Calendar

TITLE: Classical and quantum integrable systems in enumerative geometry

ABSTRACT: For more than a quarter of a century, thanks to the ideas and questions originating in modern high energy physics, there has been a very fruitful interplay between enumerative geometry and integrable system, both classical and quantum. While it impossible to summarize even the most important aspects of this interplay in one talk, I will try to highlight a few logical points with the goal to explain the place and the role of certain more recent developments.

Talk chair: Cumrun Vafa

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

For more information, please visit the event page.

Register here to attend.

We report on a new development in asymptotic Hodge theory, arising from work of Golyshev–Zagier and Bloch–Vlasenko, and connected to the Gamma Conjectures in Fano/LG-model mirror symmetry. The talk will focus exclusively on the Hodge/period-theoretic aspects through two main examples. Given a variation of Hodge structure M on a Zariski open in P^1, the periods of the limiting mixed Hodge structures at the punctures are interesting invariants of M.  More generally, one can try to compute these asymptotic invariants for iterated extensions of M by “Tate objects”, which may arise for example from normal functions associated to algebraic cycles. The main point of the talk will be that (with suitable assumptions on M) these invariants are encoded in an entire function called the motivic Gamma function, which is determined by the Picard-Fuchs operator L underlying M. In particular, when L is hypergeometric, this is easy to compute and we get a closed-form answer (and a limiting motive).  In the non-hypergeometric setting, it yields predictions for special values of normal functions; this part of the story is joint with V. Golyshev and T. Sasaki.

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

Consider a free group and its group von Neumann algebra A. Finding
criteria on the boundedness or complete boundedness of multipliers on the Lp(A) is a major subject of analysis on free groups. A remarkable result of U↵e Haagerup and his co-authors characterizes the completely bounded radial Fourier multipliers on A (i.e., for p = 1). However, the case of finite p 6= 2 is a considerably more delicate matter, as it is for abelian groups. One of very few existing significant results is that on the free Hilbert transform recently proved by Tao Mei and Eric Ricard. In this talk I will present some new work, joint with these authors. A more-detailed abstract can be found in the seminar announcement at https://mathpicture.fas.harvard.edu/seminar.

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

Economics and social science research often require analyzing datasets of sensitive personal information at fine granularity, with models fit to small subsets of the data. Unfortunately, such fine-grained analysis can easily reveal sensitive individual information. We study algorithms for simple linear regression that satisfy differential privacy, a constraint which guarantees that an algorithm’s output reveals little about any individual input data record, even to an attacker with arbitrary side information about the dataset. We consider the design of differentially private algorithms for simple linear regression for small datasets, with tens to hundreds of datapoints, which is a particularly challenging regime for differential privacy. Focusing on a particular application to small-area analysis in economics research, we study the performance of a spectrum of algorithms we adapt to the setting. We identify key factors that affect their performance, showing through a range of experiments that algorithms based on robust estimators (in particular, the Theil-Sen estimator) perform well on the smallest datasets, but that other more standard algorithms do better as the dataset size increases. See https://arxiv.org/abs/2007.05157 for more details.

Joint work with Audra McMillan, Jayshree Sarathy, Adam Smith, and Salil Vadhan.

If time permits, I will chronicle past work on differentially private linear regression, discussing previous works on distributed linear regression and hypothesis testing in the general linear model.

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

I will discuss recent work on computing the top weight cohomology of A_g for g up to 7. We use combinatorial methods coming from the relationship between the top weight cohomology of A_g and the homology of the link of the moduli space of tropical abelian varieties to carry out the computation. This is joint work with Madeline Brandt, Juliette Bruce, Melody Chan, Margarida Melo, and Corey Wolfe.

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

I will discuss the problem of understanding the collapsing behavior of Ricci-flat Kahler metrics on a Calabi-Yau manifold that admits a holomorphic fibration structure, when the Kahler class degenerates to the pullback of a Kahler class from the base. I will present recent work with Hans-Joachim Hein where we obtain a priori estimates of all orders for the Ricci-flat metrics away from the singular fibers, as a corollary of a complete asymptotic expansion.

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

TITLE: Log Calabi-Yau fibrations

ABSTRACT: Fano and Calabi-Yau varieties play a fundamental role in algebraic geometry, differential geometry, arithmetic geometry, mathematical physics, etc. The notion of log Calabi-Yau fibration unifies Fano and Calabi-Yau varieties, their fibrations, as well as their local birational counterparts such as flips and singularities. Such fibrations can be examined from many different perspectives. The purpose of this talk is to introduce the theory of log Calabi-Yau fibrations, to remind some known results, and to state some open problems.

