Department of

# Mathematics

Seminar Calendar
for Symplectic and Poisson Geometry Seminar events the year of Wednesday, September 13, 2017.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     August 2017           September 2017          October 2017
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4  5                   1  2    1  2  3  4  5  6  7
6  7  8  9 10 11 12    3  4  5  6  7  8  9    8  9 10 11 12 13 14
13 14 15 16 17 18 19   10 11 12 13 14 15 16   15 16 17 18 19 20 21
20 21 22 23 24 25 26   17 18 19 20 21 22 23   22 23 24 25 26 27 28
27 28 29 30 31         24 25 26 27 28 29 30   29 30 31



Monday, January 23, 2017

4:00 pm in 243 Altgeld Hall,Monday, January 23, 2017

#### J-holomorphic cylinders between ellipsoids

###### Ely Kerman (UIUC)

Abstract: The primary tool for detecting obstructions to symplectic embeddings are regular J-holomorphic curves in symplectic cobordisms. The more the better. In this talk, I will describe an existence theorem for such curves in dimension 4 and, time permitting, an application. This is based on joint work in progress with Richard Hind.

Monday, January 30, 2017

4:00 pm in 243 Altgeld Hall,Monday, January 30, 2017

#### Geometric Lie 2-algebras

###### Cristian Ortiz (University of Sao Paulo)

Abstract: Lie 2-algebras were introduced by Baez and Crans as groupoid objects in the category of Lie algebras. In this talk we will present a class of examples of Lie 2-algebras, those coming from sections of LA-groupoids. We will see that the category of sections of an LA-groupoid has a natural structure of Lie 2-algebra which is Morita invariant in a suitable sense. In the particular case of the tangent groupoid, one gets a Lie 2-algebra structure on the category of multiplicative vector fields on a Lie groupoid. We will explain how this can be used to introduce a geometric notion of vector field on the quotient stack of a Lie groupoid. This is joint work with James Waldron.

Monday, February 13, 2017

4:00 pm in 243 Altgeld Hall,Monday, February 13, 2017

#### Constructing A-symplectic structures

###### Ralph Klaasse (Utrecht University)

Abstract: In this talk we discuss how to construct A-symplectic structures for a Lie algebroid A by adapting Gompf-Thurston techniques to Lie algebroid morphisms. As an application we obtain both log-symplectic and stable generalized complex structures out of log-symplectic structures. In particular we define a class of maps called boundary Lefschetz fibrations and show they equip their total space with a stable generalized complex structure. This is based on joint work with Gil Cavalcanti.

Monday, February 27, 2017

4:00 pm in 243 Altgeld Hall,Monday, February 27, 2017

#### "The minimal number of periodic Reeb orbits as a cuplength

###### Jean Gutt (University of Georgia)

Abstract: I will present the recent result with P.Albers and D.Hein that every graphical hypersurface in a prequantization bundle over a symplectic manifold M pinched between two circle bundles whose ratio of radii is less than \sqrt{2} carries either one short simple periodic orbit or carries at least cuplength(M)+1 simple periodic Reeb orbits.

Monday, March 6, 2017

4:00 pm in 243 Altgeld Hall,Monday, March 6, 2017

#### Contact embeddings in three dimensions

###### Olguta Buse (Indiana University-Purdue University Indianapolis )

Abstract: We study neighborhoods of transversal knots in contact thee dimensional manifolds. By doing so, we introduce the concepts of capacity and shape for a three dimensional contact manifold $(M, \xi)$ relative to a transversal knot $K$. We will explain the connection with the existing literature and provide our main computation for the shape in the case of lens spaces $L(p,q)$ with a toric contact structure. The main tool used here are rational surgeries which will be explained through their toric interpretations based on the continuous fraction expansions of $p/q$. This is joint work with D. Gay.

Monday, March 13, 2017

4:00 pm in 243 Altgeld Hall,Monday, March 13, 2017

#### Deformations of log symplectic structures on surfaces

###### Melinda Lanius (UIUC)

Abstract: A star log symplectic bi-vector on a surface has a degeneracy loci locally modelled by a finite set of lines in the plane intersecting at a point. We will discuss two ways to capture the behaviour of their deformations: one global' and one more local' in flavor. From a global perspective, we classify all star log symplectic structures on compact surfaces up to symplectomorphism by some associated Lie algebroid de Rham cohomology classes. In a more local snap shot, we compute the Poisson cohomology of these structures and discuss the relationship of our classification and the second Poisson cohomology.

