# Seminars & Events for 2014-2015

##### Community detection with the non-backtracking operator

Community detection consists in identification of groups of similar items within a population. In the context of online social networks, it is a useful primitive for recommending either contacts or news items to users.

##### Chow rings and modified diagonals

Beauville and Voisin proved that decomposable cycles (intersections of divisors) on a projective K3 surface span a 1-dimensional subspace of the (infinite-dimensional) group of 0-cycles modulo rational equivalence. I will address the following question: what is the rank of the group of decomposable 0-cycles of a smooth projective variety?

##### Two counterexamples arising from infinite sequences of flops

I will explain how infinite sequences of flops give rise to some interesting phenomena: first, an infinite set of smooth projective varieties that have equivalent derived categories but are not isomorphic; second, a pseudoeffective divisor for which the asymptotic multiplicity along a certain subvariety is infinite, in the relative setting.

##### The construction problem for Hodge numbers

What are the possible Hodge numbers of a smooth complex projective variety? We construct enough varieties to show that many of the Hodge numbers can take all possible values satisfying the constraints given by Hodge theory. For example, there are varieties such that a Hodge number hp,0 is big and the intermediate Hodge numbers hi,p−i are small.

##### Riemann Sums and Mobius

[$S =$] square-free natural numbers. An Hilbert-Schmidt operator, [$\mathcal{A}$] , associated to the Möbius function has the property that [$\mathcal{A}: \bigcup_{0<r<\infty} l^r(S) \to \bigcap_{0<r<\infty} l^r(S),$] injectively.

##### Ancient solutions to geometric flows

We will discuss ancient or eternal solutions to geometric parabolic partial differential equations. These are special solutions that appear as blow up limits near a singularity. They often represent models of ingularities.

##### Geometry of asymptotically flat graphical hypersurfaces in Euclidean space

We consider a special class of asymptotically flat manifolds of nonnegative scalar curvature that can be isometrically embedded in Euclidean space as graphical hypersurfaces. In this setting, the scalar curvature equation becomes a fully nonlinear equation with a divergence structure, and we prove that the graph must be weakly mean convex.

##### Counting n particles in the plane

How do algebraic geometers "count" the number of ways of putting n points in the plane? I'll explain what Euler characteristic is, what a Hilbert scheme is, and how to compute the Euler characteristic of the Hilbert scheme of n points in the plane. The answer may surprise you, especially if you don't know what those words mean yet. Everything will be defined from scratch.

##### Generalized Morse-Kaktuani Flows

The Prouhet-Thue-Morse sequence and its generalizations have occurred in many settings. ``Morse-Kakutani flow'' refers to Kakutani's 1967 generalization of the Morse minimal flow (1922). These flows are $\mathbb{Z}_2$ skew products of almost one-to-one extensions of the adding machine ($x \to x+1$ on the $2$-adic completion of $\mathbb{Z}$).

##### Local and global colorability of graphs

It is shown that for any fixed c > 2 and r, the maximum possible chromatic number of a graph on n vertices in which every subgraph of radius at most r is c-colorable is equal to n^{1/(r+1)} up to a factor poly-logarithmic in n.

##### Euler Systems from Special Cycles on Unitary Shimura Varieties and Arithmetic Applications

We construct a new Euler system from a collection of special 1-cycles on certain Shimura 3-folds associated to U(2,1) x U(1,1) and appearing in the context of the Gan--Gross--Prasad conjectures.

##### Heegaard-Floer homology of algebraic links

The intersection of an algebraic plane curve singularity with a small 3-sphere is an algebraic link. I will explain how to compute the Heegaard-Floer homology of this link in terms of the homologies of certain subvarieties in the space of functions on the plane.

##### Superconformal simple type and Witten's conjecture on the relation between Donaldson and Seiberg-Witten invariants

We shall discuss two new results concerning gauge-theoretic invariants of "standard" four-manifolds, namely closed, connected, four-dimensional, orientable, smooth manifolds with $b^1=0$ and $b+\geq 3$ and odd.

##### Nonexistence results for solitons in the mean curvature flow

In this talk I will give a more refined quantitative understanding of some of the important known solitons in the n-dimensional mean curvature flow in R^{n+1}. The global estimates in question follow by iteration of monotonicity formulae - an idea and technique which, while quite elementary in nature, appears to be particularly useful in several of such situations.

##### Remarks on Prandtl boundary layers

In this talk we will review some recent results on the instability of Prandtl boundary layers which arise in the inviscid limit of incompressible Navier Stokes equations near a boundary (joint work with Y. Guo and T. Nguyen).

##### Physics-inspired algorithms and phase transitions in community detection

Detecting communities, and labeling nodes, is a ubiquitous problem in the study of networks. Recently, we developed scalable Belief Propagation algorithms that update probability distributions of node labels until they reach a fixed point.

##### Minerva Lecture I: Logic, Elliptic curves, and Diophantine stability

**Minerva Lecture I: ** An introduction to aspects of *mathematical logic* and *the arithmetic of elliptic curves *that make these branches of mathematics inspiring to each other. Specifically: algebraic curves - other than the projective line - over number fields tend to acquire no new rational points over many extension fields.

##### A rigorous result on many-body localization

A one-dimensional spin chain with random interactions exhibits many-body localization. I will discuss a proof under a physically reasonable assumption that limits the amount of level attraction in the system. The construction uses a sequence of local unitary transformations to diagonalize the Hamiltonian and connect the exact many-body eigenfunctions to the original basis vectors.

##### Extrema of the planar Gaussian Free Field: convergence of the maximum using hidden tree structures

In a recent work, Bramson, Ding and the speaker proved that the maximum of the Gaussian free field in a discrete box of side $N$, centered around its mean, converges in distribution to a shifted Gumbel. The proof uses branching random walks, modified branching random walks, and a modification of the classical second moment method. Underlying the proof is a hidden rough tree structure.

##### Minerva Lecture II: Logic, Elliptic curves, and Diophantine stability

**Minerva Lecture II: ** An introduction to aspects of *mathematical logic* and *the arithmetic of elliptic curves *that make these branches of mathematics inspiring to each other. Specifically: algebraic curves - other than the projective line - over number fields tend to acquire no new rational points over many extension fields.