# Seminars & Events for 2014-2015

##### Symplectic surfaces vs pseudoholomorphic curves

Although the first appearence of symplectic manifolds was in physics, they have played a fundamental role in pure math for the last 30 years. A big breakthrough came after the work of Mikhail Gromov, who started the study of pseudoholomorphic curves in symplectic manifolds. This make a connection with ideas coming from algebraic geometry.

##### Coloring digraphs with forbidden cycles

Let k and r be integers with k>1 and k\ge r>0. (Please click on seminar title for complete abstract.)

##### TBA - Chaudouard

##### A Morse-Bott approach to the Triangulation Conjecture

Manolescu has recently given a negative answer to the celebrated Triangulation Conjecture. His disproof relies on the construction of a new invariant of rational homology three spheres equipped with a spin structure. This is obtained by studying the Seiberg-Witten equations from the point of view of Conley index theory.

##### Equivariant structures in mirror symmetry

When a variety X is equipped with the action of an algebraic group G, it is natural to study the G-equivariant vector bundles or coherent sheaves on X. When X furthermore has a mirror partner Y, one can ask for the corresponding notion of equivariance in the symplectic geometry of Y.

##### Regularity of semi-calibrateed integral $2$-cycles

Semi-calibrated currents naturally appear when dealing with several geometric questions, some aspects of which require a deep understanding of regularity properties of semi-calibrated currents. We will focus on the case of dimension $2$, where it turns out that semi-calibrated cycles are actually pseudo holomorphic.

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

**Minerva Lecture III: ** 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.

##### Exact post-selection inference

**This is a joint PACM/ORFE Colloquium: **We describe a framework for exact post-selection inference.At the core of our framework is what we call selective inferencewhich allows us to define selective Type I and II errors for hypothesis tests and selective coverage for intervals. Several examples are discussed including the LASSO, forward stepwise, change point detection.

##### Positive cones of higher (co)dimensional numerical cycle classes

##### Special PACM Seminar: Critical points and integral geometry of smooth Gaussian random functions

**Please note special day, time and location. **In this survey talk, we describe some what might be described as the geometric theory of smooth (marginally stationary) Gaussian random functions. Beginning at the local level, the celebrated Kac-Rice formulacan be used to derive accurate approximations to the distribution of the maximum of such random functions.

##### The structure of instability in moduli theory

In many examples of moduli stacks which come equipped with a notion of stable points, one tests stability by considering "iso-trivial one parameter degenerations" of a point in the stack.

##### Extending differential forms and the Lipman-Zariski conjecture

The Lipman-Zariski conjecture states that if the tangent sheaf of a complex variety is locally free then the variety is smooth. In joint work with Patrick Graf we prove that this holds whenever an extension theorem for differential 1-forms holds, in particular if the variety in question has log canonical singularities.

##### Branching laws and period integrals for non-tempered representations

Question about decomposition of representations of groups over local fields to subgroups, such as from SO(n) to SO(n-1), has been of considerable interest in the recent past with impressive works of Waldspurger, Moeglin-Waldspurger, and Raphael Beuzart-Plessis. Corresponding global question for unitary groups has been partly settled by Wei Zhang.

##### Braid Groups and Categorification

Braid groups are fundamental objects in mathematics and categorification is the process of replacing set-theoretic theorems by category-theoretic analogues. We will connect the two by discussing a categorification of the Temperley-Lieb Algebra which results in a braid group action.

##### Burgers equation with random forcing

The Burgers equation is one of the basic nonlinear evolutionary PDEs. The study of ergodic properties of the Burgers equation with random forcing began in 1990's. The natural approach is based on the analysis of optimal paths in the random landscape generated by the random force potential.

##### Stein's conjecture and other fair representation problems

Stein's conjecture states that if an n x n matrix has entries 1...n, where each symbol appears exactly n times, then there exists a generalized diagonal where all but two symbols appear exactly once. I will talk about this conjecture and other settings in which we look for a small structure proportionally representing the bigger structure from which it is taken.

##### On the topological complexity of 2-torsion lens spaces

The topological complexity of a topological space is the minimum number of rules required to specify how to move between any two points of the space. A ``rule'' must satisfy the requirement that the path varies continuously with the choice of end points. We use connective complex K-theory to obtain new lower bounds for the topological complexity of 2-torsion lens spaces.

##### An algebro-geometric theory of vector-valued modular forms of half-integral weight

We give a geometric theory of vector-valued modular forms attached to Weil representations of rank 1 lattices. More specifically, we construct vector bundles over the moduli stack of elliptic curves, whose sections over the complex numbers correspond to vector-valued modular forms attached to rank 1 lattices.

##### Joint Columbia-IAS-Princeton Symplectic Seminar: Symplectic embeddings from concave toric domains into convex ones

**This is a Joint Columbia-IAS-Princeton Symplectic Seminar. **Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. These obstructions are known to be sharp in several interesting cases, for example for symplectic embeddings of one ellipsoid into another.

##### Joint Columbia-IAS-Princeton Symplectic Seminar: Beyond ECH capacities

**This is a Joint Columbia-IAS-Princeton Symplectic Seminar. ** ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. These obstructions are known to be sharp when the domain is a "concave toric domain" and the target is a "convex toric domain” (see previous talk).