# 30th Annual

Geometry Festival

Courant Institute of Mathematical Sciences

May 1-3, 2015

## News

*Update (2015-05-04):* Thanks to everyone who attended and made the 2015 Geometry Festival such a success! Next year's Geometry Festival will be held at Princeton; more infomation to come.

*Update (2015-04-24):* If you would like to attend the dinner,
please register by Wednesday, April 29th at noon. There will be a limited
number of dinner tickets for people who register after Wednesday on a
first-come, first-served basis.

## DATES, TIMES, AND LOCATION

The 30th Annual Geometry Festival will be held at the Courant Institute of Mathematical Sciences from Friday, May 1, to Sunday, May 3, 2015. All lectures will be in Warren Weaver Hall (Directions).

Everyone is invited to the Geometry Festival, and graduate students are especially encouraged to attend.

## REGISTRATION

A list of registered participants can be found here

## Schedule

All talks are in Warren Weaver Hall, room 109.### Friday

- 2:30-4:30Registration -- Warren Weaver, lobby
- 4:30-5:30
- Kahler-Einstein metrics along the smooth continuity method
We will discuss an equivariant version of the Yau-Tian-Donaldson conjecture, strengthening the result of Chen-Donaldson-Sun. This potentially gives new examples of Kahler-Einstein manifolds, and it can also be applied to the existence problem for Kahler-Ricci solitons. It is joint work with Ved Datar.

- 5:30-7:30Reception -- WW, 13th floor

### Saturday

- 8:30-9:30Registration and breakfast -- Warren Weaver, lobby
- 9:30-10:30
- Potential theory for nonlinear PDE's
There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation of the form \(f(D^2u) = 0\). For example, for the complex Monge-Ampère equation, it is the classical pluripotential theory, and for the real Monge-Ampère equation, it is the theory of convex functions. I will explain how these theories are defined in general. Fundamental to the analysis is a new invariant of such equations, called the

*Riesz characteristic*, which governs asymptotic behavior and singular structures. The notions of tangents to subsolutions and densities will be introduced. Results concerning the existence, uniqueness and strong uniqueness of tangents, monotonicity phenomena, the structure of sets of high density points, and the regularity of subsolutions for certain Riesz characteristics, will be presented.I will also discuss solutions to the Dirichlet Problem with Prescribed Asymptotic Singularities in the interior of the domain. In particular this constructs Green's functions and multi-pole Green's functions for these nonlinear equations. It also sharpens the results concerning points of high density.

Interesting examples include the real, complex and quaternionic Hessian equations, the \(p\)-convexity equation, and equations from calibrated geometry. In particular, this establishes a potential theory on calibrated manifolds.

- 11:00-12:00
- Existence of Lefschetz fibrations on Stein/Weinstein domains
I will describe joint work with E. Giroux in which we show that every Weinstein domain admits a Lefschetz fibration over the disk (that is, a singular fibration with Weinstein fibers and Morse singularities). We also prove an analogous result for Stein domains in the complex analytic setting. The main tool used to prove these results is Donaldson's quantitative transversality.

- 12:00-2:00Lunch -- on your own
- 2:00-3:00
- Qualitative and quantitative rectifiability
We will discuss quantitative and qualitative notions of rectifiability and show how they relate. We will compare rectifiability with respect to Hausdorff measure vs. rectifiability of a generic measure and see how the latter is a place where quantitative and qualitative play well together. Among other things, we will mention an example constructed with Garnett and Killip; work with Badger; and ongoing work with Lerman and Jones.

- 3:00-4:30Coffee break -- WW, lobby
- 4:30-5:30
- A Gromov/Thurston rigidity theorem for hyperbolic groups
The celebrated Gromov/Thurston rigidity theorem for closed hyperbolic manifolds can be stated as follows. Let \(M,N\) be closed hyperbolic manifolds of the same dimension \(n\geq 2\) and let \(f:M\to N\) be a map of degree one. If \(\operatorname{vol}(M)=\operatorname{vol}(N)\) then \(M,N\) have isomorphic fundamental groups, (and if \(n\geq 3\) then \(M,N\) are in fact isometric). We extend this result to the case that \(N\) is a \(K(G,1)\)-space for a torsion free hyperbolic group with boundary \(S^{n-1}\) and where volume is replaced by simplicial volume. We also discuss some nontrivial examples.

- 6:00Banquet -- Kimmel Center

### Sunday

- 8:30-9:30Breakfast -- Warren Weaver, lobby
- 9:30-10:30
- Almost minimizers with free boundary
In recent work with G. David, and ongoing work with G. David and M. Engelstein, we study almost minimizer for functionals which yield a free boundary, as in the work of Alt-Caffarelli and Alt-Caffarelli-Friedman. The almost minimizing property can be understood as the defining characteristic of a minimizer in a problem which explicitly takes noise into account. In this talk we will discuss regularity results for these almost minimizers and explore the structure of the corresponding free boundary. A key ingredient in the study of the 2-phase problem is the existence of almost monotone quantities.

- 11:00-12:00
- There are finitely many surgeries in Perelman's Ricci flow
Although the Ricci flow with surgery has been used by Perelman to solve the Poincaré and Geometrization Conjectures, some of its basic properties are still unknown. For example it has been an open question whether the surgeries eventually stop to occur (i.e. whether there are finitely many surgeries) and whether the full geometric decomposition of the underlying manifold is exhibited by the flow as \(t \to \infty\).

In this talk I will show that the number of surgeries is indeed finite and that the curvature is globally bounded by \(C t^{-1}\) for large \(t\). Using this curvature bound it is possible to give a more precise picture of the long-time behavior of the flow.

## ACCOMMODATIONS

We have reserved blocks of rooms at the following area hotels at reduced rates.

### Club Quarters - Wall Street

52 William Street (off Wall Street), New York, NY 10005, tel. (203) 905-2100

This is downtown, and the subway to NYU takes about 20
minutes. We have reserved a block of standard rooms with queen size
beds at the conference rate of $236 (plus tax) per night. (+$18 per
extra person per room). These rooms will be held until **April 1**. When
calling to reserve your room, please say that you are affiliated with
an NYU function, and use the group code NYU501.

### Club Quarters - World Trade Center

140 Washington Street (between Albany and Cedar), New York, NY , tel. (203) 905-2100

This is downtown, and the subway to NYU takes about 20
minutes. We have
reserved a block of standard rooms with queen size beds at the
conference rate of $266 (plus tax) per night. (+$18 per extra person
per room). These rooms will be held until **April 1**. When calling to
reserve your room, please say that you are affiliated with an NYU
function, and use the group code NYU502.

### Club Quarters - Grand Central

128 East 45th Street (corner of Lexington Avenue), New York, NY 10017, tel. (203) 905-2100

This is in midtown, and the subway to NYU takes about 20
minutes. We have
reserved a block of standard rooms with queen size beds at the
conference rate of $286 (plus tax) per night. (+$18 per extra person
per room). These rooms will be held until **April 1**. When calling to
reserve your room, please say that you are affiliated with an NYU
function, and use the group code NYU515.

We also recommend looking for rooms using one of the standard websites, such as booking.com, tripadvisor.com, expedia.com, or kayak.com.

## FINANCIAL SUPPORT

We have funds to partially support participants. Preference will be given to graduate students and postdocs who do not have access to other funding.

If you are requesting financial support,
please fill out the registration form and send an e-mail to `ryoung@cims.nyu.edu`
with your CV, list of publications,
and a statement indicating whether there is any other support available for you.
Graduate students and
postdocs requesting support should also have a letter of reference sent
to `ryoung@cims.nyu.edu`.