The purpose of this talk is to present an (apparently) new way to look at the intersection complex of a singular variety over a finite field, or, more generally, at the intermediate extension functor on pure perverse sheaves, and an application of this to the cohomology of noncompact Shimura varieties.

Let $1\le k\le n$ and $M$ be a random $n\times n$ matrix with independent uniformly random $\{\pm 1\}$-entries. We show that there exists an absolute constant $c > 0$ such that $$\mathbf{P}[\operatorname{rank}(M)\le n-k]\le \exp(-c nk).$$

For a finite Coxeter group W and a Coxeter element c of W, the c-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W. Its maximal cones are naturally indexed by the c-sortable elements of W. The main result of this paper is that the known bijection cl_c between c-sortable elements and c-clusters induces a combinatorial isomorphism of fans. In particular, the c-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for W. The rays of the c-Cambrian fan are generated by certain vectors in the W-orbit of the fundamental weights, while the rays of the c-cluster fan are generated by certain roots. For particular ("bipartite") choices of c, we show that the c-Cambrian fan is linearly isomorphic to the c-cluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map cl_c, on c-clusters by the c-Cambrian lattice. We give a simple bijection from c-clusters to c-noncrossing partitions that respects the refined (Narayana) enumeration. We relate the Cambrian fan to well known objects in the theory of cluster algebras, providing a geometric context for g-vectors and quasi-Cartan companions.

The main result is that there are infinitely many; in fact, a continuum; of closed ideals in the Banach algebra $L(L_1)$ of bounded linear operators on $L_1(0,1)$. This answers a question from A. Pietsch's 1978 book "Operator Ideals". The proof also shows that $L(C[0,1])$ contains a continuum of closed ideals. Finally, a duality argument yields that $L(\ell_\infty)$ has a continuum of closed ideals.

Horn's conjecture, which given the spectra of two Hermitian matrices describes the possible spectra of the sum, was recently settled in the affirmative. In this survey we discuss one of the many steps in this, which required us to introduce a combinatorial gadget called a {\em honeycomb}; the question is then reformulable as about the existence of honeycombs with certain boundary conditions. Another important tool is the connection to the representation theory of the group U(n), by ``classical vs. quantum'' analogies.

We note a link between combinatorial results of Bollob\'as and Leader concerning sumsets in the grid, the Brunn-Minkowski theorem and a result of Freiman and Bilu concerning the structure of sets of integers with small doubling. Our main result is the following. If eps > 0 and if A is a finite nonempty subset of a torsion-free abelian group with |A + A| <= K|A|, then A may be covered by exp(K^C) progressions of dimension [log_2 K + eps] and size at most |A|.

These are notes of lectures given at the CMI conference in August, 2014 at ICMAT in Madrid. The focus is on some mathematical questions associated to Feynman amplitudes, including Hodge structures, relations with string theory, and monodromy (Cutkosky rules).

We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A-infinity structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given a pair of smooth manifolds of the same dimension with embeddings of a submanifold B with isomorphic normal bundles, we construct a differential graded category from the singular cochains of these spaces, and prove that it is equivalent to the A-infinity category obtained by considering exact Lagrangian embeddings intersecting cleanly along B.

Let $S_0$ and $S_1$ be two homotopic, oriented 2-spheres embedded in an orientable 4-manifold $X$. After discussing several operations for modifying an immersion of a 3-manifold into a 5-manifold, we discuss the Freedman--Quinn (fq) and Stong (stong) concordance obstructions. When these are defined for the pair $S_0,S_1$, they are defined in terms of the self-intersection set of a regular homotopy from $S_0$ to $S_1$. When $S_0$ has an immersed dual sphere, we see that under some mild topological conditions on $X$, the invariants fq and stong are a complete set of concordance obstructions. This work is an adaption of the methods of Richard Stong to the context of concordances of 2-spheres.

Let $(X,d)$ be a compact metric space and $\mu$ a Borel probability on $X$. For each $N\geq 1$ let $d^N_\infty$ be the $\ell_\infty$-product on $X^N$ of copies of $d$, and consider $1$-Lipschitz functions $X^N\to\mathbb{R}$ for $d^N_\infty$. If the support of $\mu$ is connected and locally connected, then all such functions are close in probability to juntas: that is, functions that depend on only a few coordinates of $X^N$. This describes the failure of measure concentration for these product spaces, and can be seen as a Lipschitz-function counterpart of the celebrated result of Friedgut that Boolean functions with small influences are close to juntas.

This paper completes the project of constructing combinatorial models (called frameworks) for the exchange graph and g-vector fan associated to any exchange matrix B whose Cartan companion is of finite or affine type, using the combinatorics and geometry of Coxeter-sortable elements and Cambrian lattices/fans. Specifically, we construct a framework in the unique non-acyclic affine case, the cyclically oriented n-cycle. In the acyclic affine case, a framework was constructed by combining a copy of the Cambrian fan for B with an antipodal copy of the Cambrian fan for -B. In this paper, we extend this "doubled Cambrian fan" construction to the oriented n-cycle, using a more general notion of sortable elements for quivers with cycles.

