Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"NCG Learning Seminar: Spin Geometry (2)"

Time: 14:30 - 16:00

Room: MC 107

In this second talk we will discuss the idea of complexifying Clifford algebras and classifying them. We will give many examples of the Clifford algebra $Cl(s + t), s + t = n,$ on $R^n$ and see that these are actually matrices with entries from either $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. Our discussions continues as we look at Clifford modules, which are representations of a Clifford algebra, and Clifford bundles. When $M$ is a Riemannian manifold with a metric $g$, the Clifford bundle of $M$ is the Clifford bundle generated by the tangent bundle $TM$.

Geometry and Topology

Speaker: Graham Denham (Western)

"Duality properties for abelian covers"

Time: 15:30 - 16:30

Room: MC 108

In parallel with a classical definition due to Bieri and Eckmann, say an FP group G is an abelian duality group if $H^p(G,Z[G^{ab}])$ is zero except for a single integer $p=n$, in which case the cohomology group is torsion-free. We make an analogous definition for spaces. In contrast to the classical notion, the abelian duality property imposes some obvious constraints on the Betti numbers of abelian covers.

While related, the two notions are inequivalent: for example, surface groups of genus at least 2 are (Poincaré) duality groups, yet they are not abelian duality groups. On the other hand, using a result of Brady and Meier, we find that right-angled Artin groups are abelian duality groups if and only if they are duality groups: both properties are equivalent to the Cohen-Macaulay property for the presentation graph. Building on work of Davis, Januszkiewicz, Leary and Okun, hyperplane arrangement complements are both duality and abelian duality spaces. These results follow from a more general cohomological vanishing theorem, part of work in progress with Alex Suciu and Sergey Yuzvinsky.

Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western)

"NCG Learning Seminar: On the representations of Clifford algebras and spin groups"

Time: 14:30 - 16:00

Room: MC 107

One kind of representation of a spin group is obtained by restricting either a complex or real representation of the Clifford algebra to its spin group. It plays an important role in constructing generalized Dirac operators. In this talk, we will construct regular and spin representations of complex Clifford algebras. The former is a reducible representation given by Clifford multiplication on the exterior algebra and the latter is an irreducible representation on the exterior algebra of a complex polarization, also known as the polarized Fock space.

Colloquium

Speaker: Jaydeep Chipalkatti (University of Manitoba)

"The Hexagrammum Mysticum"

Time: 15:30 - 16:30

Room: MC 108

If a hexagon is inscribed in a conic, then the three points obtained by intersecting the opposite sides lie on a line. This is Pascal's theorem, first observed in 1639. By considering various pairs of sides obtained from the same six vertices, one obtains a collection of 60 such lines. This collection forms a highly intricate and symmetrical structure, usually called the 'hexagrammum mysticum'. I will explain some of the (myriad) properties of this structure, and the role of algebraic geometry and classical invariant theory in it. The pre-requisites will be kept rather modest, so as to ensure that the talk is widely accessible. 

Algebra Seminar

Speaker: Jaydeep Chipalkatti (University of Manitoba)

"On Hilbert covariants"

Time: 14:30 - 15:30

Room: MC 108

Consider a binary form

$ F = a_0 \, x_1^d + a_1 \, x_1^{d-1} \, x_2 + \dots + a_d \, x_2^d, \quad (a_i \in {\mathbf C}) $

of order $d$ in the variables $\{x_1,x_2\}$. Its Hessian is defined to be

$ \text{He} (F) = \frac{\partial^2 F}{\partial x_1^2} \frac{\partial^2 F}{\partial x_2^2} - \left(\frac{\partial^2 F}{\partial x_1 \partial x_2}\right)^2. $

It is classical that $F$ is the perfect $d$-th power of a linear form, if and only if $\text{He} (F)$ vanishes identically. Moreover, $\text{He}(F)$ is a covariant of $F$, in the sense that its construction commutes with a linear change of variables in $\{x_1,x_2\}$. Now assume that $d = r \, m$, and suppose we ask for a covariant whose vanishing is equivalent to $F$ being the perfect power of an order $r$ form. In 1885, Hilbert constructed such a covariant, to be denoted by $\mathcal{H}_{r,d}(F)$. In geometric terms, the variety of perfect powers of order $r$ forms defines a subvariety $X_r \subseteq {\mathbf P}^d$, and the coefficients of $\mathcal{H}_{r,d}$ give defining equations for this variety.

