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.

Geometry and Topology

Speaker: Paul Goerss (Northwestern)

"Diffraction and reassembly in stable homotopy theory"

Time: 15:30

Room: MC 107

The chromatic view of stable homotopy theory uses the algebraic geometry of formal groups to organize calculations and the search for large scale phenomena. One of the guiding principles, due to Hopkins, is the Chromatic Splitting Conjecture, which predicts how to rebuild stable homotopy types from simpler pictures. Recently we have seen that this conjecture is not quite true; we will discuss what goes wrong and how it might be fixed. This is joint work with Agnes Beaudry and Hans-Werner Henn, with the hard part done by Beaudry.

Homotopy Theory

Speaker: Karol Szumilo (Western)

"Formalized Homotopy Theory (part 1)"

Time: 13:30

Room: MC 107

I will present formal proofs of selected results discussed in preceding seminar talks using the Coq library UniMath.

Analysis Seminar

Speaker: Josue Rosario-Ortega (Western)

"Special Lagrangian submanifolds with edge-singularities II"

Time: 15:30

Room: MC 107

Last week we explained the geometric context of the problem of describing the moduli space of Special Lagrangian deformations of a compact manifold. The analytic details were more or less straight-forward as the elliptic theory of PDEs in a compact manifold is very complete and in some sense canonical. In this second and final part of my talk I will consider SL-submanifolds in $\mathbb{C}^{n}$, therefore the SL-submanifolds to be considered shall be non-compact and/or singular. I will survey the elliptic theories available in singular settings, and I will focus on the case of edge-singularities, the next level of singularities after conical. I will conclude the talk with a theorem describing the moduli space of SL-deformations with boundary conditions for a SL submanifold with edge-singularities.

Pizza Seminar

Speaker: Masoud Khalkhali (Western)

"Why E=mc^2 (part II)"

Time: 17:30

Room: MC 108

TBA

Geometry and Combinatorics

Speaker: Jianing Huang (Western)

"Equivariant de Rham theory: from Weil model to Cartan model"

Time: 16:00

Room: MC 105C

For a smooth manifold M with a Lie group G action, we can define equivariant cohomology based on differential forms on M. That is Weil model. This construction is analogous to Borel construction on the level of differential forms. The Cartan model is then derived from the Weil model. The Cartan model provides an explicit way to compute equivariant cohomology. We will introduce both models and prove that they are equivalent.

Noncommutative Geometry

Speaker: (Western)

"Higgs fields and symmetry breaking mechanism II"

Time: 11:30

Room: TBA

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.

Graduate Seminar

Speaker: James Richardson (Western)

"Homotopical perspective on 2-monads"

Time: 13:30

Room: MC 107

This talk will be a nontechnical introduction to the interactions between homotopy theory and 2-category theory, focusing particularly on homotopical perspectives on 2-monads. No prior experience with 2-categories will be necessary.

Colloquium

Speaker: Ugo Bruzzo (SISSA, Trieste, Italy)

"On the Noether-Lefschetz problem"

Time: 15:30

Room: MC 107

A classical result, usually ascribed to Noether and Lefschetz, states that a very general surface $X$ in complex projective 3-space $\mathbb{P}^3$ has Picard number 1 (i.e., the group of isomorphism classes of line bundles on $X$ is $\mathbb{Z}$). The problem of finding an estimate of the codimension of the loci in the moduli spaces of surfaces in $\mathbb{P}^3$ whose points correspond to surfaces with a bigger Picard group is unsuspectedly complicated.

In a first part of my talk I will review these classical results. Then I will sketch a generalization to surfaces in normal toric 3-folds.

Algebra Seminar

Speaker: Jeff Morton (University of Toledo)

"Transformation structures for 2-group actions"

Time: 16:00

Room: MC 107

2-groups, or "categorical groups", are "higher dimensional" algebraic structures which generalize groups. In particular, they can be seen as group objects in categories, and as such they are useful for describing the symmetries of categories. I will describe 2-groups and their actions, and describe how similar generalizations of the transformation groupoids associated to group actions show up in this context. Time permitting, I will outline a motivating example for this work coming from the geometry of gerbes. Based on joint work with Roger Picken (IST, Lisbon).

