Forthcoming events in this series
I will introduce the profinite completion as a way of aggregating information about the finite-sheeted covers of a 3-manifold, and discuss the state of the homeomorphism problem for 3-manifolds in this context; in particular, for geometrizable 3-manifolds.
Deformation K-theory was introduced by G. Carlsson and gives an interesting invariant of a group G encoding higher homotopy information about its representation spaces. Lawson proved a relation between this object and a homotopy theoretic analogue of the representation ring. This talk will not contain many details, instead I will outline some basic constructions and hopefully communicate the main ideas.
Abstract: Four manifolds are some of the most intriguing objects in topology. So far, they have eluded any attempt of classification and their behaviour is very different from what one encounters in other dimensions. On the other hand, Einstein metrics are among the canonical types of metrics one can find on a manifold. In this talk I will discuss many of the peculiarities that make dimension four so special and see how Einstein metrics could potentially help us understand more about four manifolds.
I plan to discuss finite time singularities for the harmonic map heat flow and describe a beautiful example of winding behaviour due to Peter Topping.
A result of Jeffrey Brock states that, given a hyperbolic 3-manifold which is a mapping torus over a surface $S$, its volume can be expressed in terms of the distance induced by the monodromy map in the pants graph of $S$. This is an abstract graph whose vertices are pants decompositions of $S$, and edges correspond to some 'elementary alterations' of those.
I will show how this theorem gives an estimate for the volume of hyperbolic complements of closed braids in the solid torus, in terms of braid properties. The core piece of such estimate is a generalization of a result of Masur, Mosher and Schleimer that train track splitting sequences (which I will define in the talk) induce quasi-geodesics in the marking graph.
In this talk we're going to discuss Hamilton's maximum principle for the Ricci flow. As an application, I would like to explain a technique due to Boehm and Wilking which provides a general tool to obtain new Ricci flow invariant curvature conditions from given ones. As we'll see, it plays a key role in Brendle and Schoen's proof of the differentiable sphere theorem.
The Ising model is a well-known statistical physics model, defined on a two-dimensional lattice. It is interesting because it exhibits a "phase transition" at a certain critical temperature. Recent mathematical research has revealed an intriguing geometry in the model, involving discrete holomorphic functions, spinors, spin structures, and the Dirac equation. I will try to outline some of these ideas.
Quiver varieties and their quantizations feature prominently in
geometric representation theory. Multiplicative quiver varieties are
group-like versions of ordinary quiver varieties whose quantizations
involve quantum groups and $q$-difference operators. In this talk, we will
define and give examples of representations of quivers, ordinary quiver
varieties, and multiplicative quiver varieties. No previous knowledge of
quivers will be assumed. If time permits, we will describe some phenomena
that occur when quantizing multiplicative quiver varieties at a root of
unity, and work-in-progress with Nicholas Cooney.
I will explain the basics of deformation quantization of Poisson
algebras (an important tool in mathematical physics). Roughly, it is a
family of associative algebras deforming the original commutative
algebra. Following Fedosov, I will describe a classification of
quantizations of (algebraic) symplectic manifolds.
In this talk I shall discuss some classical results on isometric embedding of positively/nonegatively curved surfaces into $\mathbb{R}^3$.
The isometric embedding problem has played a crucial role in the development of geometric analysis and nonlinear PDE techniques--Nash invented his Nash-Moser techniques to prove the embeddability of general manifolds; later Gromov recast the problem into his ``h-Principle", which recently led to a major breakthrough by C. De Lellis et al. in the analysis of Euler/Navier-Stokes. Moreover, Nirenberg settled (positively) the Weyl Problem: given a smooth metric with strictly positive Gaussian curvature on a closed surface, does there exist a global isometric embedding into the Euclidean space $\mathbb{R}^3$? This work is proved by the continuity method and based on the regularity theory of the Monge-Ampere Equation, which led to Cheng-Yau's renowned works on the Minkowski Problem and the Calabi Conjecture.
Today we shall summarise Nirenberg's original proof for the Weyl problem. Also, we shall describe Hamilton's simplified proof using Nash-Moser Inverse Function Theorem, and Guan-Li's generalisation to the case of nonnegative Gaussian curvature. We shall also mention the status-quo of the related problems.