Monday, March 27, 2017

4:00 pm in 243 Altgeld Hall,Monday, March 27, 2017

#### Poisson Manifolds and their Stacks

###### Joel Villatoro (UIUC)

Abstract: The categorical notion of a differentiable stack and the theory of Lie groupoids are related by the concept of Morita equivalence. To any Lie groupoid, we can associate a differentiable stack so that Morita equivalence of Lie groupoids corresponds to an isomorphism of stacks. There is also a closely related notion of Morita equivalence of Poisson manifolds. We can then ask if there is a way to associate a stack to a Poisson manifold such that a similar property holds. In this talk I will introduce a 'site' which answers this question. I will also give a few concrete examples of the kinds of geometric phenomena captured by stacks over this site.

Monday, April 3, 2017

4:00 pm in 243 Altgeld Hall,Monday, April 3, 2017

#### Vector fields on a stack form a Lie 2-algebra.

###### Eugene Lerman (UIUC)

Abstract: (joint work with Daniel Berwick-Evans). We prove that the category of vector fields on a geometric stack is a Lie 2-algebra. I will start by sketching out the definitions of a stack, a geometric stack, vector field on a stack and of a (Baez-Crans) Lie 2-algebra, which is a categorified version of a Lie algebra.

Monday, April 10, 2017

4:00 pm in 243 Altgeld Hall,Monday, April 10, 2017

#### Representations up to homotopy are weak groupoid actions

###### Seth Wolbert (UIUC)

Abstract: In this talk, I will explain how one may identify (2-term) representations up to homotopy with weak groupoid actions. I will also discuss how this naturally allows one to view VB-groupoids (vector bundles of the category of Lie groupoids) as weakly split groupoid extensions.

Monday, April 17, 2017

4:00 pm in 243 Altgeld Hall,Monday, April 17, 2017

#### Embedded Lagrangians in $\mathbb C P^2$

###### Ana Cannas da Silva (ETH Zurich)

Abstract: Weinstein's symplectic creed that "everything is a lagrangian" bolsters a central question in symplectic geometry: what lagrangians are there in a given symplectic manifold? This question comes in different flavours depending on further desired properties. We concentrate on embedded (closed) lagrangians in $\mathbb CP^2$ that sit nicely with respect to the toric structure and discuss an example that exhibits a distinguishing behavior under reduction relevant in connection with Weinstein's lagrangian composition and work of Wehrheim and Woodward in Floer theory.

Monday, April 24, 2017

3:00 pm in 243 Altgeld Hall,Monday, April 24, 2017

#### The Relative Compactification of the Universal Centralizer(CANCELED)

###### Ana Balibanu (U Chicago)

Abstract: Let $G$ be a semisimple algebraic group of adjoint type. The universal centralizer $\mathcal X$ is the family of centralizers in $G$ of regular elements in Lie($G$). This algebraic variety has a natural symplectic structure, obtained by Hamiltonian reduction from the cotangent bundle $T^ ∗G$. We introduce a relative compactification of $\mathcal X$ , in which every centralizer fiber is replaced by its closure in the wonderful compactification of $G$. We show that the symplectic structure extends to a log-symplectic structure on the boundary, using the logarithmic cotangent bundle of the wonderful compactification.

Monday, May 1, 2017

4:00 pm in 243 Altgeld Hall,Monday, May 1, 2017

#### A geometrical perspective on the quantum Fisher Information index

###### Michele Schiavina (UC Berkeley)

Abstract: Given a quantum state $\rho$ and a measurement operator $m$ one can define the classical and quantum Fisher information indices (CFI/QFI), the former depending on both $\rho$ and $m$, the latter being an intrinsic property of the quantum state. Shortly after their introduction, it was observed how the CFI is bounded by the QFI, allowing one to ask what optimal measurements can attain the bound. However, the problem of actually computing (and defining) the QFI is an obstruction that kept researchers from addressing the optimisation problem, except for simple cases. Rephrasing (finite dimensional) quantum mechanics in the geometric framework of co-adjoint orbits of the unitary group has lead to the solution of the computation problem of the quantum Fisher information, reinterpreted as a natural object on such symplectic manifolds. In this talk I will introduce the Fisher information optimisation problem, highlighting the parts where geometry has shown to be crucial, and I will describe the main construction of the Fisher information tensor and related quantities on the spaces of quantum states.

