## Stratifications, Differentiable Structures, and Poisson Geometry Seminar

### 2015.09.01

Speaker: Markus Pflaum (CU Boulder)

Title: Introduction to Poisson Geometry and Deformation Quantization

### 2015.09.08

Speaker: Matias del Hoyo (IMPA)

Title: Lie groupoids, linearization and metrics

Abstract: The Linearization Theorem for Lie groupoids provides an organizing framework for classic results on the geometry of fibrations, actions and foliations. It was conjectured by A. Weinstein and solved by N. Zung using analytical methods. In a joint work with R. Fernandes, building on previous works by M. Pflaum et al, we abord the linearization problem from a new perspective, developing a notion of metrics on Lie groupoids, obtaining a simpler proof and a stronger result. I will discuss the linearization problem for groupoids, our solution by metrics, and if time permits, future lines of research.

### 2015.09.15

Speaker: Markus Pflaum (CU Boulder)

Title: Introduction to Poisson Geometry and Deformation Quantization (Part II)

### 2015.09.22

Speaker: Markus Pflaum (CU Boulder)

Title: Foundations of Deformation Quantization (Continued)

### 2015.09.22

Speaker: Alexander Nita (CU Boulder)

Title: Fedosov's Deformation Quantization (Part I)

### 2015.09.29

Speaker: Alexander Nita(CU Boulder)

Title: Fedosov's Deformation Quantization (Part II)

### 2015.10.06

Speaker: Alexander Nita (CU Boulder)

Title: Fedosov's Deformation Quantization (Part III)

### 2015.10.13

Speaker: Jonathan Wise (CU Boulder)

Title: Differentiation and linear approximation

Abstract: We'll revisit the idea of linear approximation in a way that allows us to talk about first-order approximations of quite general mathematical objects. We'll see in a few examples how first-order deformations can be classified using cohomology.

### 2015.10.20

Speaker: Jonathan Wise (CU Boulder)

Title: Higher order derivatives and representability

Abstract: We'll study the general problem of classifying all deformations (of arbitrary orders) of a mathematical object by a commutative ring (and maybe also by a differential graded Lie algebra), culminating in the representability theorems of Artin and Schlessinger.

### 2015.10.27

Speaker: Jonathan Wise (CU Boulder)

Title: Higher order deformations

Abstract: (Part II of previous talk.)

### 2015.11.03

Speaker: Jordan Watts (CU Boulder)

Title: Poisson Reduction

Abstract: Given a mechanical system, a technique often used to simplify calculations is to reduce the number of degrees of freedom of the system by removing symmetry, thereby creating a new system with lower dimension. More geometrically, given a Lie group G and a Poisson manifold M on which G acts properly and via Poisson diffeomorphisms, the orbit space M/G inherits a natural Poisson structure.

We will prove this, and also see how the natural stratification of M by orbit-type descends to the natural stratification of M/G, but as a Poisson stratification; that is, the strata are Poisson submanifolds. This is not immediately obvious, since the strata upstairs on M generally are not Poisson submanifolds.

### 2015.11.10

Speaker: Jordan Watts (CU Boulder)

Title: Symplectic Reduction

Abstract: We will continue last week's discussion, beginning by developing the language of vector fields on subcartesian spaces. Using this language, we will outline the proof that the orbit-type stratification on the orbit space coming from a Lie group action that is proper and Poisson is in fact a Poisson stratification; that is, the strata are Poisson submanifolds.

We then specialise to the case of a Hamiltonian group action, and define symplectic reduction. Here, the symplectic quotient is a quotient space of a level set of the momentum map. We will show that in this case, the orbit-type stratification on the symplectic quotient is a symplectic stratification; that is, the strata are symplectic submanifolds.

### 2015.11.17

Speaker: Jordan Watts (CU Boulder)

Title: The de Rham Complex on Orbit Spaces and Symplectic Quotients

Abstract: This week, we look at a different problem on orbit spaces of proper Lie group actions, as well as symplectic quotients. We first recall two facts:

- If a proper action is free, the de Rham complex on the orbit space matches the de Rham complex of basic forms upstairs on the original manifold.
- It is a result of Koszul and Palais that even in the non-free case, the singular cohomology of the orbit space matches the cohomology of the basic forms upstairs.

Question: is there a de Rham complex of differential forms on the orbit space in the non-free case that yields a cohomology matching the singular cohomology? (The orbit space is not a manifold, typically.) The answer is yes, but we will not use the language of Sikorski differential spaces to answer this question (for which the answer seems to be false). Instead, we will introduce a "dual" smooth structure called diffeology, give an idea of the proof, and then consider a similar question for the symplectic quotient.

### 2016.02.03

Speaker: Lauren Grimley (Texas A&M)

Title: Hoschild cohomology of group extensions of quantum complete intersections

Abstract: Hochschild cohomology of an associative algebra has a graded ring structure and a compatible graded Lie algebra structure, making it a Gerstenhaber algebra. The structures on Hochschild cohomology reveal information about the algebra and its deformations. One class of algebras with interesting Hochschild cohomology is quantum complete intersections which are a non-commutative analog of truncated
polynomial rings. The behavior of Hochschild cohomology of this class of algebras has been shown to vary greatly based on the choice of
quantum coefficients, exhibiting behaviors unseen with commutative algebras. In this talk, we explore the behavior of Hochschild cohomology of group extensions of quantum complete intersections.

### 2016.02.03

Speaker: Timo de Wolff (Texas A&M)

Title: New certificates for nonnegativity via circuit polynomials and geometric programming

Abstract: Deciding nonnegativity of real polynomials is a key question in real algebraic geometry with crucial importance in polynomial optimization. Since this problem is NP-hard, one is interested in finding sufficient conditions (certificates) for nonnegativity, which are easier to check. Since the 19th century the standard certificates are sums of squares (SOS); see particularly Hilbert's 17th problem.