In this talk I will present a basic introduction to conformal symmetry from a physicist perspective. I will talk about infinitesimal and finite conformal transformations and the conformal group in diverse dimensions.
The moduli space of G-Higgs bundles carries a natural Hyperkahler structure, through which we can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes) with respect to each structure. Notably, these A and B-branes have gained significant attention in string theory.
We shall begin the talk by first introducing G-Higgs bundles for reductive Lie groups and the associated Hitchin fibration, and sketching how to realize Langlands duality through spectral data. We shall then look at particular types of branes (BAA-branes) which correspond to very interesting geometric objects, hyperholomorphic bundles (BBB-branes).
The presentation will be introductory and my goal is simply to sketch some of the ideas relating these very interesting areas.
I will introduce simple homotopy theory and then discuss relations between some conjectures in 2 dimensional simple homotopy theory and the 3 and 4 dimensional Poincaré conjectures.
Since its genesis in 1915, General Relativity has proven to be one of the most successful physical theories ever invented. Providing a description of the large scale structure of the universe it continues to be in agreement with all experimental tests to high accuracy. By merging Classical Mechanics and Electrodynamics to a consistent geometrical theory of space-time it has become one of the two pillars of modern theoretical physics alongside Quantum Mechanics. This talk aims to give an introduction to the ideas and concepts of General Relativity. After briefly reviewing Classical (Newtonian) Mechanics and experiments in contradiction with it the framework and axioms of General Relativity will be introduced. This will be followed by a survey on major implications of the (new) geometrical description of gravity. Finally an outlook on physics beyond General Relativity will be provided.
Lagrangian Floer cohomology categorifies the intersection number of (half-dimensional) Lagrangian submanifolds of a symplectic manifold. In this talk I will describe how and when can we define Lagrangian Floer cohomology. In the case when Floer cohomology cannot be defined I will describe an alternative invariant known as the Fukaya (A-infinity) algebra.
I will recall basic notions of operator K-theory as a non-commutative (C*-algebra) generalisation of topological K-theory. Twisted crossed products will be introduced as generalisations of group C*-algebras, and a model of Karoubi's K-theory, which makes sense for super-algebras, will be sketched. The motivation comes from physics, through the study of quantum mechanical symmetries, charged free quantum fields, and topological insulators. The relevant theorems, which are interesting in their own right but scattered in the literature, will be consolidated.
In this talk we want to discuss results by Dimca, Papadima, and Suciu about the finiteness properties of Kähler groups. Namely, we will sketch their proof that for every $2\leq n\leq \infty$ there is a Kähler group with finiteness property $\mathcal{F}_n$, but not $FP_{n+1}$. Their proof is by explicit construction of examples. These examples all arise as subgroups of finite products of surface groups and they are the first known examples of Kähler groups with arbitrary finiteness properties. The talk does not require any prior knowledge of finiteness properties or of Kähler groups.
Historically, the study of positively curved manifolds has always been challenging. There are many reasons for this, but among them is the fact that the existence of a metric of positive curvature on a manifold imposes strong topological restrictions. In this talk we will discuss some of these topological implications and we will introduce the main results in this area. We will also present some recent results that relate positive curvature to the smooth structure of the manifold.
The Calabi conjecture, posed in 1954 and proved by Yau in 1976, guaranties the existence of Ricci-flat Kahler metrics on compact Kahler manifolds with vanishing first Chern class, providing examples of the so called Calabi-Yau manifolds. The latter are of great importance to the fields of Riemannian Holonomy Groups, having Hol0 as a subgroup of SU; Calibrated Geometry, more precisely Special Lagrangian Geometry; and to String theory with the discovery of the phenomenon of Mirror Symmetry (to mention a few!). In the talk, we will discuss the necessary background to formulate the Calabi conjecture and explain some of the main ideas behind its proof by Yau, which itself is a jewel from the point of view of non-linear PDEs.
Lie algebroids are geometric structures that interpolate between finite-dimensional Lie algebras and tangent bundles of manifolds. They give a useful language for describing geometric situations that have local symmetries. I will give an introduction to the basic theory of Lie algebroids, with examples drawn from foliations, principal bundles, group actions, Poisson brackets, and singular hypersurfaces.
