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Mathias Sonnleitner, University of Passau

Covering completely symmetric convex bodies

A completely symmetric convex body is invariant under reflections or permutations of coordinates. We can bound its metric entropy numbers and consequently its mean width using sparse approximation. We provide an extension to quasi-convex bodies and present an application to unit balls of Lorentz spaces, where we can provide a complete picture of the rich behavior of entropy numbers. These spaces are compatible with sparse approximation and arise from interpolation of Lebesgue sequence spaces, for which a similar result is by now classical. Based on joint work with J. Prochno and J. Vybiral.

MC5403

Scott Wilson & Joana Cirici

Higher-homotopical BV-structures on the differential forms of symplectic and complex manifolds.

In 1985 Koszul showed that the differential forms of a symplectic manifold have an additional second order operator; part of what is now called a differential BV-algebra. Subsequent work by Getzler, Barannikov-Kontsevich, and Manin describe this structure as a (genus zero) cohomological field theory on the de Rham cohomology, i.e. an action of the compactified moduli space of (genus zero) Riemann surfaces with marked points. Such structures, also known as (formal) Frobenius manifolds, or hypercommutative algebras, have numerous connections with the A-model and mirror symmetry.

In this talk I'll explain a natural generalization of this to (almost) symplectic and complex manifolds using a higher-homotopical notion of BV-algebras. This relies on generalizations of the Kahler identities to these cases. I'll explain the setup, establish the existence of the higher-homotopy BV-structure, and give some explicit examples of almost symplectic and complex manifolds where these higher operations on cohomology are non-zero. Some examples suggest a relationship with ABC-Massey products, defined for complex manifolds.

MC 5417

Faisal Romshoo

The Ebin Slice Theorem

The Ebin Slice Theorem shows the existence of a "slice" for the action of the group of diffeomorphisms $\textrm{Diff}(M)$ on the space of Riemannian metrics $\mathcal{R}(M)$ for a closed smooth manifold $M$. We will see a proof of the existence of a slice in the finite-dimensional case and if time permits, we will go through the generalization of the proof to the infinite-dimensional setting.

MC 5417

Jashan Bal

Non-Archimedean Polish Groups

We consider the class of Polish non-Archimedean groups, those groups admitting a base at the identity of clopen subgroups. We give a complete characterization of these groups as those groups isomorphic to automorphism groups of countable, first-order structures. Time permitting, we will also discuss van Dantzig's theorem. References include Section 2.4 of Gao's IDST along with Chapter 1 of Becker and Kechris's DST of PGA.

MC 5403

William Gollinger

The Adams Spectral Sequence: Construction of the Adams Spectral Sequence

Now that we are equipped with the context of the stable homotopy category we can perform our construction. We will introduce resolutions of spectra and show how a resolution produces a spectral sequence. Identifying the terms of the resulting spectral sequence is unapproachable without additional assumptions, and we define an Adams Resolution to satisfy some homotopically exhaustive conditions. Using these conditions we can identify our E_2 page in terms of the Ext functor, and the E_\infty page in terms of the (p-completed) homotopy groups.

MC 5417

Jacques Van Wyk

An Introduction to Generalised Geometry

Generalised geometry is a field in differential and complex geometry in which one views the direct sum TM ⊕ T*M instead of TM as the bundle associated to a manifold M. Generalised geometry has seen great success in acting as a unifying framework in which structures defined on TM and T*M can be viewed as specific instances of structures defined on TM ⊕ T*M. For example, almost complex structures and pre-symplectic structures can both be viewed as generalised almost complex structures, a certain kind of automorphism of TM ⊕ T*M.

In this talk, I will give an introduction to generalised geometry. I will show TM ⊕ T*M comes with an intrinsic non-degenerate bilinear form. I will introduce the Dorfman bracket on Γ(TM ⊕ T*M), an analogue of the Lie bracket, which together with the aforementioned bilinear form gives TM ⊕ T*M the structure of a Courant algebroid. I will define generalised almost complex structures in this setting, and show how almost complex structures and pre-symplectic structures can be viewed as generalised almost complex structures. I will introduce generalised metrics and generalised connections, and if time permits, I will discuss integrability of generalised almost complex structures in terms of generalised connections, and/or discuss the analogue of the Levi-Civita connection and what complications it comes with.

MC 5417

Faisal Romshoo

Perspectives on the moduli space of torsion-free $\textrm{G}_2$-structures

Joyce showed that the moduli space of torsion-free $\textrm{G}_2$-structures for a compact $7$-manifold forms a non-singular smooth manifold. In this talk, we consider the action of gauge transformations of the form $e^{tA}$ where $A$ is a $2$-tensor, on the space of torsion-free $\textrm{G}_2$-structures. This gives us a new framework to study the moduli space. 

We will see that a $\textrm{G}_2$-structure $\widetilde{\varphi} = P^*\varphi$ acted upon by a gauge transformation $P = e^{tA}$ is infinitesimally torsion-free if and only if  $A \diamond \varphi$ is harmonic and if $A$ satisfies a ``gauge-fixing" condition, where $A \diamond \varphi$ is a special type of $3$-form. This may be the first step in giving an alternate proof of the fact that the moduli space forms a manifold in our framework of gauge transformations.

MC 5501

Keira Gunn (University of Calgary)

Some Problems on the Dynamics of Positive Characteristic Tori.

The real (or characteristic zero) torus is simply R/Z, or the "decimal part" of any real number with operations of addition and integer multiplication.  With the positive characteristic integers defined to be polynomials with coefficients from a finite field, and the positive characteristic reals their Laurent series counterparts, we can similarly construct the positive characteristic tori (each torus dependent on the choice of field).  At first glance there are many similarities to how operations work in both positive and zero characteristic, but these similarities break down quickly upon further inspection, particularly from a view of dynamics on the tori.

In this talk, we will discuss results on some orbital sets and dynamics formulae on the positive characteristic tori, including the Artin-Mazur zeta function and analogous hypothesis for Furstenberg's Orbital Theorem.

MC5403

Utkarsh Bajaj

Klein's icosahedral function

Can we define a rational function on the sphere? Sure we can. Can we define a rational function on the sphere so that it is invariant under the rotational symmetries under the icosahedron? Yes - by embedding the icosahedron in the Riemann sphere (and then doing some algebra). We then show how this beautiful function reveals connections between the symmetries of the icosahedron and the E8 lattice  - the lattice that gives the most efficient packing of spheres in 8 dimensions!

MC 5501

Filip Milidrag

The Classification of Irreducible Discrete Reflection Groups

In this talk we will make a correspondence between irreducible discrete reflection groups and associated connected Coxeter diagrams. Then we will use this to classify all connected Coxeter diagrams and by extension every irreducible discrete reflection group.

MC 5501