From a directed graph one can generate an algebra which captures the movements along the graph. One such algebras are Leavitt path algebras. Despite being introduced only 10 years ago, Leavitt path algebras have arisen in a variety of different contexts as diverse as analysis, symbolic dynamics, noncommutative geometry and representation theory. In fact, Leavitt path algebras are algebraic counterpart to graph C*-algebras, a theory which has become an area of intensive research globally. There are strikingly parallel similarities between these two theories. Even more surprisingly, one cannot (yet) obtain the results in one theory as a consequence of the other; the statements look the same, however the techniques to prove them are quite different (as the names suggest, one uses Algebra and other Analysis). These all suggest that there might be a bridge between Algebra and Analysis yet to be uncovered. In this talk, we introduce Leavitt path algebras and try to classify them by means of (graded) Grothendieck groups. We will ask nice questions!
The goal is to compare Riemannian and Lorentzian geometries and see what one loses and wins when going from the Riemann to Lorentz. Essentially, one loses compactness and ellipticity, but wins causality structure and mathematical and physical situations when natural Lorentzian metrics emerge.
Constant mean curvature (CMC) tori in \(\mathbb{S}^3\), \(\mathbb{R}^3\) or \(\mathbb{H}^3\) are in bijective correspondence with spectral curve data, consisting of a hyperelliptic curve, a line bundle on this curve and some additional data, which in particular determines the relevant space form. This point of view is particularly relevant for considering moduli-space questions, such as the prevalence of tori amongst CMC planes and whether tori can be deformed. I will address these questions for the spherical and Euclidean cases, using Whitham deformations.
In this talk I will introduce the notion of parahoric groups, a loop group analogue of parabolic subgroups. I will also discuss a global version of this, namely parahoric bundles on a complex curve. This leads us to a problem concerning the behaviour of invariant polynomials on the dual of the Lie algebra, a kind of "parahoric invariant theory". The key to solving this problem turns out to be the Kazhdan-Lusztig map, which assigns to each nilpotent orbit in a semisimple Lie algebra a conjugacy class in the Weyl group. Based on joint work with Masoud Kamgarpour and Rohith Varma.
Weyl character formula describes characters of irreducible representations of compact Lie groups. This formula can be obtained using geometric method, for example, from the Atiyah-Bott fixed point theorem or the Atiyah-Segal-Singer index theorem. Harish-Chandra character formula, the noncompact analogue of the Weyl character formula, can also be studied from the point of view of index theory. We apply orbital integrals on K-theory of Harish-Chandra Schwartz algebra of a semisimple Lie group \(G\), and then use geometric method to deduce Harish-Chandra character formulas for discrete series representations of \(G\). This is work in progress with Peter Hochs.
We prove a bubble-neck decomposition and an energy quantisation result for sequences of Willmore surfaces immersed into \(\mathbb{R}^{m\ge3}\) with uniformly bounded energy and non-degenerating conformal structure. We deduce the strong compactness (modulo the action of the Moebius group) of closed Willmore surfaces of a given genus below some energy threshold. This is joint-work with Tristan Rivière (ETH Zürich).
Hilbert schemes of points on the total spaces of the line bundles O(-n) on P1 (desingularizations of toric singularities of type (1/n)(1,1)) can be given an ADHM description, and as a result, they can be realized as varieties of quiver representations.
I will motivate the construction of pseudodifferential algebra bundles arising in index theory, and also outline the construction of general pseudodifferential algebra bundles (and the associated sphere bundles), showing that there are many that are purely infinite dimensional that do not come from usual constructions in index theory. I will also discuss characteristic classes of such bundles. This is joint work with Richard Melrose.
The elusive Weyl fermion was recently realised as quasiparticle excitations of a topological semimetal. I will explain what a semi-metal is, and the precise mathematical sense in which they can be "topological", in the sense of the general theory of topological insulators. This involves understanding vector bundles with singularities, with the aid of Mayer-Vietoris principles, gerbes, and generalised degree theory.
