# Seminars & Events for 2012-2013

##### V-soliton equations, symplectic reductions and Kahler-Ricci flow

It has been widely recognized by now that in order to understand the geometry of Kahler manifolds under the Ricci flow one needs to understand a geometric-analytic version of Mori's Minimal Model Program.

##### Homological Mirror Symmetry for a Calabi-Yau hypersurface in projective space

We prove homological mirror symmetry for a smooth Calabi-Yau hypersurface in projective space. In the one-dimensional case, this is the elliptic curve, and our result is related to that of Polishchuk-Zaslow; in the two-dimensional case, it is the K3 quartic surface, and our result reproduces that of Seidel; and in the three-dimensional case, it is the quintic three-fold.

##### Long-time analysis of 3 dimensional Ricci flow

**Please note special time for this seminar.** It is still an open problem how Perelman's Ricci flow with surgeries behaves for large times. For example, it is unknown whether surgeries eventually stop to occur and whether the full geometric decomposition of the underlying manifold is exhibited by the flow as $t \to \infty$.

##### Low-regularity local wellposedness of Chern-Simons-Schroedinger

The Chern-Simons-Schroedinger model in two spatial dimensions is a covariant NLS-type problem that is $L^2$ critical. We prove that, with respect to the heat gauge, this problem is locally well-posed for initial data that is small in $H^s$, $s > 0$. This work is joint with Baoping Liu and Daniel Tataru.

##### Inaugural Minerva Lectures II: How to use linear algebraic groups

How to use linear algebraic groups

##### Flows and Decompositions of Games: Harmonic and Potential Games

We introduce a novel flow representation for finite games in strategic form. Based on this representation, we develop a canonical direct sum decomposition of an arbitrary game into three components, which we refer to as the potential, harmonic and nonstrategic components.

##### The fixed point of parabolic renormalization

**Please note special day (Tuesday). **I will present our joint work with O. Lanford on the parabolic renormalization operator acting on the space of simple parabolic analytic germs.

##### Derived categories of coherent sheaves on K3 surfaces

I will survey what is known and what is not known about the derived category of coherent sheaves on K3 surfaces. In particular, I shall explain how to rephrase a conjecture of Bridgeland describing the group of autoequivalences in terms of more conventional moduli spaces and period domains.

##### Inaugural Minerva Lectures III: Counting solutions mod p and letting p tend to infinity

Counting solutions mod p and letting p tend to infinity

##### Belyi's Theorem

Belyi's Theorem provides a surprising and fairly recently-proved connection between the analytic and arithmetic theory of Riemann surfaces/algebraic curves.

##### Directed width parameters: algorithms and structural properties

Treewidth of an undirected graph measures how close the graph is to being a tree. Several problems that are NP-hard on general graphs are solvable in polynomial time on graphs with bounded treewidth. Motivated by the success of treewidth, several directed analogues of treewidth have been introduced to measure the similarity of a directed graph to a directed acyclic graph (DAG).

##### Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices

We show that the variance of centred linear statistics of eigenvalues of GUE matrices remains bounded for large $n$ for some classes of test functions less regular than Lipschitz functions. This observation is suggested by the limiting form of the variance (which has previously been computed explicitly), but it does not seem to appear in the literature.

##### On the parity of coefficients of modular forms.

Recently Nicolas and Serre have determined the structure of the Hecke algebra acting on modular forms of level 1 modulo 2, and Serre has conjectured the existence of a universal Galois representation over

##### A New Spectral Sequence in Khovanov Homology

Khovanov homology is an invariant of links L in S^3which categorifies the Jones polynomial. In this talk, I will describe a new spectral sequence in Khovanov homology, the link forgetful spectral sequence. The spectral sequence starts at Khovanov homology and converges to the Khovanov homology of the disjoint union of the components of the link -- that is, it forgets the linking between components. The discovery of this construction was partly motivated by looking for an analog in Khovanov homology of the component-forgetting spectral sequence in link Floer homology. As an application, building on results of Kronheimer-Mrowka and Hedden-Ni, I will prove that Khovanov homology detects the unlink. Joint work with Josh Batson.

##### Densities in Geometry, Including Isoperimetric Problems and the Poincaré Conjecture

Densities or weights play an important role throughout geometry, including Perelman's proof of the Poincaré Conjecture, Lawlor's new, elegant proof of the Double Bubble Theorem, and the isoperimetric problem.

##### On the symplectic invariance of log Kodaira dimension

Every smooth affine variety has a natural symplectic structure coming from some embedding in complex Euclidean space. This symplectic form is a biholomorphic invariant. An important algebraic invariant of smooth affine varieties is log Kodaira dimension. One can ask, to what extent is this a symplectic invariant? We show some partial symplectic invariance results for smooth affine varieties of dimension less than or equal to 3.

##### The large box limit for 2D NLS

I will report on work in progress, in collaboration with Zaher Hani and Erwan Faou. Starting from 2D NLS set on the torus, we derive, in the appropriate weakly nonlinear regime, a new asymptotic model set in the whole space. The limiting system has very striking properties and is related to weak turbulence questions.

##### Minerva Lectures I - The virtual Haken conjecture: An overview of 3-manifold topology

Waldhausen conjectured in 1968 that every aspherical 3-manifold has a finite-sheeted cover which is Haken (contains an embedded essential surface). Thurston conjectured that hyperbolic 3-manifolds have a

finite-sheeted cover which fibers over the circle. The first lecture will be an overview of 3-manifold topology in order to explain the meaning of Waldhausen's virtual Haken conjecture and Thurston's virtual fibering conjecture, and how they relate to other problems in 3-manifold theory. The second lecture will give some background on geometric group theory, including the topics of hyperbolic groups and CAT(0) cube complexes after Gromov, and explain how the above conjectures may be reduced to a conjecture of Dani Wise in geometric group theory. The third lecture will discuss the proof of Wise's conjecture, that cubulated hyperbolic groups are virtually special. Part of this result is joint work with Daniel Groves and Jason Manning. We will attempt to make these lectures accessible to a general mathematical audience at the level of a colloquium talk.

##### Parabolic Molecules: Curvelets, Shearlets, and Beyond

Anisotropic representation systems such as shearlets and curvelets have had a significant impact on applied mathematics in the last decade.

##### A generic Nakano vanishing theorem

The genenic vanishing theorem of Green-Lazarsfeld says, roughly speaking, that the cohomology of a generic topologically trivial line bundle on a compact Kaehler manifold vanishes below a certain degree that depends only on the Albanese mapping.