Colloquium

Speaker: Chris Kapulkin (Western)

"Formalization of Mathematics and the Univalent Foundations"

Time: 14:00 - 15:00

Room: MC 107

I will give an introduction to the Univalent Foundations, a new approach to foundations of mathematics, proposed by Voevodsky based on ideas from homotopy theory. The Univalent Foundations are meant to more closely (than set theory) reflect our intuitions about mathematical objects, while also making computer-aided formal verification of proofs essentially straightforward. As an example, I will present the development of category theory in the new foundations (jww Ahrens and Shulman) and will contrast it with that within set theory. I will also discuss possible applications to formal verification of cryptographic standards.

Analysis Seminar

Speaker: Daniel Burns (University of Michigan)

"Canonical complexifications, affine varieties, and eigenfunctions of the Laplacian"

Time: 15:30 - 16:30

Room: MC 108

Let $M$ be a real analytic Riemannian manifold. An adapted complex structure on $TM$ is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. Lempert-Szoke and Guillemin-Stenzel have given canonical methods to construct adapted complex structures in neighborhoods of the zero section, equipped with a solution of the Homegeous Complex Monge-Ampere equation (HCMA) related to the geometry of $M$. These complex manifolds are called Grauert tubes. This structure is called entire if the structure may be extended to the whole of $TM$. We describe a circle of problems related to determining whether an entire Grauert tube is an affine algebraic manifold with ring of polynomials intrinsically distinguished by the HCMA exhaustion. We also discuss the relationship of this construction to the Paley-Wiener type theorem of Boutet de Monvel, and the relationship to eigenfunctions of the Laplace operator on $M$. Finally, we discuss the smallest dimensional cases, namely the two sphere, and invariant metrics on the three sphere, thought of as $SU(2)$.

Speaker's web page: https://lsa.umich.edu/math/people/faculty/dburns.html

Pizza Seminar

Speaker: Matthias Franz (Western)

"The impossibility of elementary integration"

Time: 17:30 - 18:30

Room: MC 107

In Calculus courses we learn methods to integrate functions of a real variable. Sometimes they work, but other times they seem to fail. I will present a result (due to Liouville) that indeed the integral of many functions cannot be expressed "in elementary terms", i.e., in terms of exponentials, logarithms and trigonometric functions. Specifically, this includes the Gaussian distribution (Bell curve) and the logarithmic integral used for prime number counting.

There will be pizza after the talk.

Algebra Seminar

Speaker: Chris Hall (Western)

"d-matchings polynomials"

Time: 14:30 - 15:30

Room: MC 107

A celebrated result of Marcus, Spielman, and Srivas asserts that there exist bipartite Ramanujan graphs of every degree and number of vertices. In joint work with Puder and Sawin, my coauthors and I gave a significant generalization of their result. To prove our result we introduced a family of polynomials one can attach to a finite undirected graph. We call them d-matchings polynomials because when d = 1 one obtains the so-called matchings polynomial of a graph. In this talk I will define these polynomials and give some of the remarkable properties they satisfy.

Analysis Seminar

Speaker: Yael Karshon (University of Toronto)

"Diffeological, Frolicher, and differential spaces"

Time: 15:30 - 16:30

Room: MC 108

Differential calculus on Euclidean spaces can be generalized in different ways. A diffeological structure on an (arbitrary) set is given by maps from open subsets of Euclidean spaces to the set (satisfying some axioms); a differential structure is given by maps from the set to R; a Frolicher structure is given by maps from R to the set as well as maps from the set to R. We describe relations between these structures and present examples, including good subsets, bad quotients, and infinite dimensional spaces of smooth maps.

Speaker's web page: http://www.math.toronto.edu/karshon/

Dept Oral Exam

Speaker: Nadia Alluhaibi (Western)

"On Vector-Valued Automorphic Forms On Bounded Symmetric Domains"

Time: 14:00 - 15:30

Room: MC 107

The main points of this talk are as follows:

- constructions of vector-valued automorphic forms on bounded symmetric domains via Poincare series

- vector-valued automorphic forms associated to submanifolds of the complex unit ball - studying the behavior of asymptotics of the inner product of two Poincare series associated to submanifolds of the complex unit ball, for large weights, with examples.