Much effort has been devoted to generalizing the
Calderón-Zygmund theory in harmonic analysis from Euclidean
spaces to metric measure spaces, or spaces of homogeneous type.
Here the underlying space \(\mathbb{R}^n\) with Euclidean metric
and Lebesgue measure is replaced by a set \(X\) with general
metric or quasi-metric and a doubling measure. Further, one can
replace the Laplacian operator that underpins the
Calderón-Zygmund theory by more general operators \(L\)
satisfying heat kernel estimates.
I will present recent joint work with P. Chen, X.T. Duong,
J. Li and L.X. Yan along these lines. We develop the theory of
product Hardy spaces \(H^p_{L_1,L_2}(X_1 \times X_2)\), for \(1
\leq p < \infty\), defined on products of spaces of homogeneous
type, and associated to operators \(L_1\), \(L_2\) satisfying
Davis-Gaffney estimates. This theory includes definitions of
Hardy spaces via appropriate square functions, an atomic Hardy
space, a Calderón-Zygmund decomposition, interpolation
theorems, and the boundedness of a class of Marcinkiewicz-type
spectral multiplier operators.
One can use the symplectic form to construct an elliptic complex replacing the de Rham complex. Then, under suitable curvature conditions, one can form coupled versions of this complex. Finally, on complex projective space, these constructions give rise to a series of elliptic complexes with geometric consequences for the Fubini-Study metric and its X-ray transform. This talk, which will start from scratch, is based on the work of many authors but, especially, current joint work with Jan Slovak.
In étale topology, instead of considering open subsets of a space, we consider étale neighbourhoods lying over these open subsets. In this talk, I define an étale analog of dynamical systems: to understand a dynamical system \(f:(X,\Omega )\to(X,\Omega )\), we consider other dynamical systems lying over it. I then propose to use this to resolve the following two questions:
Question 1: What should be the topological entropy of a dynamical system \((f,X,\Omega )\) when \((X,\Omega )\) is not a compact space?
Question 2: What is the relation between topological entropy of a rational map or correspondence (over a field of arbitrary characteristic) to the pullback on cohomology groups and algebraic cycles?
The set of meromorphic functions on an elliptic curve naturally possesses the structure of a complex manifold. The component of degree 3 functions is 6-dimensional and enjoys several interesting complex-analytic properties that make it, loosely speaking, the opposite of a hyperbolic manifold. Our main result is that this component has a 54-sheeted branched covering space that is an Oka manifold.
Given a twist over an etale groupoid, one can construct an associated C*-algebra which carries a good deal of geometric and physical meaning; for example, the K-theory group of this C*-algebra classifies D-brane charges in string theory. Twisted vector bundles, when they exist, give rise to particularly important elements in this K-theory group. In this talk, we will explain how to use the classifying space of the etale groupoid to construct twisted vector bundles, under some mild hypotheses on the twist and the classifying space. My hope is that this talk will be accessible to a broad audience; in particular, no prior familiarity with groupoids, their twists, or the associated C*-algebras will be assumed. This is joint work with Carla Farsi.
Over the past 30 years the Chern-Simons functional for connections on G-bundles over three-manfolds has lead to a deep understanding of the geometry of three-manfiolds, as well as knot invariants such as the Jones polynomial. Here we study this functional for three-manfolds that are topologically given as the total space of a principal circle bundle over a compact Riemann surface base, which are known as Seifert manifolds. We show that on such manifolds the Chern-Simons functional reduces to a particular gauge-theoretic functional on the 2d base, that describes a gauge theory of connections on an infinite dimensional bundle over this base with structure group given by the level-k affine central extension of the loop group LG. We show that this formulation gives a new understanding of results of Beasley-Witten on the computability of quantum Chern-Simons invariants of these manifolds as well as knot invariants for knots that wrap a single fiber of the circle bundle. A central tool in our analysis is the Caloron correspondence of Murray-Stevenson-Vozzo.
