## Francis Brown : Periods, Galois theory and particle physics: Galois theory and transcendence

- Gergen Lectures ( 302 Views )Classical Galois theory replaces the study of algebraic numbers with group theory. This idea is extremely powerful, and led to the proof of the insolubility of the general quintic equation. A deep idea, originating in the work of Grothendieck, is that Galois theory should extend to the theory of periods. I will describe a cheap way to set up such a theory and illustrate it in the case of multiple zeta values. It gives rise to a symmetry group which respects the algebraic identities satisfied by these numbers and explains their underlying structure.

## Francis Brown : Periods, Galois theory and particle physics: Applications

- Gergen Lectures ( 286 Views )In the final lecture, I will propose how the Galois theory of periods should lead to a classification of periods by types. When applied to the set of Feynman integrals occurring in particle physics, experiments suggest the emergence of a `cosmic? Galois group of symmetries acting on the constants of high-energy physics.

## Christopher Hacon : Birational geometry in characteristic $p>5$

- Gergen Lectures ( 285 Views )After the recent exciting progress in understanding the geometry of algebraic varieties over the complex numbers, it is natural to try to understand the geometry of varieties over an algebraically closed field of characteristic $p>0$. Many technical issues arise in this context. Nevertheless, there has been much recent progress. In particular, the MMP was established for 3-folds in characteristic $p>5$ by work of Birkar, Hacon, Xu and others. In this talk we will discuss some of the challenges and recent progress in this active area.

## Andrei Zelevinsky : Cluster algebras via quivers with potentials

- Gergen Lectures ( 283 Views )This lecture ties together the strands developed in the first two lectures. We discuss a recent proof (due to H. Derksen, J. Weyman, and the speaker) of a series of conjectures on cluster algebras by means of the machinery of quivers with potentials. An important ingredient of our argument is a categorification of cluster algebras using quiver Grassmannians, a family of projective algebraic varieties that are a far-reaching generalization of ordinary Grassmannians. Generalizing an idea due to P. Caldero, F. Chapoton and B. Keller, we show that the Euler characteristics of these varieties carry crucial information about the structure of cluster algebras.

## Robert V. Kohn : A Variational Perspective on Wrinkling Patterns in Thin Elastic Sheets: What sets the local length scale of tensile wrinkling?

- Gergen Lectures ( 282 Views )The wrinkling of thin elastic sheets is very familiar: our skin
wrinkles, drapes have coarsening folds, and a sheet stretched
over a round surface must wrinkle or fold.

What kind of mathematics is relevant? The stable configurations of a
sheet are local minima of a variational problem with a rather special
structure, involving a nonconvex membrane term (which favors isometry)
and a higher-order bending term (which penalizes curvature). The bending
term is a singular perturbation; its small coefficient is the sheet
thickness squared. The patterns seen in thin sheets arise from energy
minimization -- but not in the same way that minimal surfaces arise
from area minimization. Rather, the analysis of wrinkling is an example
of "energy-driven pattern formation," in which our goal is to understand
the asymptotic character of the minimizers in a suitable limit (as the
nondimensionalized sheet thickness tends to zero).

What kind of understanding is feasible? It has been fruitful to focus
on how the minimum energy scales with sheet thickness, i.e. the "energy
scaling law." This approach entails proving upper bounds and
lower bounds that scale the same way. The upper bounds tend to be
easier, since nature gives us a hint. The lower bounds are more subtle,
since they must be ansatz-free; in many cases, the arguments used to
prove the lower bounds help explain "why" we see particular patterns.
A related but more ambitious goal is to identify the prefactor as well
as the scaling law; Ian Tobasco's striking recent work on geometry-driven
wrinkling has this character.

Lecture 1 will provide an overview of this topic (assuming no background
in elasticity, thin sheets, or the calculus of variations). Lecture 2 will
discuss some examples of tensile wrinkling, where identification of the
energy scaling law is intimately linked to understanding the local
length scale of the wrinkles. Lecture 3 will discuss our emerging
undertanding of geometry-driven wrinkling, where (as Tobasco has
shown) it is the prefactor not the scaling law that explains the
patterns seen experimentally.

## Richard Schoen : Positive scalar curvature and connections with relativity

- Gergen Lectures ( 275 Views )In this series of three lectures we will describe positivity conditions on Riemannian metrics including the classical conditions of positive sectional, Ricci, and scalar curvature. We will discuss open problems and recent progress including our recent proof of the differentiable sphere theorem (joint with Simon Brendle). That proof employs the Ricci flow, so we will spend some time explaining that technique. Finally we will discuss problems related to positive scalar curvature including some high dimensional issues which occur in that theory. If time allows we will describe recent progress on black hole topologies. These lectures, especially the first two, are intended for a general audience.

## Alice Guionnet : The spectrum of non-normal matrices, II: the Brown measure.

- Gergen Lectures ( 268 Views )In this talk, which is a continuation of Wednesday's lecture, we shall describe the natural candidate for the limit of the empirical measure of the eigenvalues of non-normal matrices, the so-called Brown measure. We will give some details about how to prove convergence towards such a limit, but also discuss the instability of such convergence.

## Leo P. Kadanoff : Drips and Jets: Singularities, Topology Changes, and Scaling for Fluid Interfaces

- Gergen Lectures ( 38 Views )We investigate the behavior of the interface between two fluids. We are interested in the singularities which develop when the bridge connecting two pieces of fluid goes to zero thickness. One physical situation is Hele-Shaw flow: two fluids are trapped between parallel glass plates and feel frictional forces. Another situation is a cylindrically symmetrical stream, for example, in a dripping faucet. In a third case, a fluid in an electric or magnetic field which pulls off a piece of fluid ending in a sharp point. The last case has an interface between a light and a heavier fluid being sucked up as in a drinking straw. At a critical value of the sucking, a very thin bridge of the heavy fluid is formed. Analytical, numerical, and experimental methods are used to describe what happens right around the pinchoff of these bridges. Much of the resulting behavior can be understood via scaling arguments.

## Gang Tian : Geometry and Analysis of low-dimensional manifolds

- Gergen Lectures ( 37 Views )In this series of talks, I will focus on geometry and analysis of manifolds of dimension 2, 3 or 4. The first talk is a general introduction of this series. I will start the talk by reviewing some classical theories on Riemann surfaces and their recent variations in geometric analysis. Then we survey some recent progress on 3- and 4-manifolds. I hope that this talk will show some clues how geometric equations can be applied to studying geometry of underlying spaces. In the second talk, I will discuss recent works on the Ricci flow and its application to the geometrization of 3-manifolds, in particular, I will briefly discuss Perelman's work towards the Poincare conjecture. In last talk, I will discuss geometric equations in dimension 4 and how they can be applied to studying geoemtry of underlying 4-spaces. Some recent results will be discussed and some open problems will be given.