Spectral networks are certain collections of paths on a Riemann surface, introduced by Gaiotto, Moore, and Neitzke to study BPS states in certain N=2 supersymmetric gauge theories. They are interesting geometric objects in their own right, with a number of mathematical applications. In this talk I will give an introduction to what a spectral network is, and describe the "abelianization map" which, given a spectral network, produces nice "spectral coordinates" on the appropriate moduli space of flat connections. I will show that coordinates obtained in this way include a variety of previously known special cases (Fock-Goncharov coordinates and Fenchel-Nielsen coordinates), and mention at least one reason why generalising them in this way is of interest.
We give an overview of Kitaev's lattice model in the setting of an arbitrary finite group G (where $G = Z_{2}$ is the famous Toric Code). We also exhibit the connection this model has with so-called 123-TQFTs (topological quantum field theories), making use of ideas coming from higher gauge theory and Hopf algebra representations.
Topological insulators are a type of system in condensed matter physics that exhibit a robustness that physicists like to call topological. In this talk I will give a definition of a subclass of such systems: gapped, free fermions. We will look at how such systems, as shown by Kitaev, can be classified in terms of topological K-groups by using the Clifford module model for K-theory as introduced by Atiyah, Bott and Shapiro. I will be using results from Wednesday's JTGT, where I'll give a quick introduction to topological K-theory.
Since their introduction in the context of symplectic geometry, moment maps and symplectic quotients have been generalized in many different directions. In this talk I plan to give an introduction to the notions of hyperkähler moment map and hyperkähler quotient through two examples, apparently very different, but related by the so called ADHM construction of instantons; the moduli space of instantons and a space of complex matrices arising from monads.
The first half of this talk will be an introduction to the wonderful world of Higgs bundles. The last half concerns Fourier--Mukai transforms, and we will discuss how to merge the two concepts by constructing a Fourier--Mukai transform for Higgs bundles. Finally we will discuss some properties of this transform. We will along the way discuss why you would want to transform Higgs bundles.
I plan to give a non technical introduction (i.e. no prerequisites required apart basic differential geometry) to some analytic aspects of the theory of harmonic maps between Riemannian manifolds, motivate it by briefly discussing some relations to other areas of geometry (like minimal submanifolds, string topology, symplectic geometry, stochastic geometry...), and finish by talking about the heat flow approach to the existence theory of harmonic maps with some open problems related to my research.
Following last week's talk on Beilinson-Bernstein localisation theorem, we give basic notions in deformation quantisation explaining how this theorem can be interpreted as a quantised version of the Springer resolution. Having attended last week's talk will be useful but not necessary.
In the sixties Griffiths constructed a holomorphic map, known as the local period map, which relates the classification of smooth projective varieties to the associated Hodge structures. Fiorenza and Manetti have recently described it in terms of Schlessinger's deformation functors and, together with Martinengo, have started to look at it in the context of Derived Deformation Theory. In this talk we propose a rigorous way to lift such an extended version of Griffiths period map to a morphism of derived deformation functors and use this to construct a period morphism for global derived stacks.
A universal D-module of dimension n is a rule assigning to every family of smooth $n$-dimensional varieties a family of D-modules, in a compatible way. This seems like a huge amount of data, but it turns out to be entirely determined by its value over a single formal disc. We begin by recalling (or perhaps introducing) the notion of a D-module, and proceed to define the category $M_n$ of universal D-modules. Following Beilinson and Drinfeld we define the Gelfand-Kazhdan structure over a smooth variety (or family of varieties) of dimension $n$, and use it to build examples of universal D-modules and to exhibit a correspondence between $M_n$ and the category of modules over the group-scheme of continuous automorphisms of formal power series in $n$ variables
We will talk about the Beilinson-Bernstein localization theorem, which is a major result in geometric representation theory. We will try to explain the main ideas behind the theorem and this will lead us to some geometric constructions that are used in order to produce representations. Finally we will see how the theorem is demonstrated in the specific case of the Lie algebra sl2
This talk will give an introduction to generalized complex geometry, where complex and symplectic structures are particular cases of the same structure, namely, a generalized complex structure. We will also talk about a sister theory, generalized complex geometry of type Bn, where generalized complex structures are defined for odd-dimensional manifolds as well as even-dimensional ones.
