Comprehensive Exam Presentation

Speaker: Shahab Azarfar (Western)

"A Report on Quanta of Geometry"

Time: 16:00

Room: MC 108

We try to investigate a generalization of the Heisenberg commutation re- lation [p; q] = ô€€€i}, introduced by Chamseddine, Connes and Mukhanov as \the one-sided and the two-sided quantization equations", which captures the geometry. The momentum variable p is encoded by the Dirac operator and the analogue of the position variable q is the Feynman slash of real scalar elds over a closed even-dimensional spin manifold. Existence of a solution of the one-sided equation implies that the manifold decomposes into a dis- connected sum of spheres of unit volume which represent quanta of geometry. The two-sided equation, as the rened version of the one-sided equation by involving the real structure on a spin manifold, implies the quantization of the volume of the spin manifold.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Yang-Mills equations III: BPST instantons"

Time: 11:30

Room: MC 107

I shall give a detailed construction of the BPST instantons.

Homotopy Theory

Speaker: Mitchell Riley (Western)

"Homotopy n-Types (part 2)"

Time: 13:30

Room: MC 107

Continuing the previous talk, I will present some properties of (-1)- and 0-types; propositions and sets.

Analysis Seminar

Speaker: Octavian Mitrea (Western)

"Polynomial Convexity"

Time: 15:30

Room: MC 107

We introduce polynomially convex subsets of the n-dimensional Euclidean complex space and expose some of their key properties. We discuss the presence of an analytic structure in the polynomially convex hull of a compact set, Rossi's local maximum principle and Oka's characterization theorem. This talk is given in fulfillment of the requirements for Part II of the PhD comprehensive examination.

Comprehensive Exam Presentation

Speaker: Octavian Mitrea (Western)

"Polynomial Convexity"

Time: 15:30

Room: MC 107

We introduce polynomially convex subsets of the n-dimensional Euclidean complex space and expose some of their key properties. We discuss the presence of an analytic structure in the polynomially convex hull of a compact set, Rossi's local maximum principle and Oka's characterization theorem.

Noncommutative Geometry

Speaker: Shahab Azarfar (Western)

"Volume Quantization from Spin Geometry II"

Time: 11:00

Room: MC 107

We try to investigate a generalization of the Heisenberg commutation relation ${[p,q]=-i \hslash}$, introduced by Chamseddine, Connes and Mukhanov as ``the one-sided and the two-sided quantization equations'', which captures the geometry. The momentum variable $p$ is encoded by the Dirac operator and the analogue of the position variable $q$ is the Feynman slash of real scalar fields over a closed even-dimensional spin manifold. Existence of a solution of the one-sided equation implies that the manifold decomposes into a disconnected sum of spheres of unit volume which represent quanta of geometry. The two-sided equation, as the refined version of the one-sided equation by involving the real structure on a spin manifold, implies the quantization of the volume of the spin manifold.

Graduate Seminar

Speaker: Mitchell Riley (Western)

"Combinatorial Games"

Time: 13:30

Room: MC 108

In this talk we will introduce the theory combinatorial games, a simple mathematical structure with incredibly rich algebraic properties. As well as containing all real numbers, the class of games contains all ordinals, a collection of infinitesimals and plenty in between. The study of combinatorial games can be applied directly to the analysis of actual strategy games, including Chess and Go. If time permits, we will use the techniques of the talk to analyse a curious chess endgame.

Comprehensive Exam Presentation

Speaker: Ahmed Ashraf (Western)

"Characterizing f-vector"

Time: 14:30

Room: MC 108

Given a d-dimensional convex polytope, its k-th face number is the number of (k 􀀀-1)-dimensional faces it has. The f-vector of a polytope is the sequence of its face numbers. Beside Euler's formula, these numbers satisfy further equalities and inequalities. Characterization of f-vector of d-dimensional convex polytope is already known for d ≤ 3 . For d ≥ 3, we do not have a complete answer, but g-theorem gives us a characterization for simplicial (and dually simple) case. Here we review g-theorem and its various proofs.

