Noncommutative Geometry

Speaker: (Western)

"Path Integrals 1"

Time: 11:00

Room: MC 108

Comprehensive Exam Presentation

Speaker: Dinesh Valluri (Western)

"Localization and Bott residue formula"

Time: 11:00

Room: MC 107

In this talk we give a brief introduction to Equivariant intersection theory and explain the localization formula. We will derive a version of Bott's residue formula using the localization theorem and compute an example to illustrate its use in enumerative geometry.

Dept Oral Exam

Speaker: Mitsuru Wilson (Western)

"Gauss-Bonnet-Chern Type Theorems for Noncommutative Spheres"

Time: 11:00

Room: MC 107

In noncommutative geometry, a framework in classical geometry need not have a trivial generalization. In my defence, I will introduce pseudo-Riemannian calculus of modules over noncommutative algebras in order to investigate as to what extent the differential geometry of classical Riemannian manifolds can be extended to a noncommutative setting. In this framework, it is possible to prove an analogue of the Levi-Civita theorem, which states that there exists at most one connection, which satisfies torsion-free condition and metric compatible condition on a given smooth manifold. More significantly, the corresponding curvature operator has the same symmetry properties as in the classical curvature tensors. We consider a pseudo-Riemannian calculus over the noncommutative 3-sphere and the noncommutative 4-sphere to explicitly determine the torsion-free and metric compatible connection, and compute its scalar curvature. Lastly, I will also prove a Gauss-Bonnet-Chern type theorem for the noncommutative 4-sphere, which is the main result of my main thesis.

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"Feynman-Kac formula 4"

Time: 13:00

Room: MC 108

Dept Oral Exam

Speaker: Ivan Kobyzev (Western)

"Algebraic stacks and equivariant K-theory"

Time: 11:00

Room: MC 107

We give a definition of a root stack and describe its most basic properties. We describe the algebraic K-theory of a root stack. Sufficient conditions for a quotient stack to be a root stack are given. When these 2 results are combined immediate applications to equivariant K-theory are obtained. For example, we extend the work of Ellingsgurd and Lonsted to higher dimensions and to higher K-groups.

Noncommutative Geometry

Speaker: (Western)

"Path Integrals 2"

Time: 11:00

Room: MC 106

Homotopy Theory

Speaker: Dan Christensen (Western)

"Computation and data structures in Coq"

Time: 13:30

Room: MC 107

I will give an overview of how to do computations in Coq, including non-trivial recursive computations. I will also describe data structures such as lists and trees, and show how they can be manipulated.

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"Feynman-Kac formula 5"

Time: 11:00

Room: MC 108

Comprehensive Exam Presentation

Speaker: Sergio Chaves (Western)

"Conjugation Spaces"

Time: 14:30

Room: MC 107

Let X be a topological space and \tau an involution of X; \tau induces an action of the group G = { id, \tau} on X. A particular case is when X is a complex manifold and \tau is the complex conjugation; there some algebraic relations between the rings H*(X), H*(X^G) and H*_G(X) occur, where X^G denotes the fixed point subspace and H*_G(X) the G-equivariant cohomology ring of X. Such relations motivate the definition of cohomology frame and allow us to generalize the notion of complex conjugation to another topological spaces. Therefore, we say that a Conjugation Space is a space with involution which admits a cohomology frame. In this talk the notion of conjugation space is introduced, and some properties and examples of such spaces are presented.

Dept Oral Exam

Speaker: Sajad Sadeghi (Western)

"On Logarithmic Sobolev Inequality For the Noncommutative Two Torus and the Scalar Curvature Formula For the Noncommutative Three Torus"

Time: 10:00

Room: MC 106

I will first give an overview of the noncommutative geometry. Then I will discuss

the classic Sobolev type inequalities and also the logarithmic Sobolev inequality and will compare them.

Moreover, one of the main results, which is the logarithmic Sobolev inequality on the noncommutative

two torus (NCT2), will be be proved for a class of elements in NCT2. Afterwards, I will introduce a family of

spectral triples, each of which represents a class of conformaly perturbed metrics on the noncommutative

three torus (NCT3) as a noncommutative compact spin manifold. Then using Connes' pseudodifferential

calculus, I will define the scalar curvature of NCT3 equipped with the mentioned conformaly perturbed

metrics and will compute it by an explicit formula.