In this talk we introduce *polynomials supported on circuits*. For this class, nonnegativity is characterized by an invariant, which can be derived immediately from the initial polynomial. In consequence, we obtain an *entirely new class* of nonnegativity certificates, which are *independent* of SOS certificates.

Our certificates crucially extend geometric programming approaches for the computation of lower bounds in polynomial optimization. Particularly, for polynomials with simplex Newton polytope our approach is significantly faster and often yields better than bounds than semidefinite programming, which is the standard method for polynomial optimization.

These results generalize earlier works by Fidalgo, Ghasemi, Kovacec, Marshall, and Reznick. The talk is based on joint work with Sadik Iliman.

### 2016.02.09

Speaker: Markus Pflaum (CU Boulder)

Title: Introduction to stratified spaces (part I)

### 2016.02.23

Speaker: Markus Pflaum (CU Boulder)

Title: Introduction to stratified spaces (part II)

### 2016.03.01

Speaker: Jordan Watts (CU Boulder)

Title: Tame circle actions

Abstract: A famous question of Dusa McDuff, often referred to as the "McDuff Conjecture", is whether there exists a non-Hamiltonian symplectic circle action with isolated fixed points on a compact symplectic
manifold. Susan Tolman recently answered this question in the affirmative, constructing a 6-dimensional such space with exactly
32 fixed points. A crucial ingredient to this construction involves Hamiltonian circle actions on complex manifolds and orbifolds in which
the interaction between the complex structure and the symplectic form is fairly weak. Specifically, versions of Sjamaar's holomorphic slice
theorem, the birational equivalence theorem of Guillemin and Sternberg, as well as reduction, cutting, and blow-up (all of which work in the
Kaehler world) are required in this weaker setting.

All of these theorems and constructions are extended to this weaker setting in joint work by Tolman and myself. In particular, by weak, we mean that if xi is the vector field induced by the circle action, omega the symplectic form, and J the complex structure, then plugging in xi and J(xi) into omega results in a positive number on the complement of the fixed point set. This condition is sufficient for all of the theorems and constructions above except for the blow-up (which also requires tameness at the point to be blown-up).

In this talk, I will focus on the holomorphic slice theorem about a fixed point, reduction, and (time-permitting) the birational equivalence theorem proven in the joint paper.

### 2016.03.09

Speaker: João Nuno Mestre (Utrecht University)

Title: Measures and densities on differentiable stacks

Abstract: We explain how an extension of Haefliger's approach to transverse measures for foliations allows us to define and study measures and geometric measures (densities) on differentiable stacks. The abstract theory works for any differentiable stack, but it becomes very concrete for proper stacks - for example, when computing the volume associated with a density, we recover the explicit formulas that are taken as definition by Weinstein. This talk is based on joint work with Marius Crainic.

### 2016.03.15

Speaker: Christopher Seaton (Rhodes College)

Title: Constructing symplectomorphisms between singular linear symplectic quotients

Abstract: Let G be a compact Lie group and V a unitary representation of G. Treating V as a real symplectic manifold and choosing the homogeneous quadratic moment map, we can form the symplectic quotient, the quotient M0=Z/G where Z is the zero fiber of the moment map. The space M0 has a variety of structures, including that of a symplectic stratified space and a semialgabraic set, and its algebra of smooth functions has a graded subalgebra R[M0] of regular functions defined as a quotient of the real polynomial invariants of the G-action on V.

In this talk, we will discuss the use of the polynomial invariants of the G-representation to construct symplectomorphisms and distinguish between symplectic quotients. In particular, we will study the problem of which symplectic quotients are symplectomorphic or diffeomorphic to symplectic orbifolds. We demonstrate that in some cases, R[M0] has many of the nice properties of a ring of orbifold invariants and describe explicit computations of invariants of R[M0]. Along the way, we will see some recent results in computational invariant theory that are relevant in other contexts.

(Joint work with Joshua Cape, Carla Farsi, Hans-Christian Herbig, Daniel Herden, Ethan Lawler, and Gerald Schwarz)

### 2016.04.19

Speaker: Laura Scull (Fort Lewis College)

Title: Atlases for ineffective orbifolds

Abstract: Effective orbifolds were originally defined via charts and atlases, analogous to manifolds. In the current literature, they are often defined instead via certain topological groupoids, which gives a more convenient category for studying them. These two approaches are known to be equivalent and produce the same objects. More recently, applications arising in geometry and physics have led to an interest in more general ineffective orbifolds. However, the current generalizations of the atlas definition do not correspond to the objects created by ineffective topological groupoids. I will discuss a project, joint with D. Pronk and M. Tommasini, where we develop an alternate atlas definition for ineffective orbifolds. Our atlases generalize the existing effective atlas definition, and our definition leads to the same objects as those defined by ineffective topological groupoids.

### 2016.04.26

Speaker: Dorette Pronk (Dalhousie University)

Title: Mapping spaces for orbifolds

Abstract: For this talk we will consider orbifolds as Lie groupoids with certain properties with
generalized maps between them. The generalized maps are obtained by taking the bicategory of fractions of the 2-category of orbifold groupoids with groupoid homomorphisms. The fractions are needed because we want to consider Morita equivalences as invertible
We will first consider what generalized maps between orbifolds are and how we can think of them in terms of a generalization of an atlas refinement together with a map on the refinement. We will then discuss the structure of the 2-cells between them (the arrows of the mapping groupoid) and find that this is simpler than what one would expect for an arbitrary bicategory of fractions, and we will give a nice geometric interpretation. We will end by calculating various examples, and in particular showing that the two ineffective groupoids that Laura Scull presented last week are indeed not Morita equivalent.