In (1+1)-d TQFT, products and coproducts are associated to pairs of pants decompositions of Riemann surfaces. We consider a toy model in dimension (0+1) consisting of specific broken paths in a Lie group. The products and coproducts are constructed by a Brownian motion average of holonomy along these paths with respect to a connection on an auxiliary bundle. In the trivial case over the torus, we (seem to) recover the Hopf algebra structure on the symmetric algebra. In the general case, we (seem to) get deformations of this Hopf algebra. This is a preliminary report on joint work with Michael Murray and Raymond Vozzo.
The main result I will discuss is non-vanishing of the image of the index map from the \(G\)-equivariant K-homology of a \(G\)-manifold \(X\) to the K-theory of the C*-algebra of the group \(G\). The action of \(G\) on \(X\) is assumed to be proper and cocompact. Under the assumption that the Kronecker pairing of a K-homology class with a low-dimensional cohomology class is non-zero, we prove that the image of this class under the index map is non-zero. Neither discreteness of the locally compact group \(G\) nor freeness of the action of \(G\) on \(X\) are required. The case of free actions of discrete groups was considered earlier by B. Hanke and T. Schick.
In this talk we present an overview of the current research in mean curvature flow and fully nonlinear curvature flows with free boundaries, with particular focus on our own results. Firstly we consider the scenario of a mean curvature flow solution with a ninety-degree angle condition on a fixed hypersurface in Euclidean space, that we call the contact hypersurface. We prove that under restrictions on either the initial hypersurface (such as rotational symmetry) or restrictions on the contact hypersurface the flow exists for all times and converges to a self-similar solution. We also discuss the possibility of a curvature singularity appearing on the free boundary contained in the contact hypersurface. We extend some of these results to the setting of a hypersurface evolving in its normal direction with speed given by a fully nonlinear functional of the principal curvatures.
It is well-known that orbifolds can be represented by a special kind of Lie groupoid, namely those that are étale and proper. Lie groupoids themselves are one way of presenting certain nice differentiable stacks. In joint work with Ray Vozzo we have constructed a presentation of the mapping stack \(\mathrm{Hom}(\mathrm{disc}(M),X)\), for \(M\) a compact manifold and \(X\) a differentiable stack, by a Fréchet-Lie groupoid. This uses an apparently new result in global analysis about the map \(C^\infty(K_1,Y) \to C^\infty(K_2,Y)\) induced by restriction along the inclusion \(K_2 \to K_1\), for certain compact \(K_1\), \(K_2\). We apply this to the case of \(X\) being an orbifold to show that the mapping stack is an infinite-dimensional orbifold groupoid. We also present results about mapping groupoids for bundle gerbes.
When the metric on a Riemannian manifold is perturbed in a rough (merely bounded and measurable) manner, do basic estimates involving the Hodge Dirac operator \(D = d+d^*\) remain valid? Even in the model case of a perturbation of the euclidean metric on \(\mathbb{R}^n\), this is a difficult question. For instance, the fact that the \(L^2\) estimate \(\|Du\|_2 \sim \|\sqrt{D^{2}}u\|_2\) remains valid for perturbed versions of \(D\) was a famous conjecture made by Kato in 1961 and solved, positively, in a ground breaking paper of Auscher, Hofmann, Lacey, McIntosh and Tchamitchian in 2002. In the past fifteen years, a theory has emerged from the solution of this conjecture, making rough perturbation problems much more tractable. In this talk, I will give a general introduction to this theory, and present one of its latest results: a flexible approach to \(L^p\) estimates for the holomorphic functional calculus of \(D\). This is joint work with D. Frey (Delft) and A. McIntosh (ANU).
I will begin this talk by showing how to obtain the Betti numbers of certain smooth complex projective varieties by counting points over a finite field. For singular or non-compact varieties this motivates us to consider the "virtual Hodge numbers" encoded by the "Hodge-Deligne polynomial", a refinement of the topological Euler characteristic. I will then discuss the computation of Hodge-Deligne polynomials for certain singular character varieties (i.e. moduli spaces of flat connections).