Geometry and Topology

Speaker: Mattia Talpo (UBC)

"Parabolic sheaves, root stacks and the Kato-Nakayama space"

Time: 15:30

Room: MC 107

Parabolic bundles on a punctured Riemann surface were introduced by Mehta and Seshadri in the ‘80s, in relation to unitary representations of its topological fundamental group. Their definition was generalized, in several steps, to a definition over an arbitrary logarithmic scheme due to Borne and Vistoli, who also proved a correspondence with sheaves on stacks of roots. I will review these constructions, and push them further to the case of an “infinite” version of the root stacks. Towards the end I will discuss a comparison result (for log schemes over the complex numbers) between this “infinite root stack” and the so-called Kato-Nakayama space, and hint at some work in progress about relating sheaves on this latter space to parabolic sheaves with arbitrary real weights.

Noncommutative Geometry

Speaker: (Western)

"Examples of Yang-Mills Theories"

Time: 11:30

Room: MC 107

We shall give several examples of Yang-Mills Theories.

Homotopy Theory

Speaker: Cihan Okay (Western)

"Introduction to the Coq proof assistant"

Time: 13:30

Room: MC 107

This talk will be a basic interactive introduction to Coq. I will start with simple operations on natural numbers, then move on to functions on arbitrary types, and illustrate how to prove a logical statement in computer.

Analysis Seminar

Speaker: Octavian Mitrea (Western)

"Open Whitney umbrellas are locally polynomially convex"

Time: 15:30

Room: MC 107

We present the following theorem: A totally real smooth surface in $\mathbb{C}^2$ with an open Whitney umbrella at the origin, is locally polynomially convex near the singular point. This is a natural generalization of a result of Shafikov and Sukhov that addresses the same problem, but in the generic case. Our theorem establishes polynomial convexity in full generality in this context. This is joint work with Rasul Shafikov.

Colloquium

Speaker: Kumar Murty (Toronto)

"Automorphy and the Sato-Tate conjecture"

Time: 15:30

Room: MC 107

We shall give a motivated description of prime number theorems in general and the Sato-Tate conjecture in particular, and describe some recent joint work with Ram Murty.

Noncommutative Geometry

Speaker: Shahab Azarfar (Western)

"Volume Quantization from Spin Geometry (III)"

Time: 12:30

Room: MC 106

We try to investigate a generalization of the Heisenberg commutation relation ${[p,q]=-i \hslash}$, introduced by Chamseddine, Connes and Mukhanov as ``the one-sided and the two-sided quantization equations'', which captures the geometry. The momentum variable $p$ is encoded by the Dirac operator and the analogue of the position variable $q$ is the Feynman slash of real scalar fields over a closed even-dimensional spin manifold. Existence of a solution of the one-sided equation implies that the manifold decomposes into a disconnected sum of spheres of unit volume which represent quanta of geometry. The two-sided equation, as the refined version of the one-sided equation by involving the real structure on a spin manifold, implies the quantization of the volume of the spin manifold.

Algebra Seminar

Speaker: Adam Chapman (Michigan State University)

"Linkage of $p$-algebras of prime degree"

Time: 16:00

Room: MC 107

Quaternion algebras contain quadratic field extensions of the center. Given two algebras, a natural question to ask is whether they share a common field extension. This gives us an idea of how closely related those algebras are to one another. If the center is of characteristic 2 then those extensions divide into two types - the separable type and the inseparable type. It is known that if two quaternion algebras share an inseparable field extension then they also share a separable field extension and that the converse is not true. We shall discuss this fact and its generalization to $p$-algebras of arbitrary prime degree.

Noncommutative Geometry

Speaker: (Western)

"Examples of Yang-Mills Theories II"

Time: 11:30

Room: MC 107

We shall give several examples of Yang-Mills Theories.