In this paper we show that a Brownian Gibbsian line ensemble whose top curve approximates a parabola must be given by the parabolic Airy line ensemble. More specifically, we prove that if $\boldsymbol{\mathcal{L}} = (\mathcal{L}_1, \mathcal{L}_2, \ldots )$ is a line ensemble satisfying the Brownian Gibbs property, such that for any $\varepsilon > 0$ there exists a constant $\mathfrak{K} (\varepsilon) > 0$ with $$\mathbb{P} \Big[ \big| \mathcal{L}_1 (t) + 2^{-1/2} t^2 \big| \le \varepsilon t^2 + \mathfrak{K} (\varepsilon) \Big] \ge 1 - \varepsilon, \qquad \text{for all $t \in \mathbb{R}$},$$ then $\boldsymbol{\mathcal{L}}$ is the parabolic Airy line ensemble, up to an independent affine shift. Specializing this result to the case when $\boldsymbol{\mathcal{L}} (t) + 2^{-1/2} t^2$ is translation-invariant confirms a prediction of Okounkov and Sheffield from 2006 and Corwin-Hammond from 2014.

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For affine type A, we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids.

We show that every subset of SL_2(Z/pZ) grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators A of SL_2(Z/pZ), every element of SL_2(Z/pZ) can be expressed as a product of at most O((log p)^c) elements of the union of A and A^{-1}, where c and the implied constant are absolute.

We define a notion of a measured length space X having nonnegative N-Ricci curvature, for N finite, or having infinity-Ricci curvature bounded below by K, for K a real number. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein space P_2(X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdorff limits. We give geometric and analytic consequences.

It is known that some matrix integrals over U(n) satisfy an sl(2,R)-algebra of Virasoro constraints. Acting with these Virasoro generators on 2-dimensional Schur function expansions leads to difference relations on the coefficients of this expansions. These difference relations, set equal to zero, are precisely the backward and forward equations for non-intersecting random walks. The transition probabilities for these random walks appear as the coefficients of an expansion of U(n)-matrix integrals (of the type above), by inserting in the integral the product of two Schur polynomials associated with two partitions; the latter are specified by the initial and final positions of the non-intersecting random walk. An essential ingredient in this work is the generalization of the Murnaghan-Nakayama rule to the action of Virasoro on Schur polynomials.

In this paper we construct an analog of Steenrod operations in motivic cohomology and prove their basic properties including the Cartan formula, the Adem relations and the realtions to characteristic classes.

This paper contains two theorems concerning the theory of maximal Cohen--Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen--Macaulay modules $M$ and $N$ must have finite length, provided only finitely many isomorphism classes of maximal Cohen--Macaulay modules exist having ranks up to the sum of the ranks of $M$ and $N$. This has several corollaries. In particular it proves that a Cohen--Macaulay local ring of finite Cohen--Macaulay type has an isolated singularity. A well-known theorem of Auslander gives the same conclusion but requires that the ring be Henselian. Other corollaries of our result include statements concerning when a ring is Gorenstein or a complete intersection on the punctured spectrum, and the recent theorem of Leuschke and Wiegand that the completion of an excellent Cohen--Macaulay local ring of finite Cohen--Macaulay type is again of finite Cohen--Macaulay type. The second theorem proves that a complete local Gorenstein domain of positive characteristic $p$ and dimension $d$ is $F$-rational if and only if the number of copies of $R$ splitting out of $R^{1/p^e}$ divided by $p^{de}$ has a positive limit. This result generalizes work of Smith and Van den Bergh. We call this limit the $F$-signature of the ring and give some of its properties.

We obtain global well-posedness, scattering, and global $L^{10}_{t,x}$ spacetime bounds for energy-class solutions to the quintic defocusing Schr\"odinger equation in $\R^{1+3}$, which is energy-critical. In particular, this establishes global existence of classical solutions. Our work extends the results of Bourgain and Grillakis, which handled the radial case. The method is similar in spirit to the induction-on-energy strategy of Bourgain, but we perform the induction analysis in both frequency space and physical space simultaneously, and replace the Morawetz inequality by an interaction variant. The principal advantage of the interaction Morawetz estimate is that it is not localized to the spatial origin and so is better able to handle nonradial solutions. In particular, this interaction estimate, together with an almost-conservation argument controlling the movement of $L^2$ mass in frequency space, rules out the possibility of energy concentration.

In this note we prove two main results. 1. In a rigid braided finite tensor category over C (not necessarily semisimple), some power of the Casimir element and some even power of the braiding is unipotent. 2. In a (semisimple) modular category, the twists are roots of unity dividing the algebraic integer D^{5/2}, where D is the global dimension of the category (the sum of squares of dimensions of simple objects). Both results generalize Vafa's theorem, saying that in a modular category twists are roots of unity, and square of the braiding has finite order. We also discuss the notion of the quasi-exponent of a finite rigid tensor category, which is motivated by results 1 and 2 and the paper math/0109196 of this http URL and the author.