In this talk, I will outline a wholly different construction of this covariant, which leads to a generalisation called the G$\mathrm{\ddot{o}}$ttingen covariants. Moreover, we have the theorem that the ideal generated by the coefficients of the Hilbert covariant generates $X_r$ as a ${scheme}$, and not merely as a variety. This is joint work with Abdelmalek Abdesselam from the University of Virginia.

Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"NCG Learning Seminar: Isospectral and Nonisometric Domains in the Euclidean Plane"

Time: 14:30 - 16:00

Room: MC 107

In 1964, Milnor discovered flat tori in dimension 16 that are isospectral but not isometric. As amazing a result as this is, it still took about thirty years to construct isospectral plane domains that are not isometric. In this talk, I will review Sunada's method, as extended by Berard, to give an example of a pair of simply-connected nonisometric domains in the Euclidean plane that are both Dirichlet isospectral and Neumann isospectral.

Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"NCG Learning Seminar: Spin Geometry (3)"

Time: 14:30 - 15:20

Room: MC 107

We will write down a complete classification of all Clifford algebras of the form $Cl(n,0)$ and $Cl(0,n)$. From here, we will look at a theorem that relates periodicity of Clifford algebras with Bott periodicity in K-theory. Next, we will consider the Pauli-spin and Dirac matrices and see how they generate important algebras such as the real quaternion algebra, the complex quaternion algebra, the real Dirac algebra and higher order Clifford algebras. From here, we will start a discussion about the Dirac operator and Dirac equation, and subsequently introduce important examples such as the Pauli-Dirac and Dirac-Yukawa operators.

Geometry and Topology

Speaker: Paul Smith (University of Washington)

"Graded modules over path algebras of quivers."

Time: 15:30 - 16:30

Room: MC 108

We establish connections between various algebras and module categories that can be associated to a finite directed graph. These connections involve the singularity category for certain finite dimensional algebras, von Neumann regular algebras, Leavitt path algebras, $C^*$-graph algebras, AF algebras, some ideas from symbolic dynamics, and the space of Penrose tilings as a special case.

Analysis Seminar

Speaker: Hadi Seyedinejad (Western)

"Non-open complex analytic maps"

Time: 15:30 - 16:30

Room: MC 108

Fibres of a morphism between complex spaces form a family that encodes much information regarding the behaviour of the morphism. For example, knowing only about the variation of the topological dimension of the fibres suffices to determine whether a mapping is open or not (Remmert Open Mapping Theorem). This has lead us to an efficient algebraic method of testing for openness by means of the blow-up mapping, and successively, to a very efficient method of testing for flatness (joint work with Janusz Adamus). Apart from merely detecting non-openness, I am also trying to study different modes of being non-open for an analytic map, especially in the general setting of maps over a singular target.

Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western)

"NCG Learning Seminar: Spin Structures on Manifolds"

Time: 14:30 - 16:00

Room: MC 107

A spin structure on an oriented (Euclidean) vector bundle is a principal spin-bundle that is a non-trivial 2-fold covering of the oriented (orthonormal) frame bundle of $E$, denoted by $P_{SO}(E)$. An (oriented Riemannian) manifold is called spin if its tangent bundle has such a structure. It turns out that the existence of a spin structure has a topological obstruction -- namely, the (vannishing) of the second Stiefel-Whitney class of the manifold. In this talk, we will introduce spin structures and identify the obstruction for its existence in terms of the second Stiefel-Whitney class. Furthermore, some examples will be examined, including showing that for $n$-tori and for compact Riemann surfaces of genus $g$, there are exactly $2^n$ and $2^{2g}$ non-equivalent spin structures, respectively.