Sub-Riemannian geometries occur in several areas of pure and applied mathematics, including harmonic analysis, PDEs, control theory, metric geometry, geometric group theory, and neurobiology. We introduce sub-Riemannian manifolds and give some examples. Therefore we discuss some of the open problems, and in particular we focus on the Sard Theorem for the endpoint map, which is related to the study of length minimizers. Finally, we consider some recent results obtained in collaboration with E. Le Donne, R. Montgomery, P. Pansu and D. Vittone.
Using an earlier result, joint with Quillen, I will formulate a gap labelling conjecture for magnetic Schrodinger operators with smooth aperiodic potentials on Euclidean space. Results in low dimensions will be given, and the formulation of the same problem for certain non-Euclidean spaces will be given if time permits. This is ongoing joint work with Moulay Benameur.
In this talk, I will present recent results, join with Tien-Cuong Dinh and Viet-Anh Nguyen, on counting periodic points of plane Cremona maps (i.e. birational maps of \(\mathbb{P}^2\)). The tools used include a Lefschetz fixed point formula of Saito, Iwasaki and Uehara for birational maps of surface whose fixed point set may contain curves; a bound on the arithmetic genus of curves of periodic points by Diller, Jackson and Sommerse; a result by Diller, Dujardin and Guedj on invariant (1,1) currents of meromorphic maps of compact Kahler surfaces; and a theory developed recently by Dinh and Sibony for non proper intersections of varieties. Among new results in the paper, we give a complete characterisation of when two positive closed (1,1) currents on a compact Kahler surface behave nicely in the view of Dinh and Sibony's theory, even if their wedge intersection may not be well-defined with respect to the classical pluripotential theory. Time allows, I will present some generalisations to meromorphic maps (including an upper bound for the number of isolated periodic points which is sometimes overlooked in the literature) and open questions.
Consider a function from the circle to itself such that the derivative is greater than one at every point. Examples are maps of the form \(f(x) = mx\) for integers \(m > 1\). In some sense, these are the only possible examples. This fact and the corresponding question for maps on higher dimensional manifolds was a major motivation for Gromov to develop pioneering results in the field of geometric group theory. In this talk, I'll give an overview of this and other results relating dynamical systems to the geometry of the manifolds on which they act and (time permitting) talk about my own work in the area.
I will describe new joint work with Franc Forstneric (arXiv:1602.01529). This work brings together four diverse topics from differential geometry, holomorphic geometry, and topology; namely the theory of minimal surfaces, Oka theory, convex integration theory, and the theory of absolute neighborhood retracts. Our goal is to determine the rough shape of several infinite-dimensional spaces of maps of geometric interest. It turns out that they all have the same rough shape.
Orientifold string theories are quantum field theories based on the geometry of a space with an involution. T-dualities are certain relationships between such theories that look different on the surface but give rise to the same observable physics. In this talk I will not assume any knowledge of physics but will concentrate on the associated geometry, in the case where the underlying space is a (complex) elliptic curve and the involution is either holomorphic or anti-holomorphic. The results blend algebraic topology and algebraic geometry. This is mostly joint work with Chuck Doran and Stefan Mendez-Diez.
For an elliptic operator on a compact manifold acted on by a compact Lie group, the Atiyah-Segal-Singer fixed point formula expresses its equivariant index in terms of data on fixed point sets of group elements. This can for example be used to prove Weyl's character formula. We extend the definition of the equivariant index to noncompact manifolds, and prove a generalisation of the Atiyah-Segal-Singer formula, for group elements with compact fixed point sets. In one example, this leads to a relation with characters of discrete series representations of semisimple Lie groups. (This is joint work with Hang Wang.)