Colloquium

Speaker: Blake Madill (Waterloo)

"On some techniques in modern radical theory"

Time: 15:30 - 16:30

Room: MC 107

The notion of a radical of a ring was first suggested by Wedderburn in 1908 and first used by Koethe in 1930. We say a class of rings C is a radical class of rings if it is homomorphically closed, the sum of all ideals in C of a ring R, denoted by C(R), is again in C, and C(R/C(R))=(0). We call C(R) the radical of R associated to the class C. The idea of radical theory is to collect ring theoretical information in C(R) and be able to then give more information about R. In this talk we will discuss some of the classical results in radical theory and describe some of modern techniques being used in current research. In particular, techniques from graded-ring theory and theoretical computer science will be discussed. This talk will be accessible to a general mathematical audience

Algebra Seminar

Speaker: Hadi Seyedinejad (Western)

"Effective extensions of the ring of polynomials in real algebraic geometry"

Time: 14:30 - 15:30

Room: MC 107

Some key tools, such as the Nullstellensatz, of course fail in algebraic geometry over the reals. However, having less of such machinery that works universally (as in the case over an algebraically closed field) does not make real algebraic geometry any less fascinating. One needs instead to come up with the right tools that are tailored to a phenomenon of interest. We are interested in 'irreducible components' of real algebraic sets, particularly as they relate to the sheets of Nash on such sets. As it turns out, we need to extend the classical ring of polynomials and obtain a wider class of functions to work with. We compare different approaches and present our point of focus, namely the ring of arc-analytic functions, in which even the Nullstellensatz is proven to hold.

Noncommutative Geometry

Speaker: Yang Liu (Max Planck Institute (Bonn))

"Scalar curvature in the conformal geometry of Connes-Landi deformation"

Time: 12:00 - 13:30

Room: MC 106

A general question behind the talk is to explore a good notion for intrinsic curvature in the framework of noncommutative geometry started by Alain Connes in the 80’s. It has only recently begun (2014) to be comprehended via the intensive study of modular geometry on the noncommutative two tori. In this talk, we will focus on a class of noncommutative manifolds obtained by deforming certain Riemannian manifolds along a torus action. I will explain how to formulate some basic notions in Riemannian geometry that are often described in local charts (such as the metric tensor, scalar curvature) using the language of functional analysis so that they will survive in the noncommutative setting. The highlight is that under a noncommutative conformal change of metric, we found not only the conformal change of the scalar curvature in Riemannian geometry but also some exciting new features: the quantum part of the curvature which is hidden in the commutative setting. What is more striking is that the quantum part of the curvature is defined by certain entire functions which play a prominent role in many other areas in mathematics (e.g. in the theory of characteristic classes).

Noncommutative Geometry

Speaker: Yang Liu (Max Planck Institute (Bonn))

"Scalar curvature in the conformal geometry of Connes-Landi deformation"

Time: 14:00 - 16:00

Room: MC 108

A general question behind the talk is to explore a good notion for intrinsic curvature in the framework of noncommutative geometry started by Alain Connes in the 80’s. It has only recently begun (2014) to be comprehended via the intensive study of modular geometry on the noncommutative two tori. In this talk, we will focus on a class of noncommutative manifolds obtained by deforming certain Riemannian manifolds along a torus action. I will explain how to formulate some basic notions in Riemannian geometry that are often described in local charts (such as the metric tensor, scalar curvature) using the language of functional analysis so that they will survive in the noncommutative setting. The highlight is that under a noncommutative conformal change of metric, we found not only the conformal change of the scalar curvature in Riemannian geometry but also some exciting new features: the quantum part of the curvature which is hidden in the commutative setting. What is more striking is that the quantum part of the curvature is defined by certain entire functions which play a prominent role in many other areas in mathematics (e.g. in the theory of characteristic classes).