Algebra Seminar

Speaker: Claudio Quadrelli (Western and Milano-Bicocca)

"Rigid fields and $p$-Galois groups: easy solutions for different equations"

Time: 14:30 - 15:30

Room: MC 108

Let $p$ be an odd prime. A field $F$ is said to be $p$-rigid if certain conditions on the cyclic algebras constructed over $F$ are satisfied. $p$-rigid fields have been studied through the last decades. In this talk I will present the properties of $p$-rigid fields together with new characterizations of such fields and their $p$-Galois groups (proved in joint work with S. Chebolu and J. Minac). In particular, given a field $F$, it is possible to detect whether $F$ is $p$-rigid simply by small quotients of $G_F(p)$, or by the cohomological dimension of $G_F(p)$, or by the $\mathbb{F}_P$-cohomology ring of $G_F(p)$, where $G_F(p)$ is the maximal pro-$p$ Galois group of $F$. In this case it is also possible to describe completely and explicitly every $p$-extension of $F$ (in a rather nice way) and every $p$-Galois group of $F$. Our results extend (and simplify) some previous results obtained by R. Ware, A. Engler and J. Koenigsmann; and are related to some important work of I. Efrat, A. Topaz, and others; and last but not least, they provide a new point of view upon such topics.

Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"NCG Learning Seminar: Isospectral Plane Domains that are not Isometric (2)"

Time: 14:30 - 16:00

Room: MC 107

Because our construction is done via Riemannian orbifolds, we will continue from last week by discussing some important theory and examples related to the Riemannian geometry of orbifolds. In fact, we will see some very special examples that show not all orbifolds are constructed via a group action on a manifold. We will then look at some planar isospectral domains that were constructed in 1994 by Buser and Conway as a segue to proving our main theorem about isospectrality and nonisometry of the two plane domains constructed by Gordon, Webb and Wolpert.

Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"NCG Learning Seminar: Isospectral and Nonisometric Plane Domains (3)"

Time: 14:30 - 16:00

Room: MC 107

As Sunada originally thought of his famous theorem using zeta functions and trace formulae, we will outline how he proved that the almost conjugacy of subgroups implies isospectrality of the manifolds. We will then continue with some important theory and examples of Riemannian orbifolds and construct Buser's example from 1987 of two isospectral, nonisometric flat (not planar) two-dimensional manifolds with boundary, embedded in $\mathbb{R}^{3}$. Finally, we will construct the important planar domain examples by Gordon, Webb and Wolbert in 1992. In addition, we will provide explicit pictures of these domains and subsequent isometric plane domains as constructed by Buser and Conway in 1994.

Geometry and Topology

Speaker: Timo Schurg (University of Bonn)

"Derived Algebraic Cobordism"

Time: 15:30 - 16:30

Room: MC 108

I will introduce an extension of Levine and Morel's algebraic cobordism to (quasi-projective) derived schemes. This extension will have the additional utility of naturally encompassing virtual fundamental classes and virtual Gysin homomorphisms. Time permitting, I will discuss the isomorphism with a different (obvious) extension of algebraic cobordism to derived schemes that shows any quasi-smooth projective variety is cobordant to a smooth projective variety.

Analysis Seminar

Speaker: Wayne Grey (Western)

"Mixed-norm L^P spaces"

Time: 15:30 - 16:30

Room: MC 108

Mixed-norm L^P spaces were described by Benedek and Panzone in 1961, but are connected to Littlewood's earlier 4/3 inequality and recent work on the Bohnenblust-Hille inequality. Among these properties are mixed-norm versions of Holder's inequality and Minkowski's integral inequality, which can work together to simplify certain proofs. Minkowski's integral inequality already has a mixed-norm character, and its general mixed-norm extension (described by Fournier in 1987) allows embeddings among various mixed norm spaces on a product measure space. Much of the talk should be clear with a basic background in integration and conventional L^p spaces

Noncommutative Geometry

Speaker: Asghar Ghorbanppour (Western)

"NCG Learning Seminar: Dirac and Generalized Dirac Operators"

Time: 14:30 - 16:00

Room: MC 107

For any manifold $M$ with a spin structure $P_{spin}(X)$, we can define a canonical first order operator, known as the Dirac operator, acting on a spinor bundle. The spinor bundle is a bundle associated to the spin structure and the representation of spin group, coming from an irreducible representation of the Clifford algebra. The existence of an irreducible real Clifford module is equivalent to the existence of a spin structure on the manifold; however, we can always construct a Dirac bundle, which is (not necessarily irreducible) a $Cl(X)$-module with compatible metric and connection. Similarly, we may define a differential operator known as the generalized Dirac operator. In this talk, after some general discussion about Clifford bundles and Dirac bundles, we will focus on two important examples of the generalized Dirac operator: the de Rham complex and, in the $4k$-dimensional case, the signature complex.

