Calendar

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

TITLE: Area-minimizing integral currents and their regularity

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

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

For more information, please visit the event page.

Register here to attend.

TITLE: The origins of Langlands’ conjectures

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

Talk chair:  Wilfried Schmid

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

For more information, please visit the event page.

Register here to attend.