proof of the invariance of the quantum q-difference module under variation of GIT.

https://harvard.zoom.us/j/97335783449?pwd=S3U0eVdyODFEdzNaRXVEUTF3R3NwZz09

Random field Ising model is a canonical example to study the effect of disorder on long range order. In 70’s, Imry-Ma predicted that in the presence of weak disorder, the long-range order persists at low temperatures in three dimensions and above but disappears in two dimensions. In this talk, I will review mathematical development surrounding this prediction, and I will focus on recent progress on exponential decay (joint with Jiaming Xia) and correlation length in dimension two (joint with Mateo Wirth), a new proof for long range order in dimension three (joint with Zijie Zhuang) and a recent general inequality for the Ising model (joint with Jian Song and Rongfeng Sun) which has implications for random field Ising model.

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

Peking University

In this talk, we determine when there is a Brill–Noether curve of given degree and given genus that passes through a given number of general points in any projective space.

Given a grid diagram for a knot or link K in the three-sphere, we construct a spectrum whose homology is the knot Floer homology of K. We conjecture that the homotopy type of the spectrum is an invariant of K. Our construction does not use holomorphic geometry, but rather builds on the combinatorial definition of grid homology. We inductively define models for the moduli spaces of pseudo-holomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions. (This is joint work with Sucharit Sarkar.)

For details, please visit:

http://www.ims.cuhk.edu.hk/cgi-bin/SeminarAdmin/bin/Web

http://www.ims.cuhk.edu.hk/activities/seminar/joint-dg-seminar/

Zoom link: https://cuhk.zoom.us/j/99199383345

(Meeting ID: 991 9938 3345; Passcode: 20220216)

In 1960’s, Narasimhan and Seshadri discovered the equivalence between irreducible unitary flat bundles and stable bundles of degree $0$ on compact Riemann surfaces. In 1980’s, Donaldson, Uhlenbeck and Yau generalized it to the equivalence between irreducible Hermitian-Einstein bundles and stable bundles on smooth projective varieties. This is a surprising bridge connecting differential geometry and algebraic geometry. Since then, many interesting generalizations have been studied.
In this talk, we would like to review a stream in the study of such correspondences for Higgs bundles, integrable connections, $D$-modules and periodic monopoles.

https://harvard.zoom.us/j/95767170359

(Password: cmsa)

In this sequence of talks, I will describe our work with Alexander Frenkel and Stephen Randall, in which we presented a novel topological quantum gravity, relating three previously unrelated fields:  Topological quantum field theories (of the cohomological type), the theory of Ricci flows on Riemannian manifolds, and nonrelativistic quantum gravity.  The remarkable richness of results produced in the recent decades by mathematicians studying the Ricci flow promises to shed new light on the physics of the path integral in quantum gravity (at least in the topological regime).  In the opposite direction, the techniques of quantum field theory and path integrals may end up putting some of the mathematical results in the Ricci flow theory in a new perspective as well.

https://harvard.zoom.us/j/977347126
Password: cmsa

Hyperbolic manifolds are a class of Riemannian manifolds that are important in mathematics and physics, playing a prominent role in topology, number theory, and string theory. Associated with a given hyperbolic metric is a sequence of numbers corresponding to the discrete eigenvalues of the Laplace-Beltrami operator. While these eigenvalues usually cannot be calculated exactly, they can be found numerically and must also satisfy various bounds. In this talk, I will discuss a new approach for finding numerical bounds on the eigenvalues of closed hyperbolic manifolds using general consistency conditions and semidefinite programming, inspired by the approach of the conformal bootstrap from physics. Although these bootstrap bounds follow from seemingly trivial consistency conditions, they are surprisingly strong and are sometimes almost saturated by actual manifolds; for example, one such bound implies that the first nonzero eigenvalue of a closed hyperbolic surface must be less than 3.83890, and this is very close to being saturated by a particular genus-2 surface called the Bolza surface. I will show how to derive this and other bounds and will discuss some possible future directions for this approach.

