Hyperplane arrangements are a classical meeting point of topology, combinatorics and representation theory. Generalizing to arrangements of linear subspaces of arbitrary codimension, the theory becomes much more complicated. However, a crucial observation is that many natural sequences of arrangements seem to be defined using a finite amount of data.

In this talk I will describe a notion of 'finitely generation' for collections of arrangements, unifying the treatment of known examples. Such collections turn out to exhibit strong forms of stability, both in their combinatorics and in their cohomology representation. This structure makes the appearance of representation stability transparent and opens the door to generalizations

Random matrix theory began with the study, by Wigner in the 1950s, of high-dimensional matrices with i.i.d. entries (up to symmetry).  The empirical law of eigenvalues demonstrates two key phenomena: bulk universality (the limit empirical law of eigenvalues doesn't depend on the laws of the entries) and concentration (the convergence is robust and fast).

Several papers over the last decade (initiated by Bryc, Dembo, and Jiang in 2006) have studied certain special random matrix ensembles with structured correlations between some entries. The limit laws are different from the Wigner i.i.d. case, but each of these models still demonstrates bulk universality and concentration.

In this lecture, I will talk about very recent results of mine and my students on these general phenomena:

Bulk universality holds true whenever there are constant-width independent bands, regardless of the correlations within each band.  (Interestingly, the same is not true for independent rows or columns, where universality fails.)  I will show several examples of such correlated band matrices generalizing earlier known special cases, demonstrating how the empirical law of eigenvalues depends on the structure of the correlations.

At the same time, I will show that concentration is a more general phenomenon, depending not on the the structure of the correlations but only on the sizes of correlated partition blocks. Under  some regularity assumptions, we find that Gaussian concentration occurs in NxN ensembles so long as the correlated blocks have size smaller than N^2/log(N).

The distribution of descents in certain conjugacy classes of S_n have been previously studied, and it is shown that its moments have interesting properties. This paper provides a bijective proof of the symmetry of the descents and major indices of matchings (also known as fixed point free involutions) and uses a generating function approach to prove an asymptotic normality theorem for the number of descents in matchings.

Seidel's Lagrangian Dehn twist exact sequence has been a
cornerstone of the theory of Fukaya categories.  In the last decade,
Huybrechts and Thomas discovered a new autoequivalence in the derived
cateogry of coherent sheaves using the so-called "projective objects", which
are presumably mirrors of Lagrangian projective spaces.   On the other hand,
Seidel's construction of Lagrangian Dehn twists as symplectomorphisms can be
easily generalized to Lagrangian projective spaces.  The induce
auto-equivalence on Fukaya categories are conjectured to be the mirror of
Huybrechts-Thomas's auto-equivalence on B-side.  

This remains open until recently, and I will explain my joint work with
Cheuk-Yu Mak on the solution to this conjecture using the technique of
Lagrangian cobordisms.  Moreover, we will explain a recent progress, again
joint with Cheuk-Yu Mak, on pushing this further to Lagrangian embeddings of
finite quotients of rank-one symmetric spaces, leading to another new class
of auto-equivalences, which are different from the classical spherical
twists only in coefficients of finite characteristics.

Forcing and elementary embeddings are central topics in set theory. Most of what set theorists have focused on are the study of forcing and elementary embeddings over models of ZFC. In this talk, we focus on forcing and elementary embeddings over models of the Axiom of Determinacy (AD). In particular, we focus on answering the following questions: work in V which models AD. Let P be a forcing poset and g ⊆ P be V -generic.

1) Does V [g] model AD?

2) Is there an elementary embedding from V to V [g]?

Regarding question 1, we want to classify what forcings preserve AD. We show that forcings that add Cohen reals, random reals, and many other well-known forcings do not preserve AD. Regarding question 2, an analogous statement to the famous Kunen’s theorem for models of ZFC, can be shown: suppose V = L(X) for some set X and V models AD, then there is no elementary embedding from V to itself. We conjecture that there are no elementary embeddings from V to itself. We present some of the results discussed above. There is still much work to do to completely answer questions 1 and 2. This is an ongoing joint work with D. Ikegami.