Colloquium

Speaker: Eric Schippers (University of Manitoba)

"A correspondence between conformal field theory and Teichmuller theory"

Time: 15:30 - 16:30

Room: MC 108

Teichmuller space is a moduli space of Riemann surfaces, where two Riemann surfaces are equivalent if they are biholomorphic and are homotopically related in a certain sense. It can be thought of as the space of local deformations of Riemann surfaces. Conformal field theories are quantum mechanical or statistical field theories which are invariant under infinitesimal rotations and rescalings, and thus in two dimensions they are closely tied to complex analysis. A mathematical model of conformal field theory was sketched by Segal and Kontsevich. Attempts to realize this model rigorously has spawned a great deal of deep mathematics.

A certain moduli space in conformal field theory turns out to be the quotient of Teichmuller space by a discrete group action. This relation between the two moduli spaces leads to the solution of analytic problems in the rigorous formulation of conformal field theory, and new results in Teichmuller theory. In this talk, I will give a non-technical introduction to the ideas of Teichmuller theory, and sketch the correspondence between the moduli spaces. Joint work with David Radnell.

Algebra Seminar

Speaker: Nguyen Duy Tan (Western)

"Images of additive polynomials"

Time: 14:30 - 15:30

Room: MC 108

We study the image of an additive polynomial $f$ over a field $k$ of characteristic $p > 0$. We define the additive rank of $f$ over $k$ to be the smallest positive integer $r$ such that there exists an additive polynomial $g$ in $r$ variables with coefficients in $k$ which generates the same image as $f$ does. We show that over perfect fields the additive ranks of (non-zero) additive polynomials are always 1. We also show that for every positive integer $r$, there is some additive polynomial over a certain field with additive rank $r$.

Noncommutative Geometry

Speaker: Mingcong Zeng (Western)

"NCG Learning Seminar: A proof of Bott periodicity theorem (1)"

Time: 14:30 - 16:00

Room: MC 107

In this talk first I will give some examples of vector bundles on $S^2$ which appears to be very important in complex topological K-theory and show how to imagine them. Then I will define clutching map of vector bundles on $S^n$ which will be used to prove the Bott periodicity. Then we are prepared to give a definition of K-group and ring structure on it. At last is the statement of Bott periodicity which I will prove in the next week.

Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"NCG Learning Seminar: Isospectral and Nonisometric Plane Domains (4)"

Time: 14:30 - 16:00

Room: MC 107

We will construct Buser's example of two Schreier graphs that are isospectral but not isomprphic. The proof of this has a wonderful connection to representation theory, and some useful pre-trace formulae will be reviewed. We will then start the construction of isospectral, non-isometric planar domains that will be concluded in part 5 of this series of talks.

Geometry and Topology

Speaker: Derek Krepski (Western)

"Geometric quantization and group-valued moment maps"

Time: 15:30 - 16:30

Room: MC 108

Originally aimed at understanding the relationship between classical and quantum mechanics, geometric quantization is a construction whose ideas originated in representation theory, in the work of Kirillov, Kostant and Souriau during the late 1960's. From the symplectic geometry perspective, geometric quantization produces representations of compact Lie groups starting from `geometric' (i.e. Hamiltonian) group actions of such Lie groups. Some ideas surrounding this construction will be discussed, including their adaptation to the theory of 'group-valued' moment maps, which is a finite-dimensional model of the theory of Hamiltonian loop group actions. Applications in this context (re)produce so-called Verlinde formulas of conformal field theory for simply connected compact Lie groups. For non-simply connected Lie groups, Verlinde formulas have been conjectured, and this approach verifies the conjectured formulas for $G=SO(3)$.

