Noncommutative Geometry

Speaker: Travis Ens (Western)

"NCG Learning Seminar: Feynman's diagrams and Feynman's theorem (4)"

Time: 14:30

Room: MC 107

Geometry and Topology

Speaker: Steven Rayan (University of Toronto)

"Generalized holomorphic bundles on ordinary complex surfaces"

Time: 15:30

Room: MC 108

Generalized holomorphic bundles are a feature of Hitchin's generalized geometry. On an ordinary complex manifold, a generalized holomorphic bundle is not necessarily a holomorphic bundle. More generally, it is a kind of Higgs bundle -- sometimes called a co-Higgs bundle. I will discuss issues regarding stability and integrability for generalized holomorphic bundles over ordinary complex surfaces. In particular, I will construct a sequence of families of stable, integrable rank-2 generalized holomorphic bundles on CP^2, using the classical Schwarzenberger construction of ordinary holomorphic bundles.

Dept Oral Exam

Speaker: Claudio Quadrelli (Western)

"p-rigid fileds - a high cliff on the p-Galois see"

Time: 11:00

Room: MC 106

I plan to discuss my recent joint work with S. Chebolu and J. Minac. Let p be an odd prime and assume that a primitive p-th root of unity is in a field F. Then F is said to be p-rigid if only those cyclic algebras are split which are split for trivial reasons. I will present new characterizations of such fields and their Galois groups, which come from a more group-theoretical and cohomological approach. Our work extends, illustrates and simplifies some previous results and provides a new direct foundation of rigid fields which does not rely on valuation techniques. This work shows in fact how this new cohomological approach on maximal p-extensions of fields can be powerful, especially after the proof of the Milnor-Bloch-Kato conjecture.

Analysis Seminar

Speaker: Blagovest Sendov (Bulgarian Academy of Sciences)

"Hausdorff Approximations"

Time: 15:30

Room: MC 108

Let $A$ be a functional space of high or infinite dimension, $r(f,g);\; f,g\in A$ be a metric defined on $A$ and $\PP_n\subset A$ be an $n$-dimensional subset of $A$. The main goal of Approximation Theory, which is a theoretical basis for Numerical analysis and Numerical methods, is for given $f\in A$ to find a $p\in \PP_n$, such that $r(f,p)$ is as small as possible. Hausdorff Approximation (see \cite{BS}) is a part of Approximation Theory, in which to every function $f\in A$ corresponds a closed and bounded point set $\bar{f}$, and the distance between two functions $f,g\in A$ is defined as the Hausdorff distance between $\bar{f}$ and $\bar{g}$. An important fact is that the Hausdorff distance is not derived from a norm.

In this lecture, we underline the specifics of Hausdorff Approximation and formulate the most interesting results.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Applications of the Atiyah-Singer index theorem 2: Hirzebruch signature theorem (continued)"

Time: 14:30

Room: MC 107

After giving the final details of the heat equation proof, I shall give some applications. Most notably Hirzebruch's Signature theorem and the Riemann-Roch theorem for compact complex manifolds.

Noncommutative Geometry

Speaker: Travis Ens (Western)

"NCG Learning Seminar: Loop expansion of Feynman integrals, 1-particle irreducible graphs, and Cayley's tree formula"

Time: 14:30

Room: MC 107

Algebra Seminar

Speaker: Jessie Yang (McMaster)

"Initial ideals and tropical Severi varieties"

Time: 15:30

Room: MC 108

Tropical geometry is a systematic development of the fundamental concept, "degenerations". In this talk, I will make this statement precise in the algebraic view point, namely ''Initial ideals".

We apply the tropical approach to the classical objects in algebraic geometry, "Severi varieties". Severi varieties are spaces whose points correspond to the plane curves with a given number of nodal singular points. In this tropical approach, we can obtain purely combinatorial results on Severi varieties which involve subdivisions of polygons.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Applications of the Atiyah-Singer index theorem 3: Hirzebruch signature theorem (continued)"

Time: 14:30

Room: MC 107

I shall finish proof of the Hirzebruch signature theorem today, using index theorem. I shall give a few elementary applications, including divisibility results for Pontryagin numbers and a proof of the fact that CP^2 does not admit any spin structure. On Friday we shall see a more dramatic application in the talk by Mincong.