Dept Oral Exam

Speaker: Josue Rosario-Ortega (Western)

"Moduli space and deformations of Special Lagrangian submanifods with edge singularities"

Time: 11:00

Room: MC 107

Given a Calabi-Yau manifold $(M,\omega, J, \Omega)$ of complex dimension $n$, a Special Lagrangian submanifold $L\subset M$ is a real $n$ dimensional submanifold that is a Lagrangian submanifold with respect to the symplectic structure $\omega$ and minimal with respect to the Calabi-Yau metric of the ambient space. In this talk we shall consider singular Special Lagrangian submanifolds in $\mathbb{C}^n$ and their deformation theory. The type of singularities we will consider are the so called edge singularities. These are non-isolated singularities and they are obtained by the process of "edgification" of conical singularities. The deformations of a Special Lagrangian submanifold are given by solutions of a non-linear PDE. We will apply techniques from elliptic PDEs on singular spaces to describe the moduli space of Special Lagrangian deformations with edge singularities. When the obstruction space of this PDE vanishes and by imposing boundary conditions we obtain that the moduli space is a smooth manifold of finite dimension and the deformations obtained are regular solutions of an elliptic boundary value problem (on a singular space) having conormal expansions near the singular set.

Noncommutative Geometry

Speaker: (Western)

"Path Integrals 3"

Time: 15:00

Room: MC 108

Noncommutative Geometry

Speaker: Shahab Azarfar (Western)

"Wiener measure and Feynman-Kac formula"

Time: 11:00

Room: MC 108

Comprehensive Exam Presentation

Speaker: Rui Dong (Western)

"Random Non-commutative Geometries and Matrix Integrals"

Time: 11:00

Room: MC 107

The finite real spectral triples can be classified up to unitary equivalence according to Krajewski diagrams, and if each data except the Dirac operator $D$ of a finite real spectral triple is fixed, which is called a "fermion space", then it is easy to show that the set $\mathcal{G}$ of all the Dirac operators over this fermion space forms a vector space. If $\mathcal{G}$ is enriched with some measure, then we can consider the integral over $\mathcal{G}$. Here I am going to consider only a special kind of finite real triple: the type $(p, q)$ fuzzy space. And I will try to compute the integral $\int_{\mathcal{G}}e^{-\mathrm{Tr}D^{2}}\mathrm{d}D$ for the easiest type $(1, 0)$ fuzzy space.

Geometry and Combinatorics

Speaker: Matthias Franz (Western)

"Equivariant cohomology of smooth toric varieties"

Time: 15:50

Room: MC 108

I will present Brion's proof that the equivariant cohomology of a (sufficiently) smooth toric variety is given by the piecewise polynomials on the associated fan or, equivalently, by the Stanley-Reisner ring of the fan. If time permits, I will discuss how to obtain the non-equivariant cohomology as a consequence, both in the equivariantly formal and the general case.

Noncommutative Geometry

Speaker: Shahab Azarfar (Western)

"Wiener measures and Feynman-Kac formula"

Time: 15:00

Room: MC 108

Comprehensive Exam Presentation

Speaker: Jasmin Omanovic (Western)

"Cohomological invariants, quadratic forms and central simple algebras with involution"

Time: 13:00

Room: MC 107

The study of quadratic forms seems far removed from the study of (central) simple algebras, and in general, this is indeed the case. However, if we assume the central simple algebra carries an involution (such as matrix algebras and quaternion algebras) then we have a different story. In this talk, we will study the relationship between algebras with an involution and quadratic forms (assuming characteristic is not 2). In particular, we will discuss the relevance of cohomological invariants as a classification tool in the study of quadratic forms, in hopes of trying to extend what we have learned to central simple algebras with involution.

Noncommutative Geometry

Speaker: (Western)

"Zeta regularized determinants"

Time: 11:00

Room: MC 108

I will show how to compute the zeta regualrized determinants of differential operators, notably Jacobi operators in differential geometry and the Sturm-Liouville operators on the line. Note: this talk was cancelled last week. So this is the first part.

Homotopy Theory

Speaker: Mitchell Riley (Western)

"(2)-Category Theory in HoTT"

Time: 13:30

Room: MC 107

In this talk I will describe the development of category theory within HoTT and what it means for a category to be univalent.

Geometry and Combinatorics

Speaker: Matthias Franz (Western)

"Equivariant cohomology of smooth toric varieties, II"

Time: 16:00

Room: MC 108

We continue by proving Brion's theorem identidying the equivariant cohomology of a smooth toric variety with the ring of piecewise polynomials on the associated fan. Then we compare piecewise polynomials to the Stanley-Reisner ring. We conclude with some remarks about non-equivariant cohomology.

Noncommutative Geometry

Speaker: (Western)

"Zeta regularized determinants II"

Time: 11:00

Room: MC 107