---

https://harvard.zoom.us/j/99651364593?pwd=Q1R0RTMrZ2NZQjg1U1ZOaUYzSE02QT09

Password: cmsa

For an elliptic curve E,  Kolyvagin used Heegner points to construct special Galois cohomology classes valued in the torsion points of E. Under the conjecture that not all of these classes vanish, he showed that they encode the Selmer rank of E. I will explain a proof of new cases of this conjecture that builds on prior work of Wei Zhang. The proof naturally leads to a generalization of Kolyvagin’s work in a complimentary “definite” setting, where Heegner points are replaced by special values of a quaternionic modular form. I’ll also explain an “ultrapatching” formalism which simplifies the Selmer group arguments required for the proof.

Suppose you write down a general polynomial in x, y, z and consider the surface of all points where it vanishes. What can you say about the family of lines contained in this surface? Are there no lines, a finite number of lines, infinitely many? We’ll derive an expected dimension for the family of lines depending on the degree of the polynomial (and generalize this to more variables). In the case of cubic surfaces, we’ll discuss some more subtle questions regarding the geometry of lines over the real numbers. This story motivates some results, joint with Isabel Vogt, about a closely related problem concerning bitangents (lines that are tangent twice) to a plane quartic. There will be many examples and “hands on demos.”

Finite order Markov models are theoretically well-studied models for dependent data.  Despite their generality, application in empirical work when the order is larger than one is quite rare.  Practitioners avoid using higher order Markov models because (1) the number of parameters grow exponentially with the order, (2) the interpretation is often difficult. Mixture of transition distribution models (MTD)  were introduced to overcome both limitations. MTD represent higher order Markov models as a convex mixture of single step Markov chains, reducing the number of parameters and increasing the interpretability. Nevertheless, in practice, estimation of MTD models with large orders are still limited because of curse of dimensionality and high algorithm complexity. Here, we prove that if only few lags are relevant we can consistently and efficiently recover the lags and estimate the transition probabilities of high order MTD models. Furthermore, we show that using the selected lags we can construct non-asymptotic confidence intervals for the transition probabilities of the model. The key innovation is a recursive procedure for the selection of the relevant lags of the model.  Our results are  based on (1) a new structural result of the MTD and (2) an improved martingale concentration inequality. Our theoretical results are illustrated through simulations.

Zoom ID: 950 2372 5230 (Password: cmsa)

‘t Hooft anomalies of anomalous systems can be described via anomaly inflow by invertible theories living in one dimension higher. Thanks to this it is possible to provide a general method to determine modular transformations of anomalous 2d fermionic CFTs with general discrete symmetry group $G^f$. As a by-product, one is able to determine explicit combinatorial expressions of spin-cobordism invariants in terms of Dehn-surgery representation of 3-manifolds. The same techniques also provide a method for evaluating the map from the group classifying free fermionic anomalies to the group of anomalies in interacting theories. As examples, we work out the details for some symmetry groups, including non-abelian ones, and, as an application, we use these results to bootstrap the spectrum of the theories with a given anomaly.

https://harvard.zoom.us/j/977347126
Password: cmsa

Given a link in the thickened annulus, you can construct an associated link in a closed 3-manifold through a double branched cover construction. In this talk we will see that perspective on annular links can be applied to show annular Khovanov homology detects certain braid closures. Unfortunately, this perspective does not capture all information about annular links. We will see a shortcoming of this perspective inspired by the wrapping conjecture of Hoste-Przytycki. This is partially joint work with Fraser Binns.

https://people.math.harvard.edu/~jxwang/seminar/ for more information

We construct a certain type of Gauged Linear Sigma Model quasimap invariants that generalize the original ones and are easier to compute. Higgs-Coulomb correspondence provides identification of generating functions of our invariants with certain analytic functions that can be represented as generalized inverse Mellin transforms. Analytic continuation of these functions provides wall-crossing results for GLSM and generalizes Landau-Ginzburg/Calabi-Yau correspondence. The talk is based on a joint work in progress with Melissa Liu.

https://harvard.zoom.us/j/97335783449?pwd=S3U0eVdyODFEdzNaRXVEUTF3R3NwZz09

In the study of Ising models on large locally tree-like graphs, in both rigorous and non-rigorous methods one is often led to understanding the so-called belief propagation distributional recursions and its fixed point (also known as Bethe fixed point, cavity equation etc). In this work we prove there is at most one non-trivial fixed point for Ising models with zero and certain random external fields.