Colloquium

Speaker: Arturo Pianzola (University of Alberta)

"Why the cylinder is a straight line"

Time: 15:30

Room: MC 108

Why the cylinder is a straight line (thoughts on a modern interpretation of affine Kac-Moody Lie algebras)

Noncommutative Geometry

Speaker: Mingcong Zeng (Western)

"NCG Learning Seminar: An exotic differential structure on $S^7$"

Time: 10:30

Room: MC 107

One interesting application of Hirzebruch signature theorem is the construction of exotic differential structure on $S^7$. In this talk I will first show the construction of the exotic $S^7$, which is the sphere bundle of the 4 dimensional vector bundle over $S^4$ by using of the first Pontrjagin class and Euler class. Then for proving it has an exotic differential structure, we use Hirzebruch signature theorem to construct a invariant and compute this invariant for standard $S^7$ and our sphere bundle to see they are different.

Noncommutative Geometry

Speaker: Travis Ens (Western)

"NCG Learning Seminar: Path Integrals in Quantum Mechanics"

Time: 14:30

Room: MC 107

Using the theorems we have proven for finite dimensional integrals as motivation, I will define the Euclidean correlation functions for a quantum mechanical particle moving in an arbitrary smooth potential in terms of a sum over graphs and give a derivation of the Feynman rules for this simple system.

Analysis Seminar

Speaker: Adam Coffman (Indiana University - Purdue University Fort Wayne)

"Weighted projective spaces and a generalization of Eves' Theorem"

Time: 15:30

Room: MC 108

The cross-ratio is an interesting quantity in elementary geometry because it is invariant under projective transformations. I will propose a new generalization of the cross-ratio, although showing whether the new expression gives more information than previously known invariants requires an analysis of rational functions on real and complex weighted projective spaces. This talk is based on an article appearing soon in the Journal of Mathematical Imaging and Vision, and it will be accessible to students.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"A path integral proof of the Atiyah-Singer index theorem for Dirac operators"

Time: 14:30

Room: MC 107

This heuristic `physical proof' is due to E. Witten from 1980's and motivated the later rigorous heat equation proofs. I shall first review the Feynman path integral formalism for supersymmetric quantum mechanics. This formalism will next be applied to the Hamiltonian defined by the Dirac operator of a spin manifold, and after some non-trivial manipulations within the path integral, will lead to a proof of the index formula.I shall recall all needed background material from physics and geometry.

Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western)

"NCG Learning Seminar: Applications of the Atiyah-Singer Index theorem 3: the Hirzebruch-Riemann-Roch Theorem"

Time: 10:30

Room: MC 107

Following the previous talks on the Atiyah-Singer index theorem by Masoud,we will prove another important special case, namely the Hirzebruch-Riemann-Roch theorem. This theorem gives the holomorphic Euler characteristic of a holomorphic vector bundle over a compact Kähler manifold in terms of the Todd class of the manifold and the Chern character of the vector bundle. It will be shown how in the case of a holomorphic line bundle over a Riemann surface this reduces to the classical Riemann-Roch theorem.

Noncommutative Geometry

Speaker: Travis Ens (Western)

"NCG Learning Seminar: Path Integrals in Quantum Mechanics (2)"

Time: 14:30

Room: MC 107

Using the theorems we have proven for finite dimensional integrals as motivation, I will define the Euclidean correlation functions for a quantum mechanical particle moving in an arbitrary smooth potential in terms of a sum over graphs and give a derivation of the Feynman rules for this simple system.

Analysis Seminar

Speaker: Purvi Gupta (University of Michigan)

"Some generalizations of Hartogs' lemma on analytic continuation"

Time: 15:30

Room: MC 108

It is well known that there exist domains in C^n, n>1, such that all functions holomorphic therein extend holomorphically past the boundary. In this talk, we shall examine this phenomenon for certain refinements of the fundamental example of Hartogs. We shall look at a generalization of Hartogs' construction discovered by E .M. Chirka. Finally, we shall provide a partial answer to a related question raised by Chirka. There will be plenty of pictures, and very little familiarity with several complex variables will be required.

Colloquium

Speaker: Mahir Can (Tulane University)

"Orbits of a solvable group"

Time: 15:30

Room: MC 108

Our purpose in this general audience talk is twofold. First is to explain why solvable groups are quite necessary for studying geometry and representation theory. Second is to report on some recent progress of ours on the combinatorics of the orbits of a solvable subgroup of SL(n) acting on certain symmetric spaces and on their compactifications.

Noncommutative Geometry

Speaker: Alan Lai (Caltech)

"Spectral Action on $SU(2)$"

Time: 11:00

Room: MC 107

On a compact Lie group, there exists a 1-paramemter family of Dirac operators which interpolates the geometric Dirac operator (Levi-Civita), algebraic Dirac operator (cubic of Kostant), and the trivial Dirac operator (used in LQG). The spectral action of this family of operators is computed for $SU(2)$.