Quivers are directed graphs which can be thought of as "space" in noncommutative geometry. In this talk, we will try to establish a link between noncommutative geometry and its commutative counterpart. We will show how one can construct (differential graded) quivers which are "equivalent" (in the sense of derived category of representations) to vector bundles on smooth varieties.
Ricci solitons were introduced by Richard Hamilton in the 80's and they are a generalization of the better know Einstein metrics. During this talk we will define the notion of Ricci soliton and I will try to convince you that these metrics arise "naturally" in a number of different settings. I will also present various examples and talk a bit about some symmetry properties that Ricci solitons have.
Note: This talk is meant to be introductory and no prior knowledge about Einstein metrics will be assumed (or necessary).
We give a short exposition on the zeta determinant for a Laplace - type operator on a closed Manifold as first described by Ray and Singer in their attempt to find an analytic counterpart to R-torsion.
TQFTs have received widespread attention in recent years. In mathematics
for example due to Lurie's proof of the cobordism hypothesis. In physics
they are used as toy models to understand structure, especially
boundaries and defects.
I will present a lattice construction of 2d Spin TFT. This mostly
motivated as both a toy model and stepping stone for a mathematical
construction of rational conformal field theories with fermions.
I will first describe a combinatorial model for spin surfaces that
consists of a triangulation and a finte set of extra data. This model is
then used to construct TFT correlators as morphisms in a symmetric
monoidal category, given a Frobenius algebra as input. The result is
shown to be independent of the triangulation used, and one obtains thus
a 2dTFT.
All results and constructions can be generalised to framed surfaces in a
relatively straightforward way.
In this talk, we will review the classical Bass-Serre theory of groups acting on trees and introduce its real version, Rips' theory. If time permits, I will briefly discuss some higher dimensional spaces that are currently being investigated, namely cubings and real cubings.
I will introduce star products and formal connections and describe approaches to the problem of finding a trivialization of the formal Hitchin connection, using graph-theoretical computations.
We motivate and dicuss the Beilinson Theorem for sheaves on projective spaces. Hopefully we see some examples along the way.
By using the Weil-Gel'fand-Zak transform of Faddeev's quantum dilogarithm,
Andersen and Kasheav have proposed a new state-integral model for the
Andersen--Kashaev TQFT, where the circle valued state variables live on
the edges of oriented levelled shaped triangulations. I will look at a
couple of examples which give an idea of how the theories are coupled.
In this talk, I will describe how the eigenvalues of the Atkin operator on overconvergent modular forms might be related to the classical study of the Laplacian on certain manifolds. The goal is to phrase everything geometrically, so as to maximally engage the audience in discussion on possible approaches to study the spectral flow of this operator.
Motivated by the study of PDEs, we introduce the notion of a D-module on a variety X and give the basics of three perspectives on the theory: modules over the sheaf of differential operators on X; quasi-coherent modules with flat connection; and crystals on X. This talk will assume basic knowledge of algebraic geometry (such as rudimentary sheaf theory).
Consider a smooth, complex projective variety X inside P^n and an action of a reductive linear algebraic group G inside GL(n+1,C). On the one hand, we can view this as an algebra-geometric set-up and use geometric invariant theory (GIT) to construct a quotient variety X // G, which parameterises `most' of the closed orbits of X. On the other hand, X is naturally a symplectic manifold, and since G is reductive we can take a maximal real compact Lie subgroup K of G and consider the symplectic reduction of X by K with respect to an appropriate moment map. The Kempf-Ness theorem then says that the results of these two constructions are homeomorphic. In this talk I will define GIT and symplectic reduction and try to sketch the proof of the Kempf-Ness theorem.
In the talk we will discuss Quillen's construction of a determinant line bundle associated to a family of Cauchy-Riemann operators. I will first of all try to convince you why this is a cool thing and mention some of the many different applications. The bulk of the talk will be focused on constructing the line bundle, its hermitian metric and calculating the curvature. Hopefully a talk accessible to many.