Algebra Seminar

Speaker: Reading Week

"(No Seminar)"

Time: 16:00

Room: MC 107

Geometry and Topology

Speaker: Lennart Meier (Bonn)

"Homotopy theory of relative categories"

Time: 15:30

Room: MC 107

Relative categories are maybe the most naive model for abstract homotopy theory (just categories with a subcategory of "weak equivalences"). Barwick and Kan showed that the category of relative categories has a model structure, Quillen equivalent to the Joyal model structure on simplicial set, which has infinity-categories as fibrant objects. We will show that model categories define fibrant relative categories and also discuss other aspects of the homotopy theory of relative categories.

Noncommutative Geometry

Speaker: (Western)

"Examples of Yang-Mills Theories III"

Time: 11:30

Room: MC 107

We shall give several examples of Yang-Mills Theories.

Homotopy Theory

Speaker: Marco Vergura (Western)

"Equivalences and the Univalence Axiom (part 1)"

Time: 13:30

Room: MC 107

We introduce Voevodsky's Univalence Axiom and see some of its consequences in type theory. We also start studying various definition of type-theoretic equivalences.

Geometry and Combinatorics

Speaker: Sergio Chaves (Western)

"The Borel construction"

Time: 16:00

Room: MC 105C

Let $X$ be a topological space with an action of a topological group $G$. We want to relate to $X$ an algebraic object that reflects both the topology and the action of the group. The first candidate is the cohomology ring $H^*(X/G)$: however, if the action is not free, the space $X/G$ may have some pathology. The Borel construction allows to replace $X$ by a topological space $X'$ which is homotopically equivalent to $X'$ and the action of $G$ on $X'$ is free.

Noncommutative Geometry

Speaker: Shahab Azarfar (Western)

"Volume Quantization from Spin Geometry IV"

Time: 11:30

Room: MC 107

We try to investigate a generalization of the Heisenberg commutation relation ${[p,q]=-i \hslash}$, introduced by Chamseddine, Connes and Mukhanov as ``the one-sided and the two-sided quantization equations'', which captures the geometry. The momentum variable $p$ is encoded by the Dirac operator and the analogue of the position variable $q$ is the Feynman slash of real scalar fields over a closed even-dimensional spin manifold. Existence of a solution of the one-sided equation implies that the manifold decomposes into a disconnected sum of spheres of unit volume which represent quanta of geometry. The two-sided equation, as the refined version of the one-sided equation by involving the real structure on a spin manifold, implies the quantization of the volume of the spin manifold.

Graduate Seminar

Speaker: Nicholas Meadows (Western)

"Algebraic Surfaces"

Time: 13:30

Room: MC 108

The purpose of this talk will be to illustrate how various abstract techniques from algebraic geometry (i.e. cohomology, Riemann Roch) can be used to study algebraic surfaces. Algebraic surfaces are smooth projective varieties over \mathbb{C} of dimension 2. After reviewing the basics of linear systems and divisors on surfaces, we will study morphisms determined by linear systems on the Hirzebruch surfaces, a particularly nice class of algebraic surfaces. Depending on time, other applications and results will be described, such as the relation of Hirzebruch surfaces to the Enriques-Kodaira classification or the classication of degree n-1 nondegenerate surfaces in P^{n}.

Algebra Seminar

Speaker: Caroline Junkins (Western)

"Schubert cycles and subvarieties of generalized Severi-Brauer varieties"

Time: 16:00

Room: MC 107

For an algebraic variety X over an arbitrary field F, a classical question asks whether X has a K-point for a given field extension K/F. When X is a generalized Severi-Brauer variety, we may extend this question to ask not only about K-points, but about K-forms of any closed Schubert subvariety. In this talk, we consider an algebraic form of this question concerning data from the underlying central simple algebra of X. We then discuss applications to the Grothendieck group and Chow group of X, generalizing a result of N. Karpenko for usual Severi-Brauer varieties. This is part of ongoing work with D. Krashen and N. Lemire.

