Geometry and Topology

Speaker: Hiraku Abe (McMaster)

"Flat families of Hessenberg varieties with an application to Newton-Okounkov bodies"

Time: 15:30

Room: MC 107

Hessenberg varieties are subvarieties of the full flag variety. In this talk, I will concentrate on Lie type A. I will talk about a flat degeneration of a regular semisimple Hessenberg variety to a regular nilpotent Hessenberg variety, and I will explain how we can use this flat family to compute some Newton-Okounkov bodies of the Peterson variety of dimension 2. Along the way, we will also see that any regular nilpotent Hessenberg variety is a local complete intersection; this is a generalization of a result in Erik Insko’s PhD thesis. This is a joint work with Lauren DeDieu, Federico Galetto, and Megumi Harada.

Colloquium

Speaker: Alejandro Morales (UCLA)

"Hook formulas for Standard Young tableaux of skew shape"

Time: 14:30

Room: MC 107

Counting linear extensions of a partial order (linear orders compatible with the partial order) is a classical and computationally difficult problem in enumeration and computer science. A family of partial orders that are prevalent in enumerative and algebraic combinatorics come from Young diagrams of partitions and skew partitions. Their linear extensions are called standard Young tableaux. The celebrated hook-length formula of Frame, Robinson and Thrall from 1954 gives a product formula for the number of standard Young tableaux of partition shape. No such product formula exists for skew partitions.

In 2014, Naruse announced a formula for skew shapes as a positive sum of products of hook-lengths using ”excited diagrams” of Ikeda-Naruse, Kreiman, Knutson-Miller-Yong in the context of equivariant cohomology. We prove Naruse’s formula algebraically and combinatorially in several different ways. Also, we show how excited diagrams give asymptotic results and product formulas for the enumeration of certain families of skew tableaux. Lastly, we give analogues of Naruse's formula in the context of equivariant K-theory.

This is joint work with Igor Pak and Greta Panova.

Analysis Seminar

Speaker: Daniel Burns (University of Michigan)

"[Cancelled]"

Time: 15:30

Room: MC 108

Cancelled due to road conditions.

Colloquium

Speaker: Kiumars Kaveh (University of Pittsburg)

"Algebraic geometry, convex geometry and computational algebra"

Time: 15:30

Room: MC 107

We begin with a brief introduction to Grobner theory and tropical geometry. Grobner bases are one the most fundamental tools in computational algebra. Tropical geometry can be described as a piecewise linear version of algebra/algebraic geometry and comes from looking at a variety from "infinity". It has many applications in different areas such as phylogenetics and optimization. We then talk about new results (joint with Chris Manon) about far extending Grobner theory concepts and doing algorithmic computations in general algebras equipped with valuations (in particular coordinate rings of varieties). In particular, this makes a direct connection between tropical geometry and recently emerged theory of Newton-Okounkov bodies. A central problem in this web of ideas is "degenerating" a given variety to a toric variety. There are many connections with other areas such as applied algebra, symplectic geometry (Hamiltonian systems) and representation theory (reductive group actions).

Colloquium

Speaker: Chris Kapulkin (Western)

"Formalization of Mathematics and the Univalent Foundations"

Time: 14:00

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

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

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

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

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

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

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

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

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

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).

Algebra Seminar

Speaker: Adam Chapman (Tel-Hai Academic College)

"Linked fields of characteristic 2 and their u-invariant"

Time: 14:30

Room: MC 107

The u-invariant of a field is the maximal dimension of a nonsingular anisotropic quadratic form over that field, whose order in the Witt group of the field is finite. By a classical theorem of Elman and Lam, the u-invariant of a linked field of characteristic different from 2 can be either 0, 1, 2, 4 or 8. The analogous question in the case of characteristic 2 remained open for a long time. We will discuss the proof of the equivalent statement in characteristic 2, recently obtained in joint work by Andrew Dolphin and the speaker.

Basic Notions Seminar

Speaker: Masoud Khalkhali (Western)

"What is Spectral Geometry?"

Time: 15:30

Room: MC 107

Spectral geometry, among other things, asks the question `can one hear the shape of a drum?' To a mathematical object, say a Riemannian manifold, one can attach its spectrum and one is interested to know to what extent the object can be recovered from its spectrum. The spectral information can be encoded in terms of zeta functions, heat trace, or wave trace. Isometry invariants like volume and total scalar curvature can be obtained as special values of the spectral zeta function (Weyl's law). I shall give a quick introduction to these ideas and will end by giving the first example of two isospectral manifolds which are not isometric. The example, due to Milnor (using some deep work of Ernst Witt based on the theory of modular forms), exhibits two 16 dimensional flat tori which are isospectral but not isometric. This talk will be accessible to all grad students.