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.

Simplified as it is, the Hubbard model embodies much of the complexity of the `strong correlation problem’ and has established itself as a paradigmatic model in the field. In this talk, I will argue that several key aspects of its physics in two dimensions can now be established beyond doubt, thanks to the development of controlled and accurate computational methods. These methods implement different and complementary points of view on the quantum many-body problem. Along with pushing forward each method, the community has recently embarked into a major effort to combine and critically compare these approaches, and in several instances a consistent picture of the physics has emerged as a result. I will review in this perspective our current understanding of the emergence of a pseudogap in both the weak and strong coupling regimes. I will present recent progress in understanding how the pseudogap phase may evolve into a stripe-dominated regime at low temperature, and briefly address the delicate question of the competition between stripes and superconductivity. I will also emphasize outstanding questions which are still open, such as the possibility of a Fermi surface reconstruction without symmetry breaking. Whenever possible, connections to the physics of cuprate superconductors will be made. If time permits, I may also address the question of Planckian transport and bad metallic transport at high temperature.

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

Hierarchical spin glasses such as the generalised random energy model are known to faithfully model typical energy landscapes in the classical theory of mean-field spin glasses. Their built-in hierarchical structure is known to emerge spontaneously in the spin-glass phase of, e.g., the Sherrington-Kirkpatrick model. In this talk, I will review recent results on the effects of a transversal magnetic field on such hierarchical quantum spin glasses.
In particular, I will present a formula of Parisi-type for their free energy which allows to make predictions about the phase diagram.

Zoom: https://harvard.zoom.us/j/98520388668

We review how historically the problem of string phenomenology lead theoretical physics first to algebraic/diffenretial geometry, and then to computational geometry, and now to data science and AI.
With the concrete playground of the Calabi-Yau landscape, accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades, we show how the latest techniques in machine-learning can help explore problems of physical and mathematical interest, from geometry, to group theory, to combinatorics and number theory.

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

The well-known classical Eichler-Shimura relation for modular curves asserts that the Hecke operator $T_p$ is equal, as an algebraic correspondence over the special fiber, to the sum of Frobenius and Verschebung. Blasius and Rogawski proposed a generalization of this result for general Shimura varieties with good reduction at $p$, and conjectured that the Frobenius satisfies a certain Hecke polynomial. I will talk about a recent proof of this conjecture for Shimura varieties of Hodge type, assuming a technical condition on the unramified sigma-conjugacy classes in the associated Kottwitz set.

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

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

I will argue that the orders that are present in high temperature superconductors naturally arise with the same strength and are better regarded as intertwined rather than competing. I illustrate this concept in the context of the orders that are present in the pair-density-wave state and the phase diagrams that result from this analysis.

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

TITLE: Homotopy spectra and Diophantine equations

ABSTRACT: For a long stretch of time in the history of mathematics, Number Theory and Topology formed vast, but disjoint domains of mathematical knowledge.

Origins of number theory can be traced back to the Babylonian clay tablet Plimpton 322 (about 1800 BC)  that contained a list of integer solutions of the “Diophantine” equation $a^2+b^2=c^2$: archetypal theme of number theory, named after Diophantus of Alexandria (about 250 BC).

Topology was born much later, but arguably, its cousin — modern measure theory, — goes back  to Archimedes, author of Psammites (“Sand Reckoner”), who was approximately a contemporary of Diophantus.

In modern language, Archimedes measures the volume of observable universe by counting the number of small grains of sand necessary to fill this volume. Of course, many qualitative geometric models and quantitative estimates of the relevant distances precede his calculations. Moreover, since the estimated numbers of grains of sands are quite large (about $10^{64}$), Archimedes had to invent and describe a system of notation for large numbers going far outside the possibilities of any of the standard ancient systems.

The construction of the first bridge between number theory and topology  was accomplished only about fifty years ago: it is the theory of spectra in stable homotopy theory.

In particular, it connects $Z$, the initial object in the theory of commutative rings, with the sphere spectrum $S$.

This connection poses the challenge: discover a new information in number theory using the developed independently machinery of homotopy theory.

In this this talk based upon the authors’ (Yu. Manin and M. Marcolli) joint research project, I suggest to apply homotopy spectra to the problem of distribution of rational points upon algebraic manifolds.

Talk chair: Michael Hopkins

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

For more information, please visit the event page.

Register here to attend.