Algebra Seminar

Speaker: Detlev Hoffmann (Dortmund)

"Sums of squares in commutative rings"

Time: 14:30

Room: MC 108

Sums of squares in rings have been studied by numerous authors in the past. Typical questions are: Which elements in a ring can be written as sums of squares? If an element in a ring can be represented as a sum of squares, how many squares are needed for such a representations. We study these questions for arbitrary commutative rings, in particular in the case where $-1$ can be written as a sum of $n$ squares for some positive integer $n$. Such rings are called rings of finite level at most $n$. We derive estimates in terms of $n$ for other invariants pertaining to sums of squares such as the sublevel and the Pythagoras number. We give some examples and pose some open questions. This is joint work with David Leep.

Noncommutative Geometry

Speaker: Travis Ens (Western)

"NCG Learning Seminar: Path Integrals in Quantum Mechanics (3)"

Time: 14:30

Room: MC 107

By transforming to momentum space, the integrals used to compute the Feynman weight of a graph can be simplified. After carrying out this process, I will compute the partition function of two simple systems, quantum mechanics on a circle and circle-valued quantum mechanics. Finally I will discuss how these methods easily generalize to quantum field theory in the case of a free scalar bosonic field.

Analysis Seminar

Speaker: Remus Floricel (University of Regina)

"Asymptotic properties of quasi-shift endomorphisms"

Time: 15:30

Room: MC 108

A quasi-shift endomorphism is a unital normal *-endomorphism acting on a von Neumann algebra, of which tail and fixed point algebras coincide. Our purpose, in this presentation, is to discuss several asymptotic characterizations of quasi-shifts associated with representations of Cuntz algebras. Joint work with T. Wood.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"A path integral proof of the Atiyah-Singer index theorem for Dirac operators"

Time: 14:30

Room: MC 107

I shall first derive the semi-classical approximation of Feynman path integrals and give several examples. This involves regularized determinants of differential operators in terms of spectral zeta functions. This idea will then be applied to a supersymmetric quantum system defined by the Dirac operator of a closed spin manifold and after some non-trivial algebraic manipulations will lead to a proof the Atiyah-Singer index theorem.

Dept Oral Exam

Speaker: Hadi Seyedinejad (Western)

""On degeneracies in the family of fibres of a holomorphic mapping""

Time: 16:00

Room: MC 108

Failure of some (important) properties of a holomorphic mapping manifest themselves as degeneracies in the family of fibres of the mapping. Among these properties, openness and flatness are our main object of interest. The first goal in my thesis is to develop criteria such that first, they effectively (i.e., computationally) detect such degeneracies in the family of fibres, and second, they are applicable to the case of mappings with singular targets. For flatness, no such algorithms that work in the general setting of a singular target were known before. We prove that a mapping (with locally irreducible target) is flat (open) if and only if the fibre above the origin of the pullback by the blowing-up is not an (isolated) irreducible component. Algebraically, this flatness criterion reduces to the following straightforward prescription: compute the the local ring of the pullback, and check to see if the exceptional divisor is not a zero divisor. The second goal is to characterize different modes of such degeneracies. We take an invariant, called verticality index, as a gauge which measures the level of non-openness for mappings. We obtain some results about verticality index, specially on its behaviour and computation over singular targets.

Algebra Seminar

Speaker: Jochen Gärtner (Heidelberg)

"The Fontaine-Mazur Conjecture and tamely ramified p-adic representations"

Time: 15:30

Room: MC 108

If $k$ is a number field, $G_k=\mathrm{Gal}(\overline{k}|k)$ its absolute Galois group and $p$ a prime number, $p$-adic Galois representations $\rho: G_k\to GL_n(\mathbb{Q}_p)$ naturally arise in algebraic geometry, coming from the action of $G_k$ on etale cohomology groups of varieties defined over $k$. Fontaine and Mazur make a fundamental conjecture giving a precise characterization of those $p$-adic representations 'coming from algebraic geometry' in the above sense. In this talk we discuss consequences of the Fontaine-Mazur Conjecture for representations which are unramified at primes above $p$. After recalling results due to N. Boston and K. Wingberg providing evidence for the conjecture in the unramified case, we report on recent work on tamely ramified pro-$p$-extensions of $\mathbb{Q}$ by J. Labute.