As a concrete example, consider a sample A of Ising model on a rooted tree (regular, Galton-Watson, etc). Let B be a noisy version of A obtained by independently perturbing each spin as follows: Bv equals to Av with some small probability δ and otherwise taken to be a uniform +-1 (alternatively, 0). We show that the distribution of the root spin Aρ conditioned on values Bv of all vertices v at a large distance from the root is independent of δ and coincides with δ=0. Previously this was only known for sufficiently low-temperature” models. Our proof consists of constructing a metric under which the BP operator is a contraction (albeit non-multiplicative). I hope to convince you our proof is technically rather simple.

This simultaneously closes the following 5 conjectures in the literature: uselessness of global information for a labeled 2-community stochastic block model, or 2-SBM (Kanade-Mossel-Schramm’2014); optimality of local algorithms for 2-SBM under noisy side information (Mossel-Xu’2015); independence of robust reconstruction accuracy to leaf noise in broadcasting on trees (Mossel-Neeman-Sly’2016); boundary irrelevance in BOT (Abbe-Cornacchia-Gu-P.’2021); characterization of entropy of community labels given the graph in 2-SBM (ibid).

Joint work with Qian Yu (Princeton).

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

I will report on a project which aims to encode the DT invariants of a CY3 triangulated category in a geometric structure on its stability space. I will focus on a class of categories whose stability spaces were studied in previous joint work with Ivan Smith, and which correspond in physics to theories of class S[A1]. I will describe the resulting geometric structures using a kind of complexified Hitchin system parameterising bundles on curves equipped with pencils of flat connections.

Zoom Link: https://cuhk.zoom.us/j/94410733353

Meeting ID: 944 1073 3353
Passcode: 20220223

For details, please visit:

http://www.ims.cuhk.edu.hk/cgi-bin/SeminarAdmin/bin/Web

http://www.ims.cuhk.edu.hk/activities/seminar/joint-dg-seminar/

In this sequence of talks, I will describe our work with Alexander Frenkel and Stephen Randall, in which we presented a novel topological quantum gravity, relating three previously unrelated fields:  Topological quantum field theories (of the cohomological type), the theory of Ricci flows on Riemannian manifolds, and nonrelativistic quantum gravity.  The remarkable richness of results produced in the recent decades by mathematicians studying the Ricci flow promises to shed new light on the physics of the path integral in quantum gravity (at least in the topological regime).  In the opposite direction, the techniques of quantum field theory and path integrals may end up putting some of the mathematical results in the Ricci flow theory in a new perspective as well.

https://harvard.zoom.us/j/977347126
Password: cmsa

In this talk we describe a new method to study the singular set in the obstacle problem. This method does not depend on monotonicity formulae and works for fully nonlinear elliptic operators. The result we get matches the best-known result for the case of Laplacian.

This is based on joint works with Ovidiu Savin from Columbia University.

Zoom ID: 950 2372 5230 (Password: cmsa)

The spontaneous emergence of collective flows is a generic property of active fluids and often leads to chaotic flow patterns characterized by swirls, jets, and topological

disclinations in their orientation field. I will first discuss two examples of these collective features helping us understand biological processes:

1. (i) to explain the tortoise & hare story in bacterial competition: how motility of Pseudomonas aeruginosa bacteria leads to a slower invasion of bacteria colonies, which are individually faster, and
2. (ii) how self-propelled defects lead to finding an unanticipated mechanism for cell death.

I will then discuss various strategies to tame, otherwise chaotic, active flows, showing how hydrodynamic screening of active flows can act as a robust way of controlling and guiding active particles into dynamically ordered coherent structures. I will also explain how combining hydrodynamics with topological constraints can lead to further control of exotic morphologies of active shells.

Three-dimensional (3d) gapped topological phases with fractional excitations are divided into two subclasses: One has topological order with point-like and loop-like excitations fully mobile in the 3d space, and the other has fracton order with point-like excitations constrained in lower-dimensional subspaces. These exotic phases are often studied by exactly solvable Hamiltonians made of commuting projectors, which, however, are not capable of describing those with chiral gapless surface states. Here we introduce a systematic way, based on cellular construction recently proposed for 3d topological phases, to construct another type of exactly solvable models in terms of coupled quantum wires with given inputs of cellular structure, two-dimensional Abelian topological order, and their gapped interfaces. We show that our models can describe both 3d topological and fracton orders and even their hybrid and study their universal properties such as quasiparticle statistics and topological ground-state degeneracy.

https://harvard.zoom.us/j/977347126
Password: cmsa