Stanford

We show the primes have level of distribution 66/107 using triply well-factorable weights. This gives the highest level of distribution for primes in any setting, improving on the prior record level 3/5 of Maynard. We also extend this level to 5/8, assuming Selberg's eigenvalue conjecture. As a result, we obtain new upper bounds for twin primes and for Goldbach representations of even numbers $a$. For the Goldbach problem, this is the first use of a level of distribution beyond the 'square-root barrier', and leads to the greatest improvement on the problem since Bombieri--Davenport from 1966.

This event will take place in L1 and on Teams. A link will be available 30 minutes before the session begins. 

It's hard to believe: I've spent nearly 40 years in STEM. In that time, much changed: we changed from typewriters to PCs, from low performance to high  performance computing, from data-supported research to data-driven research, from traditional languages such as Fortran to a plethora of programming environments. And the rate of change seems to increase constantly. Some things have stayed more or less the same, such as the (lack of) diversity of the STEM community, the level of stress and the struggles we all experience (and the joys!). In this talk, I will reflect on those years, on lessons learned and not learned or unlearned, on things I wish I understood 40 years ago, and on things I still don't understand.

Margot is a professor at Stanford University in the Department of Energy Resources Engineering (ERE) and the Institute of Computational & Mathematical Engineering (ICME). Margot was born and raised in the Netherlands. Her STEM education started in 1982. In 1990 she received a MSc in applied mathematics at Delft University and then left her home country to search for sunnier and hillier places. She moved to Colorado and a year later to California to join the PhD program in Scientific Computing and Computational Mathematics at Stanford. During her PhD, Margot spent several quarters at Oxford University (with very good memories). Before returning to Stanford as faculty member in ERE, Margot spent 5 years as lecturer at the University of Auckland, New Zealand. From 2010-2018, Margot was the director of ICME. During this directorship, she founded the Women in Data Science initiative, which is now a global organization in over 70 countries. From 2015-2020, Margot was also the Senior Associate Dean of Educational Affairs at Stanford's school of Earth, Energy & Environmental Sciences. Currently, Margot still co-directs WiDS and is the Chair of the Board of SIAM. She has since moved back to the mountains (still sunny too) and now lives in Bend, Oregon.

Let us fix a prime $p$. The Erdős-Ginzburg-Ziv problem asks for the minimum integer $s$ such that any collection of $s$ points in the lattice $\mathbb{Z}^n$ contains $p$ points whose centroid is also a lattice point in $\mathbb{Z}^n$. For large $n$, this is essentially equivalent to asking for the maximum size of a subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero.

In this talk, we discuss a new upper bound for this problem for any fixed prime $p\geq 5$ and large $n$. Our proof uses the so-called multi-colored sum-free theorem which is a consequence of the Croot-Lev-Pach polynomial method, as well as some new combinatorial ideas.

Amplitudes in planar N=4 SYM are dual to light-like polygonal Wilson-loop expectation values. In many cases their perturbative expansion can be expressed in terms of multiple polylogarithms which also obey certain single-valuedness conditions or branch cut restrictions. The rigidity of this function space, together with a few other conditions, allows one to construct the six-point amplitude -- or hexagonal Wilson loop -- through at least five loops, and the seven-point amplitude through 3.5 loops. Then one can "fold" the polygonal Wilson loops into multiple degenerate configurations, expressing the limiting behavior in terms of simpler function spaces, and learning in the process about how amplitudes factorize.
 

Given some class of "geometric spaces", we can make a ring as follows. Additive structure: when U is an open subset a space X,  [X] = [U] + [X - U]. Multiplicative structure:  [X][Y] = [XxY]. In the algebraic setting, this ring (the "Grothendieck ring of varieties") contains surprising structure, connecting geometry to arithmetic and topology.  I will discuss some remarkable
statements about this ring (both known and conjectural), and present new statements (again, both known and conjectural).  A motivating example will be polynomials in one variable. This is joint work with Melanie Matchett Wood.

We first present the localized virtual cycles by cosections of obstruction sheaves constructed by Kiem and Li. This construction has two kinds of applications: one is define invariants for non-proper moduli spaces; the other is to reduce the obstruction classes. We will present two recent applications of this construction: one is the Gromov-Witten invariants of stable maps with fields (joint work with Chang); the other is studying Donaldson-Thomas invariants of Calabi-Yau threefolds (joint work with Kiem).

We first present the localized virtual cycles by cosections of obstruction sheaves constructed by Kiem and Li. This construction has two kinds of applications: one is define invariants for non-proper moduli spaces; the other is to reduce the obstruction classes. We will present two recent applications of this construction: one is the Gromov-Witten invariants of stable maps with fields (joint work with Chang); the other is studying Donaldson-Thomas invariants of Calabi-Yau threefolds (joint work with Kiem).

String theory suggests the simultaneous presence of many ultralight axions possibly populating each decade of mass down to the Hubble scale 10^-33eV. Conversely the presence of such a plenitude of axions (an "axiverse") would be evidence for string theory, since it arises due to the topological complexity of the extra-dimensional manifold and is ad hoc in a theory with just the four familiar dimensions. We investigate how upcoming astrophysical experiments will explore the existence of such axions over a vast mass range from 10^-33eV to 10^-10eV. Axions with masses between 10^-33eV to 10^-28eV cause a rotation of the CMB polarization that is constant throughout the sky. The predicted rotation angle is of order \alpha~1/137. Axions in the mass range 10^-28eV to 10^-18eV give rise to multiple steps in the matter power spectrum, that will be probed by upcoming galaxy surveys and 21 cm line tomography. Axions in the mass range 10^-22eV to 10^-10eV affect the dynamics and gravitational wave emission of rapidly rotating astrophysical black holes through the Penrose superradiance process. When the axion Compton wavelength is of order of the black hole size, the axions develop "superradiant" atomic bound states around the black hole "nucleus". Their occupation number grows exponentially by extracting rotational energy from the ergosphere, culminating in a rotating Bose-Einstein axion condensate emitting gravitational waves. This mechanism creates mass gaps in the spectrum of rapidly rotating black holes that diagnose the presence of axions. The rapidly rotating black hole in the X-ray binary LMC X-1 implies an upper limit on the decay constant of the QCD axion f_a