Monday, September 11, 2017

3:00 pm in 243 AH,Monday, September 11, 2017

#### Relative Floer theory and wall-crossing

###### James Pascaleff (UIUC)

Abstract: Given a monotone Lagrangian L in a monotone symplectic manifold, there is a function known as the disc potential that encodes counts of maslov index 2 discs with boundary on L. The behavior of this potential under certain geometric transformations of the Lagrangian (mutations) is governed by what is known as a "wall-crossing formula." In this talk I will present a new, simple argument that allows one to prove such formulas in a general setting. The main new ingredient is a reformulation of the problem in terms of relative Floer theory. This is joint work with Dmitry Tonkonog.

Monday, September 18, 2017

3:00 pm in 243 Altgeld Hall,Monday, September 18, 2017

#### Non-associative local Lie groupoids

###### Daan Michiels (UIUC)

Abstract: Historically, local Lie groups were studied before Lie groups. Surprisingly, not every local Lie group can be embedded into a Lie group, and a theorem by Mal'cev shows that the obstructions to such embeddings can be understood in terms of the associativity of the local Lie group. In this talk, I will explain Mal'cev's theorem and show how it generalizes to the groupoid setting. Then, I will point out an intimate connection between this result, and the theory of integrability of Lie algebroids.

Monday, October 16, 2017

3:00 pm in 243 Altgeld Hall,Monday, October 16, 2017

#### Stability of relative equilibria and isomorphic vector fields

###### Stefan Klajbor (UIUC)

Abstract: Isomorphic vector fields were introduced by Hepworth in his study of vector fields on differentiable stacks. While originating in the realm of stacks, this notion can be useful in the study of equivariant dynamics on manifolds. In particular, we argue in favor of the usefulness of replacing an equivariant vector field by an isomorphic one in order to study the nonlinear stability of relative equilibria. We use this idea to obtain a criterion for nonlinear stability. As an application, we sketch how to use this criterion to obtain Montaldi and Rodrı́guez-Olmos’s criterion for stability of Hamiltonian relative equilibria on symplectic manifolds.

Monday, November 6, 2017

3:00 pm in 243 AH,Monday, November 6, 2017

#### The partial compacti cation of the universal centralizer

###### Ana Balibanu (Harvard)

Abstract: Let $G$ be a semisimple algebraic group of adjoint type. The universal centralizer is the family of centralizers in $G$ of regular elements in $\text{Lie}(G)$, parametrized by their conjugacy classes. It has a natural symplectic structure, obtained by Hamiltonian reduction from the cotangent bundle $T^*G$. We consider a partial compactification of the universal centralizer, where each centralizer fiber is replaced by its closure inside the wonderful compactification of $G$. We show that the symplectic structure extends to a log-symplectic Poisson structure on the partial compactification, through a Hamiltonian reduction of the logarithmic cotangent bundle of the wonderful compactification.

Monday, November 13, 2017

3:00 pm in 243 Altgeld Hall,Monday, November 13, 2017

#### Symplectic groupoids up to homotopy

###### Cristian Ortiz (University of Sao Paulo)

Abstract: Symplectic groupoids correspond to the global counterpart of Poisson manifolds and play a role in generalized notions of moment maps. In this seminar, we will introduce symplectic groupoids "up to homotopy" and we will show the relation between these objects and symplectic stacks. We will also describe Lagrangian morphisms in this context and we will present examples arising in connection with moment maps in Dirac geometry.

Monday, December 11, 2017

3:00 pm in 243 Altgeld Hall,Monday, December 11, 2017

#### Post-Lie algebroids and Lie algebra actions

###### Ari Stern (Washington University in St. Louis)

Abstract: A fundamental problem in numerical analysis is the approximation of flows of vector fields on manifolds. For vector fields in Euclidean space (i.e., systems of ODEs), Butcher gave a brilliant characterization of Runge-Kutta methods in terms of the algebra and combinatorics of rooted trees. Subsequent work related this to the fact that vector fields on Euclidean space form a "pre-Lie algebra". More recently, this work has been extended to "post-Lie algebroids": vector bundles whose sections form a "post-Lie algebra" with respect to a connection. We give a complete characterization of post-Lie algebroids in terms of local Lie algebra actions. As a corollary, we get a "no-go theorem" limiting the cases to which this class of numerical methods can be applied. (Joint work with Hans Munthe-Kaas and Olivier Verdier.)