Geometry and Topology

Speaker: Ben Williams (UBC)

"The EHP sequence in A1 algebraic topology"

Time: 15:30

Room: MC 107

The classical EHP sequence is a partial answer to the question of how far the unit map of the loop-suspension adjunction fails to be a weak equivalence. It can be used to move information from stable to unstable homotopy theory. I will explain why there is an EHP sequence in A1 algebraic topology, and some implications this has for the unstable A1 homotopy groups of spheres.

Noncommutative Geometry

Speaker: (Western)

"Higgs fields and symmetry breaking mechanism"

Time: 11:30

Room: MC 107

Existence of massive gauge bosons breaks down the local gauge invariance of Yang-Maills Lagrangians. In this lecture we shall look at one method to deal with this situation through the introduction of Higgs fields.

Homotopy Theory

Speaker: Marco Vergura (Western)

"Equivalences and the Univalence Axiom (part 2)"

Time: 13:30

Room: MC 107

Following Chapter 4 of the HoTT book, we continue our journey in the various characterizations of equivalences in Type Theory. We also show how Function Extensionality follows from the Univalence Axiom.

Analysis Seminar

Speaker: Josue Rosario-Ortega (Western)

"Special Lagrangian submanifolds with edge-singularities"

Time: 15:30

Room: MC 107

Given a Calabi-Yau manifold $(M,\omega,\Omega)$ of complex dimension $n$, a Special Lagrangian submanifold (SL-submanifold) $L\subset M$ is a real $n$ dimensional submanifold calibrated by $\text{Re}\:\Omega$. These type of submanifolds are Lagrangian with respect to the symplectic structure $\omega$ and minimal with respect to the Calabi-Yau metric of the ambient space. Singular SL-submanifolds are particularly important as they play a fundamental role in mirror symmetry. In this talk I will survey the results obtained in the last years on deformation and moduli spaces of SL-submanifolds with conical singularities. Moreover I will introduce SL-submanifolds with higher order singularities (in particular edge singularities) and I will explain the approach used by the speaker to study moduli spaces of such type of singularities and some results obtained about the moduli space.

Geometry and Combinatorics

Speaker: Sergio Chaves (Western)

"The Borel construction (Part 2)"

Time: 16:00

Room: TC 342

Let $X$ be a topological space with an action of a topological group $G$. We want to relate to $X$ an algebraic object that reflects both the topology and the action of the group. The first candidate is the cohomology ring $H^{*}(X/G)$: however, if the action is not free, the space $X/G$ may have some pathology. The Borel construction allows to replace $X$ by a topological space $X'$ which is homotopically equivalent to $X'$ and the action of $G$ on $X'$ is free.

Noncommutative Geometry

Speaker: Rui Dong (Western)

"Classification of Finite Real Spectral Triples"

Time: 11:30

Room: MC 107

First, I give a basic introduction to finite noncommutative spaces, and then I focus on the classification of finite real spectral triples.

Basic Notions Seminar

Speaker: Lex Renner (Western)

"Hilbert's Fourteenth Problem"

Time: 15:30

Room: MC 107

Hilbert's Fourteenth Problem asks about the finite generation of certain commutative rings. Furthermore, Hilbert's question was a major catalyst in the development of geometric invariant theory. But the basic question here makes sense more generally. I will discuss examples, successes, myths, and new approaches.

Algebra Seminar

Speaker: Christin Bibby (Western)

"Representation stability for the cohomology of arrangements"

Time: 16:00

Room: MC 107

From a root system, one may consider the arrangement of reflecting hyperplanes, as well as its toric and elliptic analogues. The corresponding Weyl group acts on the complement of the arrangement and hence on its cohomology. We consider a sequence of linear, toric, or elliptic arrangements which arise from a family of root systems of type A, B, C, or D, and we study the stability of the rational cohomology as a sequence of Weyl group representations. Our techniques combine a Leray spectral sequence argument similar to that of Church in the type A case along with $FI_W$-module theory which Wilson developed and used in the linear case.