Geometry and Topology

Speaker: Alberto Garcia Raboso (U. of Toronto)

"A twisted non-abelian Hodge correspondence"

Time: 15:30

Room: MC 107

The classical nonabelian Hodge correspondence establishes an equivalence between certain categories of flat bundles and Higgs bundles on smooth projective varieties. I will describe an extension of this result to twisted vector bundles.

Noncommutative Geometry

Speaker: (Western)

"Feynman's Theorem"

Time: 11:30

Room: MC 107

This is a quick survey of Feynman's asymptotic formula for m-point functions as a sum over graphs.

Homotopy Theory

Speaker: Karol Szumilo (Western)

"Formalized Homotopy Theory (part 2)"

Time: 13:30

Room: MC 107

I will present formal proofs of selected results discussed in preceding seminar talks using the Coq library UniMath.

Analysis Seminar

Speaker: Alexandre Sukhov (Lille)

"Singular Levi flat hypersurfaces"

Time: 15:30

Room: MC 107

In this partially expository talk we discuss some recent progress and open problems concerning real analytic Levi flat hypersurfaces with singularities. We prove that the Levi foliation of such a hypersurface extends as a holomorphic web to a full neighborhood of singularity. This is a joint work with R. Shafikov (Comment. Math. Helvet. '15).

Geometry and Combinatorics

Speaker: Jianing Huang (Western)

"Equivariant de Rham theory: from Weil model to Cartan model (part II)"

Time: 16:00

Room: MC 105C

For a smooth manifold M with a Lie group G action, we can define equivariant cohomology based on differential forms on M. That is Weil model. This construction is analogous to Borel construction on the level of differential forms. The Cartan model is then derived from the Weil model. The Cartan model provides an explicit way to compute equivariant cohomology. We will introduce both models and prove that they are equivalent.

This is the second part of this talk.

Noncommutative Geometry

Speaker: Rui Dong (Western)

"Classification of Finite Real Spectral Triples II"

Time: 12:30

Room: MC 107

In this talk, I will introduce the structure of finite real spectral triple first, and then I will focus on how to encode those data of a finite real spectral triple inside the so-called "Krajewski diagram".

Graduate Seminar

Speaker: James Richardson (Western)

"Homotopical perspective on 2-monads part II"

Time: 13:30

Room: MC 107

This talk is continuation of last week's talk with the same title.

Colloquium

Speaker: Alexandre Sukhov (Lille)

"On the Hodge conjecture for $q$-complete domains"

Time: 15:30

Room: MC 107

A model example of a $q$-complete (in the sense of Grauert) manifold is the complement of a projective variety of codimension $q$. In particular, 1-complete manifolds are Stein. Our main results states that the top non-zero integer cohomology class of such a manifold can be represented by analytic cycles. This is a joint work with F. Forstneric and J. Smrekar (Geometry & Topology '16).

Dept Oral Exam

Speaker: Fatemeh Sharifi (Western)

"Topics in Approximation Theory"

Time: 17:00

Room: MC 108

I am interested in approximation theory, in particular, on a closed subset of an open Riemann surface R. During the talk I will focus on Arakelyan's theorem and its topological conditions and explain that, despite the fact that approximation is not always possible for a closed subset E of R, one can construct a large closed subset of E on which approximation is possible in a strong sense. Also, I will talk about pole-free approximation by meromorphic functions with respect to spherical distance. Finally, I will present an extension of the Schwarz reflection principle for open bordered Riemann surfaces.

Algebra Seminar

Speaker: Cihan Okay (Western)

"Cohomology of metacyclic groups"

Time: 16:00

Room: MC 107

I will talk about mod $p$ cohomology of metacyclic groups. A metacyclic group is an extension of a cyclic group by another cyclic group. Mod $p$ cohomology rings of metacyclic groups are computed by Huebschmann using homological perturbation theory. Homological perturbation theory allows one to explicitly construct projective resolutions in an inductive fashion. I will describe this method and possibly relate it to other methods existing in the literature.

Geometry and Topology

Speaker: Steven Rayan (Toronto)

"Star-shaped quivers, hyperpolygons, and Higgs bundles"

Time: 15:30

Room: MC 107

I will discuss three closely-related moduli problems: moduli of representations of star-shaped quivers, moduli of hyperpolygons, and moduli of parabolic Higgs bundles. One theme that weaves these three problems together is complete integrability. I will discuss recent results on the topology of these moduli spaces (joint work with Jonathan Fisher) and then pose questions on the relationship between stability for Higgs bundles and stability for hyperpolygons.