For every integer \(n>1\) we construct a complex manifold of dimension n which is exhausted by an increasing sequence of biholomorphic images of \(\mathbb{C}^n\) (i.e., a long \(\mathbb{C}^n\)), but it does not admit any nonconstant holomorphic functions. We also introduce new biholomorphic invariants of a complex manifold, the stable core and the strongly stable core, and we prove that every compact strongly pseudoconvex and polynomially convex domain \(B\) in \(\mathbb{C}^n\)n is the strongly stable core of a long \(\mathbb{C}^n\); in particular, non-equivalent domains give rise to non-equivalent long \(\mathbb{C}^n\)'s. Thus, for any \(n>1\) there exist uncountably many pairwise non-equivalent long \(\mathbb{C}^n\)'s. These results answer several long standing open questions. (Joint work with Luka Boc Thaler.)
In this talk, I construct prequantum line bundles on Hitchin's moduli spaces of orientable and non-orientable surfaces and study the geometric quantisation and quantisation via branes by complexification of the moduli spaces.
The spectral ball is defined as the set of complex n by n matrices whose eigenvalues are all less than 1 in absolute value. Its group of holomorphic automorphisms has been studied over many decades in several papers and a precise conjecture about its structure has been formulated. In dimension 2 this conjecture was recently disproved by Kosinski. We not only disprove the conjecture in all dimensions but also give the best possible description of the automorphism group. Namely we explain how the invariant theoretic quotient map divides the automorphism group of the spectral ball into a finite dimensional part of symmetries which lift from the quotient and an infinite dimensional part which leaves the fibration invariant. We prove a precise statement as to how hopelessly huge this latter part is. This is joint work with R. Andrist.
Let \(G\) be a reductive complex Lie group (e.g., \(\mathrm{SL}(n,\mathbb{C}))\) and let \(X\) and \(Y\) be Stein manifolds (closed complex submanifolds of some \(\mathbb{C}^n\)). Suppose that \(G\) acts freely on \(X\) and \(Y\). Then there are quotient Stein manifolds \(X/G\) and \(Y/G\) and quotient mappings \(p_X:X\to X/G\) and \(p_Y: Y\to Y/G\) such that \(X\) and \(Y\) are principal \(G\)-bundles over \(X/G\) and \(Y/G\). Let us suppose that \(Q=X/G \cong Y/G\) so that \(X\) and \(Y\) have the same quotient \(Q\). A map \(\Phi: X\to Y\) of principal bundles (over \(Q\)) is simply an equivariant continuous map commuting with the projections. That is, \(\Phi(gx)=g \Phi(x)\) for all \(g\) in \(G\) and \(x\) in \(X\), and \(p_X=p_Y\circ \Phi\). The famous Oka Principle of Grauert says that any \(\Phi\) as above embeds in a continuous family \(\Phi_t: X \to Y\), \(t \in [0,1]\), where \(\Phi_0=\Phi\), all the \(\Phi_t\) satisfy the same conditions as \(\Phi\) does and \(\Phi_1\) is holomorphic. This is rather amazing. We consider the case where \(G\) does not necessarily act freely on \(X\) and \(Y\). There is still a notion of quotient and quotient mappings \(p_X: X\to X/\hspace{-1mm}/ G\) and \(p_Y: Y\to Y/\hspace{-1mm}/G\) where \(X/\hspace{-1mm}/G\) and \(Y/\hspace{-1mm}/G\) are now Stein spaces and parameterize the closed \(G\)-orbits in \(X\) and \(Y\). We assume that \(Q\cong X/\hspace{-1mm}/G\cong Y/\hspace{-1mm}/G\) and that we have a continuous equivariant \(\Phi\) such that \(p_X=p_Y \circ \Phi\). We find conditions under which \(\Phi\) embeds into a continuous family \(\Phi_t\) such that \(\Phi_1\) is holomorphic. We give an application to the Linearization Problem. Let \(G\) act holomorphically on \(\mathbb{C}^n\). When is there a biholomorphic map \(\Phi:\mathbb{C}^n \to \mathbb{C}^n\) such that \(\Phi^{-1} \circ g \circ \Phi \in \mathrm{GL}(n,\mathbb{C})\) for all \(g\) in \(G\)? We find a condition which is necessary and sufficient for "most" \(G\)-actions. This is joint work with F. Kutzschebauch and F. Larusson.