Analysis Seminar

Speaker: Dusty Grundmeier (University of Michigan)

"Rigidity of CR Mappings for Hyperquadrics"

Time: 15:30 - 16:30

Room: MC 108

This is joint work with Jiri Lebl and Liz Vivas. We prove that the rank of a Hermitian form on the space of holomorphic polynomials can be bounded by a constant depending only on the maximum rank of the form restricted to affine manifolds. As an application we prove a result along the lines of the Baouendi-Huang and Baouendi-Ebenfelt-Huang rigidity theorems for CR mappings between hyperquadrics. If we have a real-analytic CR mapping of a hyperquadric not equivalent to a sphere to another hyperquadric Q(A,B), then either the image of the mapping is contained in a complex affine subspace or A is bounded by a constant depending only on B.

Noncommutative Geometry

Speaker: Asghar Ghorbanppour (Western)

"NCG Learning Seminar: Spin^c Structure and Dirac Operators"

Time: 14:30 - 16:00

Room: MC 107

The irreducible (real) Clifford modules play a very important role in the theory of Dirac operators. The obstruction for the existence of such a module in real and complex case is different. For the real one, vector bundle should have vanishing second Stiefel-Whitney class, however, in the complex case the second Stiefel-Whitney class only needs to be mod 2 reduction of an integral class, equivalently, the vector bundle admits a $spin^c$ structure. In this talk we will examine the obstruction for the existence of $spin^c$ structure, then the construction of complex spinor bundle and the $spin^c$ connection on it and finally Dirac operator on the spinor fields will be discussed.

Colloquium

Speaker: Oliver Roendigs (Osnabrueck)

"The Grothedieck ring of varieties"

Time: 15:30 - 16:30

Room: MC 108

The Grothendieck ring of varieties over a field is a bookkeeping device for invariants of varieties which preserve the relation [X] = [Z]+[X-Z] whenever Z is a closed subvariety of X. Examples of such invariants include counting points if the field in question is finite, or the topological Euler characteristic if the field is the complex numbers. After introducing the Grothendieck ring and some invariants, I will discuss a certain invariant which involves the A^1-homotopy type of Morel and Voevodsky.

Algebra Seminar

Speaker: Jochen G$\mathrm{\ddot{a}}$rtner (Heidelberg)

"Higher Massey products in the cohomology of pro-$p$-extensions"

Time: 10:30 - 11:30

Room: MC 108

What do the 'picture hanging problem' and 'Borromean rings' have in common? Their solution can be described by Milnor invariants in link theory, or equivalently by higher cohomological Massey products.

As noticed by B. Mazur, M. Morishita et al, there is a remarkable analogy between the theory of links and pro-$p$-extensions of number fields with ramification restricted to a finite set of primes. We discuss this analogy and give an arithmetic interpretation of Massey products in low degrees. It turns out that certain symmetry relations in the topological world carry over to number theory in special cases only.

We report on the work on applications of higher Massey products in order to construct so-called mild pro-$p$-groups and investigate recent progress in the theory of tamely ramified pro-$p$-extensions by J. Labute and A. Schmidt.

Algebra Seminar

Speaker: Christian Maire (Universit$\mathrm{\acute{e}}$ de Franche-Comt$\mathrm{\acute{e}}$)

"Example of arithmetic mild pro-$p$-groups"

Time: 14:30 - 15:30

Room: MC 108

In this talk, we will show how to obtain mild pro-$p$-groups in the arithmetic context.

Noncommutative Geometry

Speaker: Mingcong Zeng (Western)

"NCG Learning Seminar: A proof of Bott periodicity theorem (2)"

Time: 14:30 - 16:00

Room: MC 107

This talk is dedicated to the proof of Bott periodicity. First we generalize the clutching function to vector bundles over $X \times S^2$, then we can simplify the clutching function, first to a Laurent polynomial, then to a polynomial, finally to a linear function. And by the discussion on the linear clutching function, we can finally decompose it into a vector bundle with trivial clutching function and another one with clutching function $z$. Finally, we can construct a inverse by the simplified clutching function for the external product to prove that it is an isomorphism.