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.

Noncommutative Geometry

Speaker: (Western)

"Matrix Integrals 1"

Time: 10:30

Room: MC 107

Matrix integrals play an important role in random matrix theory, 2d quantum gravity, and the topology of moduli spaces of Riemann surfaces. In this first lecture we shall look at a perturbative expansion for matrix integrals via Feynman graphical methods.

Homotopy Theory

Speaker: James Richardson (Western)

"Inductive types (part 2)"

Time: 13:30

Room: MC 107

In this talk we will discuss uniqueness results for inductive types, with particular focus on identity types.

Geometry and Combinatorics

Speaker: Dinesh Valluri (Western)

"Introduction to equivariant Chow groups (part II)"

Time: 16:00

Room: MC 108

In the last talk we introduced Equivariant Chow groups and proved that they are well defined using a certain double fibration argument. In this talk we will present the functoriality of the flat pull backs and proper pushforwards of Equivariant cycles. We will review Grothendieck's axiomatic construction of Chern classes in the classical context and use it to define 'Equivariant Chern classes'. If time permits we will compute some examples of Equivariant Chow groups and a result which allows us to compute such groups just using the Equivariant Chow groups of the maximal torus.

Graduate Seminar

Speaker: Octavian Mitrea (Western)

"Geodesic Convexity"

Time: 13:30

Room: MC 107

The classical notion of convexity of subsets of the Euclidean space metamorphoses into several distinct types of (geodesic) convexity in the more general Riemannian setting , this phenomenon, along with its geometric implications, being the main topic of this talk.

Basic Notions Seminar

Speaker: Martin Pinsonnault (Western)

"The Isoperimetric Inequality"

Time: 15:30

Room: MC 107

The circle is uniquely characterized by the property that among all simple closed plane curves of given length L, the circle of circumference L encloses maximum area. In this talk, we will recount some geometric generalizations and applications of this inequality. We shall also consider various geometric and analytic inequalities closely connected to the isoperimetric inequality, such as Brunn-Minkowski's, Wirtinger's, Poincaré's and Sobolev's inequalities.

Algebra Seminar

Speaker: Karol Szumilo (Western)

"Universal fibrations"

Time: 16:00

Room: MC 107

I will present a potential approach to construction of universal fibrations in the categories of presheaves of infinity-groupoids.

Geometry and Topology

Speaker: Tudor Dimofte (Perimeter)

"Applications of 3d gauge theory to geometric representation theory"

Time: 15:30

Room: MC 107

I will discuss some ongoing work (with Mat Bullimore, Davide Gaiotto, and Justin Hilburn) on 3d gauge theories with half-maximal supersymmetry ("N=4"). These theories are labelled by a compact group G and a quaternionic representation R. From this data one obtains many geometric objects. Among them are the "Higgs branch" of vacua (a hyperkahler quotient R///G), the "Coulomb branch" of vacua (a modification of the cotangent bundle of the dual complex torus T*(T')), deformation quantizations of these spaces, and Fukaya-like categories of modules for the quantizations. These various objects and their relations generalize many constructions in geometric representation theory, including a phenomenon called symplectic duality (studied by Braden-Licata-Proudfoot-Webster, generalizing the Koszul duality of Beilinson-Ginzburg-Soergel) and a finite version of the AGT correspondence (by Braverman-Feigin-Finkelberg-Rybnikov).

I will give an introduction to some of these ideas.

Noncommutative Geometry

Speaker: (Western)

"Matrix Integrals 2"

Time: 10:30

Room: MC 108

Matrix integrals play an important role in random matrix theory, 2d quantum gravity, and the topology of moduli spaces of Riemann surfaces. In these lectures we shall look at a perturbative expansion for matrix integrals via Feynman graphical methods.

Homotopy Theory

Speaker: Pal Zsamboki (Western)

"Higher Inductive Types (part 1)"

Time: 13:30

Room: MC 107

We introduce the first example of a higher inductive type, S^1, for which we prove a universal property. Then after discussing the interval I, we move on to spheres. We finish with finite CW complexes.

Analysis Seminar

Speaker: Andrew Zimmer (University of Chicago)

"Negatively curved metric spaces and several complex variables"

Time: 15:30

Room: MC 107

In this talk I will discuss how to use ideas from the theory of metric spaces of negative curvature to understand the behavior of holomorphic maps between bounded domains in complex Euclidean space. Every bounded domain has an metric, called the Kobayashi metric, which is distance non-increasing with respect to holomorphic maps. Moreover, this metric often satisfies well-known negative curvature type conditions (for instance, Gromov hyperbolicity or visibility) and one can then use this negative curvature to understand the behavior of holomorphic maps. I will discuss the domains where the Kobayashi metric satisfies negative curvature type conditions and how to use these conditions to prove new results. Some of this is joint work with Gautam Bharali.