Noncommutative Geometry

Speaker: (Western)

"Feynman's Theorem II"

Time: 11:30

Room: MC 107

This is a quick survey of Feynman's asymptotic formula for m-point functions as a sum over graphs.

Homotopy Theory

Speaker: Karol Szumilo (Western)

"Formalized Homotopy Theory (part 3)"

Time: 13:30

Room: MC 107

I will present formal proofs of selected results discussed in preceding seminar talks using the Coq library UniMath.

Colloquium

Speaker: Farzad Fathizadeh (California Institute of Technology)

"Modular forms in gravitational instantons"

Time: 15:00

Room: MC 108

In a succession of papers, physicists and mathematicians have achieved an explicit parameterization of Bianchi-IX gravitational instantons in terms of theta functions with characteristics. By exploiting the latter, in this talk, I will shed light on a rationality phenomena in the spectral action of SU(2)-invariant Bianchi-IX metrics. This will be done by showing that for the instantons, each term in the expansion of their spectral action gives rise to a modular form of weight 2 that can be written explicitly in terms of well-known modular forms, namely the Eisenstein series and the modular discriminant. An elegant proof of the rationality result will also be presented, which is based on expressing Seeley-de Witt coefficients as noncommutative residues of Laplacians. This talk is based on joint works with Wentao Fan and Matilde Marcolli.

Geometry and Combinatorics

Speaker: Jianing Huang (Western)

"Equivariant de Rham theory: from Weil model to Cartan model (part III)"

Time: 16:00

Room: MC 105C

For a smooth manifold M with a Lie group G action, we can define equivariant cohomology based on differential forms on M. That is Weil model. This construction is analogous to Borel construction on the level of differential forms. The Cartan model is then derived from the Weil model. The Cartan model provides an explicit way to compute equivariant cohomology. We will introduce both models and prove that they are equivalent.

This is the third and final part of this talk.

Noncommutative Geometry

Speaker: Rui Dong (Western)

"Classification of Finite Real Spectral Triples III"

Time: 11:30

Room: MC 107

In this talk, I will introduce the structure of finite real spectral triple first, and then I will focus on how to encode those data of a finite real spectral triple inside the so-called "Krajewski diagram".

Algebra Seminar

Speaker: Good Friday

"(No Seminar)"

Time: 16:00

Room: MC 107

Geometry and Topology

Speaker: Easter Monday (no talk)

"Western"

Time: 08:30

Room: MC 107

Noncommutative Geometry

Speaker: (Western)

"1particle irreducible graphs and the effective action"

Time: 11:30

Room: MC 107

I will show how the effective action can be calculated by summing over 1 particle irreducible graphs.

Homotopy Theory

Speaker: James Richardson (Western)

"Inductive types (part 1)"

Time: 13:30

Room: MC 107

In this talk we will introduce W-types and discuss several examples of inductive types.

Geometry and Combinatorics

Speaker: Dinesh Valluri (Western)

"Introduction to equivariant Chow groups"

Time: 16:00

Room: MC 105C

We will recall the notion of Chow group of a scheme briefly and motivate the need for the notion of Equivariant Chow groups. We give a definition of the latter which closely resembles the Borel construction of Equivariant Cohomology groups via certain approximation of the universal G-bundle. We will prove that such construction is well defined and see some examples. We will end the talk with a discussion of functoriality of flat pullbacks and proper pushforwards in the equivariant context.

Noncommutative Geometry

Speaker: Rui Dong (Western)

"TBA"

Time: 11:30

Room: MC 107

TBA

Basic Notions Seminar

Speaker: Masoud Khalkhali (Western)

"From Triangles to Elliptic Complexes"

Time: 15:30

Room: MC 107

The index theorem of Atiyah and Singer is a milestone of modern mathematics. This result which computes the virtual dimension of the space of solutions of an elliptic operator in topological terms, has its roots in classical results like Gauss-Bonnet and Riemann-Roch theorems. I shall trace some of these roots, going back all the way to a statement in Euclid's Elements! Any proof of the index theorem involves some heavy doses of analysis as well as geometry and topology. I shall briefly indicate the original cobordism proof, and then will focus on a more modern heat equation proof and its ramifications.