Noncommutative Geometry

Speaker: Travis Ens (Western)

"NCG Learning Seminar: Path Integrals in Quantum Mechanics (4)"

Time: 14:30

Room: MC 107

By transforming to momentum space, the integrals used to compute the Feynman weight of a graph can be simplified. After carrying out this process, I will compute the partition function of two simple systems, quantum mechanics on a circle and circle-valued quantum mechanics. Finally I will discuss how these methods easily generalize to quantum field theory in the case of a free scalar bosonic field.

Analysis Seminar

Speaker: Dayal Dharmasena (Syracuse University)

"Holomorphic Fundamental Semigroup of Riemann Domains"

Time: 15:30

Room: MC 108

Let $(W,\Pi)$ be a Riemann domain over a complex manifold $M$ and $w_0$ be a point in $W$. Let $\mathbb D$ be the unit disk in $\mathbb C$ and $\mathbb T=\partial\mathbb D$. Consider the space ${\mathcal S}_{1,w_0}({\overline {\mathbb D}},W,M)$ of continuous mappings $f$ of $\mathbb T$ into $W$ such that $f(1)=w_0$ and $\Pi\circ f$ extends to a holomorphic on $\mathbb D$ mapping $\hat f$. Mappings $f_0,f_1\in{\mathcal S}_{1,w_0}({\overline{\mathbb D}},W,M)$ are called {\it holomorphically homotopic or $h$-homotopic} if there is a continuous mapping $f_t$ of $[0,1]$ into ${\mathcal S}_{1,w_0}({\overline{\mathbb D}},W,M)$. Clearly, the $h$-homotopy is an equivalence relation and the equivalence class of $f\in{\mathcal S}_{1,w_0}({\overline{\mathbb D}},W,M)$ will be denoted by $[f]$ and the set of all equivalence classes by $\eta_1(W,M,w_0)$. \par There is a natural mapping $\iota_1:\,\eta_1(W,M,w_0)\to\pi_1(W,w_0)$ generated by assigning to $f\in{\mathcal S}_{1,w_0}({\overline{\mathbb D}},W,M)$ its restriction to $\mathbb T$. We introduce on $\eta_1(W,M,w_0)$ a binary operation $\star$ which induces on $\eta_1(W,M,w_0)$ a structure of a semigroup with unity and show that $\eta_1(W,M,w_0)$ is an algebraic biholomorphic invariant of Riemann domains. Moreover, $\iota_1([f_1]\star[f_2])=\iota_1([f_1])\cdot\iota_1([f_2])$, where $\cdot$ is the standard operation on $\pi_1(W,w_0)$. Then we establish standard properties of $\eta_1(W,M,w_0)$ and provide some examples. When $W$ is a finitely connected domain in $M=\mathbb C$ and $\Pi$ is the identity, we show that $\iota_1$ is an isomorphism of $\eta_1(W,M,w_0)$ onto the minimal subsemigroup of $\pi_1(W,w_0)$ containing holomorphic generators and invariant with respect to the inner automorphisms. In particular, we show for a general domain $W\subset\mathbb C$ that $[f_1]=[f_2]$ if and only if $\iota_1([f_1])=\iota_1([f_2])$. This is a joint work with Evgeny Poletsky.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Localization in equivariant cohomology and index formula"

Time: 14:30

Room: MC 107

The path integral formula for the index of the Dirac operator can be interpreted as a localization formula for U(1)-equivariant cohomology of the free loop space of the manifold. In this lecture I shall first recall the Cartan model of equivariant differential forms of a finite dimensional manifold and the localization formula of Berline-Vergne. We shall then see that the loop space analogue of this result will give the A hat genus. This can be regarded as the bosonic component of the index formula. The corresponding localization formula in the supersymmetric case gives the full index formula.

Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western)

"NCG Learning Seminar: Applications of the Atiyah-Singer Index theorem 4: the Hirzebruch-Riemann-Roch Theorem"

Time: 10:30

Room: MC 107

Following the previous talks on the Atiyah-Singer index theorem by Masoud, we will prove another important special case, namely the Hirzebruch-Riemann-Roch theorem. This theorem gives the holomorphic Euler characteristic of a holomorphic vector bundle over a compact Kähler manifold in terms of the Todd class of the manifold and the Chern character of the vector bundle. It will be shown how in the case of a holomorphic line bundle over a Riemann surface this reduces to the classical Riemann-Roch theorem.

Algebra Seminar

Speaker: David Riley (Western)

"On the behaviour of the Frobenius map in a noncommutative world"

Time: 14:30

Room: MC 108