Noncommutative Geometry

Speaker: Mitsuru Wilson (Western)

"A Gauss-Bonnet Theorem for the noncommutative 4-sphere"

Time: 10:30

Room: MC 107

Algebra Seminar

Speaker: Adam Topaz (UC Berkeley)

"On the Ihara/Oda-Matsumoto Conjecture"

Time: 16:00

Room: MC 107

One of the key themes in Grothendieck's "Esquisse d'un Programme" is the idea of studying absolute Galois groups via their action on objects of geometric origin. Motivated by this, the conjecture of Ihara/Oda-Matsumoto (I/OM) predicts that the Galois action on geometric fundamental groups of algebraic varieties leads to a combinatorial/topological description of absolute Galois groups. This conjecture was resolved by Pop in the 90's, and several variants/generalizations have since been formulated. In this talk, I will discuss the recent proof of the mod-ell abelian-by-central variant of this conjecture, which follows from studying the Galois action on certain lattices of geometric origin.

Geometry and Topology

Speaker: Ada Boralevi (Sissa, Trieste)

"Orthogonal and unitary tensor decomposition from an algebraic perspective"

Time: 15:30

Room: MC 107

While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. In this talk I will present an intrinsic characterization of those tensors that do, by means of polynomial equations of degree at most four. The exact degrees, and the corresponding polynomials, are different in each of three times two scenarios: ordinary, symmetric, or alternating tensors; and real-orthogonal versus complex-unitary. This is a joint project with J. Draisma, E. Horobet, and E. Robeva.

Noncommutative Geometry

Speaker: (Western)

"Matrix Integrals 3"

Time: 11:00

Room: MC 108

Homotopy Theory

Speaker: Pal Zsamboki (Western)

"Higher Inductive Types (part 2)"

Time: 13:30

Room: MC 107

We start by discussing pushouts and (-1)- and 0-truncations, then we use these to discuss quotients of sets by mere relations and algebraic structures on sets. We finish with a particular case of the flattening lemma, which says that if W is a particular higher inductive type, then the total space of the fibration corresponding to a presheaf on W is equivalent to an appropriate higher inductive type.

Noncommutative Geometry

Speaker: Mitsuru Wilson (Western)

"A Gauss-Bonnet Theorem for the Noncommutative 4-sphere II"

Time: 10:30

Room: MC 108

Colloquium

Speaker: Vitali Vougalter (Toronto)

"On the existence of stationary solutions for some systems of integro-differential equations with anomalous diffusion"

Time: 15:30

Room: MC 107

The article is devoted to the proof of the existence of solutions of a system of integro-differential equations appearing in the case of anomalous diffusion when the negative Laplacian is raised to some fractional power. The argument relies on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains along with the Sobolev inequality for a fractional Laplace operator are being used.

Algebra Seminar

Speaker: Stefan Gille (Alberta)

"On Rost nilpotence for threefolds"

Time: 10:30

Room: MC 107

After recalling the Rost nilpotence property and its meaning, I will show that for 3-dimensional smooth projective varieties over a field of characteristic 0, one has only 'to care' about 0-cycles. Using the Brown-Gersten-Quillen spectral sequence, one can get one's hands on the zero cycles of a 3-dimensional toric model allowing a proof of Rost nilpotence for such varieties.

Noncommutative Geometry

Speaker: (Western)

"Matrix Integrals IV"

Time: 13:00

Room: MC 108

I shall prove the Harer-Zagier formula and Wigner's semicircle law for random matrices. We shall then move to evaluation of matrix integrals via graphs on surfaces.

Geometry and Combinatorics

Speaker: Ahmed Ashraf (Western)

"Introduction to toric varieties"

Time: 16:00

Room: MC 108

We will introduce equivalent ways to define toric varieties and discuss their relationship.

Noncommutative Geometry

Speaker: Rui Dong (Western)

"The local Index Formula"

Time: 11:00

Room: MC 107

I am going to give a basic definition of Hochschild and cyclic cohomology first, and then I will try to give the local index formula in a more general case, if time is enough, I will give the proof of local index formula for the even case.

Graduate Seminar

Speaker: Andres Fontalvo (Western)

"Introduction to Hopf Algebras"

Time: 13:30

Room: MC 108

Given the category of modules over an algebra, we would like to know when it is a monoidal category with the usual tensor product of vector spaces. This leads to the notion of Hopf algebra.

Algebra Seminar

Speaker: Béatrice Chetard (Western)

"Group cohomology and Steenrod operations"

Time: 14:30

Room: MC 108

I will present detailed calculations of the cohomology rings of several small groups, together with the action of the Steenrod squares on said cohomology rings.