## 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.

## 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.

## Jordan S. Ellenberg : Stability and Arithmetic Counting Problems

- Gergen Lectures ( 281 Views )A big theme in contemporary number theory is "arithmetic statistics": what does the class group of a random number field look like? What do the zeroes of a random L-function look like? What does a random rational point on a variety look iike? In this talk we will explain how arithmetic statistics problems over function fields are naturally tied to topological questions about stability for homology groups of certain moduli spaces; in particular, we will explain how a stability theorem for Hurwitz spaces (moduli spaces of finite branched covers of the line) can be used to prove a version of the Cohen-Lenstra conjectures over function fields. There will be some overlap with a talk I gave at Duke in December 2009, but many things which were speculations then are theorems now.

## Luis Caffarelli : Degenerate ellipticity and the porous media equation

- Gergen Lectures ( 273 Views )In the first lecture I will give a brief discussion of local and non local diffusion and degenerate ellipticity and different local and non local models for compressible flows in porous media.

In the second and third lectures I will discuss some properties of the
(infinitesimal) porous media equation, a non local in space model and
equations with memory.

## Gerhard Huisken : Parabolic Evolution Equations for the Deformation of Hypersurfaces

- Gergen Lectures ( 37 Views )A smooth one-parameter family *F _{0} : M^{n}x [0,T) ---> (N^{n+1},g)* of hypersurfaces in a Riemannian manifold

*(N*is said to move by its curvature if it satisfies an evolution equation of the form

^{(n+1)},g)*(d/dt) F(p,t) = f(p,t) p M*

such that at each point of the surface its speed in normal direction is a function $f$ of the extrinsic curvature of the hypersurface. Examples such as the flow by mean curvature, flow by Gauss curvature or flow by inverse mean curvature arise naturally both in Differential Geometry, where they exhibit fascinating interactions between the extrinsic curvature of the surfaces and intrinsic geometric properties of the ambient manifold, and in Mathematical Physics, where they serve as models for the evolution of interfaces in phase transitions. The first lecture gives a general introduction to the main examples and phenomena and highlights some recent results. The second lecture shows how parabolic rescaling techniques can be combined with a priori estimates to study and in some cases classify possible singularities of the mean curvature flow. The series concludes with applications of hypersurface families in General relativity, including a recent proof of an optimal lower bound for the total energy of an isolated gravitating system by Huisken and Ilmanen.

^{n}, t [0,T),## 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.

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

- Gergen Lectures ( 35 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.