L6

We continue the study by Melo and Winter [arXiv:1712.01763, 2017] on the possible intersection sizes of a $k$-dimensional subspace with the vertices of the $n$-dimensional hypercube in Euclidean space. Melo and Winter conjectured that all intersection sizes larger than $2^{k-1}$ (the “large” sizes) are of the form $2^{k-1} + 2^i$. We show that this is almost true: the large intersection sizes are either of this form or of the form $35\cdot2^{k-6}$ . We also disprove a second conjecture of Melo and Winter by proving that a positive fraction of the “small” values is missing.

The classical theory of Erdős–Rényi random graphs is concerned primarily with random subgraphs of $K_n$ or $K_{n,n}$. Lately, there has been much interest in understanding random subgraphs of other graph families, such as regular graphs.

We study the following problem: Let $G$ be a $k$-regular bipartite graph with $2n$ vertices. Consider the random process where, beginning with $2n$ isolated vertices, $G$ is reconstructed by adding its edges one by one in a uniformly random order. An early result in the theory of random graphs states that if $G=K_{n,n}$, then with high probability a perfect matching appears at the same moment that the last isolated vertex disappears. We show that if $k = Ω(n)$, then this holds for any $k$-regular bipartite graph $G$. This improves on a result of Goel, Kapralov, and Khanna, who showed that with high probability a perfect matching appears after $O(n \log(n))$ edges have been added to the graph. On the other hand, if $k = o(n / (\log(n) \log (\log(n)))$, we construct a family of $k$-regular bipartite graphs in which isolated vertices disappear long before the appearance of perfect matchings.

Joint work with Roman Glebov and Zur Luria.
 

How long a monotone path can one always find in any edge-ordering of the complete graph $K_n$? This appealing question was first asked by Chvatal and Komlos in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was $n^{2/3−o(1)}$, which was proved by Milans. This talk will be
about nearly closing this gap, proving that any edge-ordering of the complete graph contains a monotone path of length $n^{1−o(1)}$. This is joint work with Bucic, Kwan, Sudakov, Tran, and Wagner.

In this talk I will discuss recent joint work with Dominik Gruber where 
we find a reasonable model for random (infinite) Burnside groups, 
building on earlier tools developed by Coulon and Coulon-Gruber.

The free Burnside group with rank r and exponent n is defined to be the 
quotient of a free group of rank r by the normal subgroup generated by 
all elements of the form g^n; quotients of such groups are called 
Burnside groups.  In 1902, Burnside asked whether any such groups could 
be infinite, but it wasn't until the 1960s that Novikov and Adian showed 
that indeed this was the case for all large enough odd n, with later 
important developments by Ol'shanski, Ivanov, Lysenok and others.

In a different direction, when Gromov developed the theory of hyperbolic 
groups in the 1980s and 90s, he observed that random quotients of free 
groups have interesting properties: depending on exactly how one chooses 
the number and length of relations one can typically gets hyperbolic 
groups, and these groups are infinite as long as not too many relations 
are chosen, and exhibit other interesting behaviour.  But one could 
equally well consider what happens if one takes random quotients of 
other free objects, such as free Burnside groups, and that is what we 
will discuss.
 

We present a construction which associates to a KdV equation the lamplighter group. 
In order to establish this relation we use automata and random walks on ultra discrete limits. 
It is also related to the L2 Betti numbers introduced by Atiyah which are homotopy 
invariants of closed manifolds.

In this talk I will articulate and contextualize the following sequence of results.

The Bruhat decomposition of the general linear group defines a stratification of the orthogonal group.
Matrix multiplication defines an algebra structure on its exit-path category in a certain Morita category of categories.  
In this Morita category, this algebra acts on the category of n-categories -- this action is given by adjoining adjoints to n-categories. 

This result is extracted from a larger program -- entirely joint with John Francis, some parts joint with Nick Rozenblyum -- which proves the cobordism hypothesis.  

The celebrated Lang-Vojta Conjecture predicts degeneracy of S-integral points on varieties of log general type defined over number fields. It admits a geometric analogue over function fields, where stronger results have been obtained applying a method developed by Corvaja and Zannier. In this talk, we present a recent result for non-isotrivial surfaces over function fields dominating a two-dimensional torus. This extends Corvaja and Zannier’s result in the isotrivial case and it is based on a refinement of gcd estimates for polynomials evaluated at S-units. This is a joint work with A. Turchet.

In 1973, Serre observed that the Hecke eigenvalues of Eisenstein series can be p-adically interpolated. In other words, Eisenstein series can be viewed as specializations of a p-adic family parametrized by the weight. The notion of p-adic variations of modular forms was later generalized by Hida to include families of ordinary cuspforms. In 1998, Coleman and Mazur defined the eigencurve, a rigid analytic space classifying much more general p-adic families of Hecke eigenforms parametrized by the weight. The local nature of the eigencurve is well-understood at points corresponding to cuspforms of weight k ≥ 2, while the weight one case is far more intricate.

In this talk, we discuss the geometry of the eigencurve at weight one Eisenstein points. Our approach consists in studying the deformation rings of certain (deceptively simple!) Artin representations. Via this Galois-theoretic method, we obtain the q-expansion of some non-classical overconvergent forms in terms of p-adic logarithms of p-units in certain number field. Finally, we will explain how these calculations suggest a different approach to the Gross-Stark conjecture.

I will give an overview of some recent progress on potential automorphy results over CM fields, that is joint work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne. I will focus on explaining an application to the generalized Ramanujan-Petersson conjecture. 

For any prime p > 3, the strongest lower bounds for the number of imaginary quadratic fields with discriminant down down to -X for which the class group has trivial (non-trivial) p-torsion are due to Kohnen and Ono (Soundararajan). I will discuss recent refinements of these classic results in which we consider the imaginary quadratic fields whose class number is indivisible (divisible) by p such that a given finite set of primes factor in a prescribed way. We prove a lower bound for the number of such fields with discriminant down to -X which is of the same order of magnitude as Kohnen and Ono's (Soundararajan's) results. For the indivisibility case, we rely on a result of Wiles establishing the existence of imaginary quadratic fields with trivial p-torsion in their class groups satisfying almost any given finite set of local conditions, and a result of Zagier which says that the Hurwitz class numbers are the Fourier coefficients of a mock modular form.