Calendar

I will describe an analogue of Saito’s theory of primitive forms for Calabi-Yau A-infinity categories. Under some conditions on the Hochschild cohomology of the category, this construction recovers the (genus zero) Gromov-Witten invariants of a symplectic manifold from its Fukaya category. This includes many compact toric manifolds, in particular projective spaces.

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

This talk aims to summarize a project I was involved in during the past 5 years, which links together many areas in math, CS and physics. I hope to explain our motivations and goals, summarize our understanding so far, as well as challenges and open problems. I plan to describe, through examples, many of the concepts they refer to, and the evolution of ideas leading to what we know. More details can be found at mathpicture.fas.harvard.edu/seminar. No special background is assumed.

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

We consider a monopolist who wishes to sell n goods to a single additive buyer, where the buyer’s valuations for the goods are drawn according to independent distributions. It is well known that—unlike in the single item case—the revenue-optimal auction (a pricing scheme) may be complex, sometimes requiring a continuum of menu entries, that is, offering the buyer a choice between a continuum of lottery tickets. It is also known that simple auctions with a finite bounded number of menu entries (lotteries for the buyer to choose from) can extract a constant fraction of the optimal revenue, as well as that for the case of bounded distributions it is possible to extract an arbitrarily high fraction of the optimal revenue via a finite bounded menu size. Nonetheless, the question of the possibility of extracting an arbitrarily high fraction of the optimal revenue via a finite menu size, when the valuation distributions possibly have unbounded support (or via a finite bounded menu size when the support of the distributions is bounded by an unknown bound), remained open since the seminal paper of Hart and Nisan (2013), and so has the question of any lower-bound on the menu-size that suffices for extracting an arbitrarily high fraction of the optimal revenue when selling a fixed number of goods, even for two goods and even for i.i.d. bounded distributions.

In this talk, we resolve both of these questions. We first show that for every n and for every ε>0, there exists a menu-size bound C=C(n,ε) such that auctions of menu size at most C suffice for extracting a (1-ε) fraction of the optimal revenue from any valuation distributions, and give an upper bound on C(n,ε), even when the valuation distributions are unbounded and nonidentical. We then proceed to giving two lower bounds, which hold even for bounded i.i.d. distributions: one on the dependence on n when ε=1/n and n grows large, and the other on the dependence on ε when n is fixed and ε grows small. Finally, we apply these upper and lower bounds to derive results regarding the deterministic communication complexity required to run an auction that achieves such an approximation.

Based upon:
* The Menu-Size Complexity of Revenue Approximation, Moshe Babaioff, Y. A. G., and Noam Nisan, STOC 2017
* Bounding the Menu-Size of Approximately Optimal Auctions via Optimal-Transport Duality, Y. A. G., STOC 2018

Yannai Gonczarowski is a postdoctoral researcher at Microsoft Research New England. His main research interests lie in the interface between the theory of computation, economic theory, and game theory—an area commonly referred to as Algorithmic Game Theory. In particular, Yannai is interested in various aspects of complexity in mechanism design (where mechanisms are defined broadly from auctions to matching markets), including the interface between mechanism design and machine learning. Yannai received his PhD from the Departments of Math and CS, and the Center for the Study of Rationality, at the Hebrew University of Jerusalem, where he was advised by Sergiu Hart and Noam Nisan, as an Adams Fellow of the Israel Academy of Sciences and Humanities. Throughout most of his PhD studies, he was also a long-term research intern at Microsoft Research in Herzliya. He holds an M.Sc. in Math (summa cum laude) and a B.Sc. in Math and CS (summa cum laude, Valedictorian) from the Hebrew University. Yannai is also a professionally-trained opera singer, having acquired a bachelor’s degree and a master’s degree in Classical Singing at the Jerusalem Academy of Music and Dance. For his doctoral dissertation, Yannai was awarded the Hans Wiener Prize of the Hebrew University of Jerusalem, the 2018 Michael B. Maschler Prize of the Israeli Chapter of the Game Theory Society, and the ACM SIGecom Doctoral Dissertation Award for 2018. Yannai is also the recipient of the ACM SIGecom Award for Best Presentation by a Student or Postdoctoral Researcher at EC’18, and of the Best Paper Award at MATCH-UP’19. His first textbook, “Mathematical Logic through Python” (Gonczarowski and Nisan), which introduces a new approach to teaching the material of a basic Logic course to Computer Science students, tailored to the unique intuitions and strengths of this cohort of students, is forthcoming in Cambridge University Press.
Zoom: https://harvard.zoom.us/j/98684244514

I’ll review the basic construction of Randall-Sundrum
braneworlds and some of their applications to formal problems in quantum
field theory. I will highlight some recent results regarding scenarios
with mismatched brane tensions. In the last part of the talk, I’ll
review how RS branes have led to exciting new results regarding
evaporation of black holes and will put emphasis on the interesting role
the graviton mass plays in these discussions.

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

Graph-structured data is ubiquitous throughout the natural and social sciences, from telecommunication networks to quantum chemistry. Building relational inductive biases into deep learning architectures is crucial if we want systems that can learn, reason, and generalize from this kind of data. Recent years have seen a surge in research on graph representation learning, most prominently in the development of graph neural networks (GNNs). Advances in GNNs have led to state-of-the-art results in numerous domains, including chemical synthesis, 3D-vision, recommender systems, question answering, and social network analysis. In the first part of this talk I will provide an overview and summary of recent progress in this fast-growing area, highlighting foundational methods and theoretical motivations. In the second part of this talk I will discuss fundamental limitations of the current GNN paradigm and propose open challenges for the theoretical advancement of the field.

Zoom: https://harvard.zoom.us/j/91458092166?pwd=RnFwelNhakw1b3B6dy9UMkt6T2xoQT09

The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of p-adic groups.

I will provide an overview of our understanding of the representations of p-adic groups, with an emphasis on recent progress.

I will also outline how new results about the representation theory of p-adic groups can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p, which is joint work with Sug Woo Shin. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group.

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

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

We consider the problem of finding 3 positive integers (a,b,c) such that a/(b+c)+b/(c+a)+c/(a+b)=4. We will examine how the problem became a viral post on Facebook, solve the problem using elliptic curves, and examine some modern developments when the number 4 is replaced by N.

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

Quantum entanglement has played a key role in studying emergent phenomena in strongly-correlated many-body systems.  Remarkably, The entanglement properties of the ground state encodes information on the nature of excitations.  Here we introduce two new entanglement measures $g(A:B)$ and $h(A:B)$ which characterizes certain tripartite entanglement between $A$, $B$, and the environment.  The measures are based off of the entanglement of purification and the reflected entropy popular among holography.  For 1D states, the two measures are UV insensitive and yield universal quantities for symmetry-broken, symmetry preserved, and critical phases.  We conclude with a few remarks regarding applications to 2D phases.

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