Noncommutative Geometry

Speaker: (Western)

"Double matrix integrals"

Time: 11:30

Room: MC 108

Homotopy Theory

Speaker: Andres Felipe Fontalvo Orozco (Western)

"Higher Inductive Types and the Univalence Axiom in Action"

Time: 13:30

Room: MC 107

Selected topics in chapter 8 of the HoTT book will be presented to illustrate how homotopy theory can be developed within type theory.

Geometry and Combinatorics

Speaker: Ahmed Ashraf (Western)

"The orbit cone correspondence"

Time: 16:00

Room: MC 108

We will study the orbits of T-action on the toric variety X defined by a fan. The main aim is to show the orbits are in bijective correspondence with the cones in the fan defining X.

Noncommutative Geometry

Speaker: Rui Dong (Western)

"Local index formula II"

Time: 11:30

Room: MC 108

Algebra Seminar

Speaker: Marina Palaisti (Western)

"One of the Bloch-Kato Conjectures"

Time: 14:30

Room: MC 108

We tell the story of the development of the Bloch-Kato Conjecture. In particular, we concentrate on the case of global fields. This is a Part II of the PhD comprehensive examination presentation.

Comprehensive Exam Presentation

Speaker: Marina Palaisti (Western)

"One of the Bloch-Kato Conjectures"

Time: 14:30

Room: MC 108

We tell the story of the development of the Bloch-Kato Conjecture. In particular, we concentrate on the case of global fields. This is a Part II of the PhD comprehensive examination presentation.

Noncommutative Geometry

Speaker: (Western)

"Random matrix theory: eigenvalue spacing distributions and Wigner's surmise"

Time: 11:00

Room: MC 106

Algebra Seminar

Speaker: Michael Chapman (Ben-Gurion University of the Negev)

"Filtrations of free groups and Magnus theory, Part I"

Time: 14:30

Room: MC 107

We make a systematic study of filtrations of a free group F defined as products of powers of the lower central series of F. Under some assumptions on the exponents, we characterize these filtrations in terms of the group algebra, the Magnus algebra of non-commutative power series, and linear representations by upper-triangular unipotent matrices. These characterizations generalize classical results of Grun, Magnus, Witt, and Zassenhaus from the 1930's, as well as later results on the lower p-central filtration and the p-Zassenhaus filtrations. We derive alternative recursive definitions of such filtrations, extending results of Lazard.

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"An introduction to Feynman-Kac formula I"

Time: 11:00

Room: MC 108

Colloquium

Speaker: Peter LeFanu Lumsdaine (Stockholm University)

"Homotopy Type Theory: an introduction and survey"

Time: 15:30

Room: MC 107

Around ten years ago, it was discovered that Martin-Lof's Intensional Type Theory - a logical system designed in the 1970's as a more constructive and computational alternative to ZFC-style foundations - admits quite unexpected interpretations in simplicial sets and other homotopy-theoretic settings. More than this: under this interpretation, the logic turned out to be remarkably powerful for expressing and reasoning with standard homotopy-theoretic properties and constructions, in very elementary ways. This connection has since proved extremely fruitful, and the resulting programme of work has become known as *Homotopy Type Theory* or *Univalent Foundations*.

In this talk, I will give a general introduction to type theory, and a survey of the recent homotopically-influenced developments - in particular, of the connections with $\infty$-toposes in the sense of Rezk/Lurie, for which it is hoped that type theory can provide an "internal language", in a certain precise sense.

Algebra Seminar

Speaker: Michael Chapman (Ben-Gurion University of the Negev)

"Filtrations of free groups and Magnus theory, Part II"

Time: 14:30

Room: MC 108

We make a systematic study of filtrations of a free group F defined as products of powers of the lower central series of F. Under some assumptions on the exponents, we characterize these filtrations in terms of the group algebra, the Magnus algebra of non-commutative power series, and linear representations by upper-triangular unipotent matrices. These characterizations generalize classical results of Grun, Magnus, Witt, and Zassenhaus from the 1930's, as well as later results on the lower p-central filtration and the p-Zassenhaus filtrations. We derive alternative recursive definitions of such filtrations, extending results of Lazard.

Noncommutative Geometry

Speaker: Rui Dong (Western)

"Matrix integrals from spectral triples"

Time: 11:00

Room: MC 106

Geometry and Combinatorics

Speaker: Ahmed Ashraf (Western)

"Toric Varieties from Polytopes"

Time: 16:00

Room: MC 108

We will finish of the construction of a toric variety from a fan. The orbits of $T$-action on the toric variety $X$ defined by a fan $\Sigma$ are in bijective correspondence with cones of $\Sigma$. Then we give a couple of examples of toric varieties arising from normal fans of polytopes.

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"Feynman-Kac formula II"

Time: 11:00

Room: MC 108

Noncommutative Geometry

Speaker: (Western)

"Matrix integrals and planar diagrams"

Time: 11:00

Room: MC 108

Colloquium

Speaker: Aleks Essex (Electrical Engineering, Western)

"Banks to Ballots: The Mighty Discrete Logarithm and its Many Uses"

Time: 14:30

Room: MC 107

Although crypto systems based on the hardness of solving discrete logarithms date back 40 years, they are more prevalent today than ever. The elliptic curve Diffie-Hellman protocol, for example, accounts for 90% of public-key agreements on the internet, meaning your device probably used discrete log based cryptography to fetch these very words. In this talk we’ll examine some of the more complex applications of discrete logarithm based cryptography, including homomorphic encryption, zero-knowledge proofs, distributed decryption, and secure multi-party protocols. Finally, we’ll tie these cryptographic primitives together with a case study of a cryptographically end-to-end verifiable internet voting system.

Homotopy Theory

Speaker: Christian Sattler (University of Leeds)

"Algebraic model structures via "glueing""

Time: 13:30

Room: MC 107

Consider a Grothendieck topos with a functorial cylinder with connections and a choice of generating cofibrations. Together, this induces a naive notion of fibration. We give sufficient coditions under which this gives rise to an (algebraic) model structure. The key point will be the glueing construction of Cohen et al., a recent combinatorial discovery that allows for constructively extending fibrations along trivial cofibrations without resorting to a theory of minimal fibrations.

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"Feynman-Kac formula 3"

Time: 11:00

Room: MC 108

Colloquium

Speaker: Chris Hall (University of Wyoming)

"Expander and Ramanujan Graphs"

Time: 14:30

Room: MC 107

I will define expander and Ramanujan graphs and give a variety of existence results. Most notably I will state a joint result of D. Puder, W. Sawin, and myself on so-called Ramanujan covers of graphs. Our result generalizes one of several recent celebrated joint results of A. Marcus, D. Spielman, and N. Srivastava.

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