Geometry and Topology

Speaker: Matthias Franz (Western)

"Equivariant cohomology and syzygies"

Time: 15:30

Room: MC 107

The GKM method is a powerful way to compute the equivariant (and ordinary) cohomology of many spaces with torus actions. So far it has been applied to so-called equivariantly formal $T$-spaces, which include compact Hamiltonian $T$-manifolds.

In this talk I will explain that the GKM method is valid for a much larger class of $T$-spaces. The explanation is based on a new interpretation of a sequence originally due to Atiyah and Bredon, and involves the notion of syzygies as used in commutative algebra. I will also exhibit a surprising relation between the equivariant Poincaré pairing and the GKM description.

This is joint work with Chris Allday and Volker Puppe.

Analysis Seminar

Speaker: Seyed Mohammad Hadi Seyedinejad (Western)

"Testing local regularity of complex analytic mappings by fibred powers, II"

Time: 14:40

Room: MC 107

This two-session talk will be concerned with holomorphic mappings between complex analytic sets (or more generally, analytic spaces). Local regularity of such a mapping can be measured by uniformity (or lack thereof) of the family of its fibres. In the first part of the talk, we will discuss the general idea of testing local regularity (like openness or flatness) by passing to fibred powers of a given map. The second session will be devoted to a recent joint work with Janusz Adamus: We establish an analytic version of flatness descent to prove a criterion for flatness of a holomorphic mapping with singular target. Previously, the best analogous result had been known only for the case of smooth targets.

Graduate Seminar

Speaker: Zack Wolske (Western)

"Techniques in Algebraic Number Theory"

Time: 16:30

Room: MC 107

We introduce some standard techniques in algebraic number theory to investigate solutions of polynomials. If we consider the integers mod p, and our polynomial has no solution there (local), then it has no integer solution (global). But if there are solutions for every p, can we find a global solution? More generally, we can ask for rational solutions, and consider completions of the rationals as localizations. This is called the Hasse local-global principle. We will introduce and use Henselian lifting, the class number, and the Minkowski bound to give an example of a polynomial which does not satisfy the Hasse principle.

Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"Einstein Manifolds and Distinct 7-Manifolds Admitting Positively Curved Riemannian Structures"

Time: 10:30

Room: MC 108

An Einstein manifold is a smooth manifold whose Ricci tensor is proportional to the metric. Many homogeneous spaces can be realized as Einstein manifolds, and have been widely studied for general existence and nonexistence of Einstein metrics. In this talk we will give examples of homogeneous and Einstein manifolds and discuss some of the general underlying theory related to these spaces. We will also briefly discuss how this can be extended to the noncommutative case. Finally, we will show that if we are given a closed, connected, one-dimensional subgroup H of SU(3) that has no nonzero fixed points, then SU(3)/H admits an SU(3)-invariant Riemannian structure of strictly positive curvature. This result was first proven in 1975 by Aloff and Wallach, and it was here that the famous Aloff-Wallach spaces were introduced.

Noncommutative Geometry

Speaker: Mohammad Hassanzadeh (UNB)

"A new class of ASYD modules for Hopf cyclic cohomology"

Time: 13:30

Room: MC 108

We show that the category of coefficients for Hopf cyclic cohomology has two proper subcategories where one of them is the category of stable anti Yetter-Drinfeld modules. Generalizations of suitable coefficients for Hopf cyclic cohomology are introduced. The notion of stable anti Yetter-Drinfeld modules is extended based on underlying symmetries. We show that the new introduced categories for coefficients of Hopf cyclic cohomology and the category of stable anti-Yetter-Drinfeld modules are all different. (This is joint work with Bahram. Rangipour and Dan. Kucerovsky )

Algebra Seminar

Speaker: Marcy Robertson (Western)

"Introduction to derived Hall algebras"

Time: 14:30

Room: MC 107

Roughly speaking, the Hall algebra $H(A)$ of a (small) Abelian category $A$ is the algebra of finitely supported functions on the moduli space of objects of $A$ (i.e. the set of isoclasses of objects of $A$ with the discrete topology). Interest in Hall algebras exploded in the early 1990's when Ringel discovered that the Hall algebra associated to the category of $F_q$-representations of a Dynkin quiver $Q$ provides a realization of the positive part of the (quantized) enveloping algebra of the (simple) complex Lie algebra associated to the same Dynkin diagram. To\"{e}n and Bergner have used the theory of model categories to obtain Hall algebras on triangulated categories. In this talk we will survey these constructions and, time permitting, explain some open problems in this area which are being studied via homotopy theory.

Geometry and Topology

Speaker: Dan Christensen (Western)

"The homotopy theory of smooth spaces"

Time: 15:30

Room: MC 107

I will describe some categories of "smooth spaces" which generalize the notion of manifold. The generalizations allow us to form smooth spaces consisting of subsets and quotients of manifolds, as well as loop spaces and other function spaces. In more technical language, these categories of smooth spaces are complete, cocomplete and cartesian closed. I will give examples, discuss possible applications and explain what we have learned about the homotopy theory of these categories. This is work in progress with Enxin Wu.

Analysis Seminar

Speaker: Damir Kinzebulatov (Toronto)

"Oka-Cartan type theory for some subalgebras of holomorphic functions on coverings of complex manifolds"

Time: 14:30

Room: MC 107

We develop the basic elements of complex function theory within certain subalgebras of holomorphic functions on coverings of complex manifolds (including holomorphic extension from complex submanifolds, properties of divisors, corona type theorem, holomorphic analogue of Peter-Weyl approximation theorem, Hartogs type theorem, characterization of the uniqueness sets, etc). Our model examples are: (1) subalgebra of Bohr's holomorphic almost periodic functions on tube domains (i.e. the uniform limits of exponential polynomials) (2) subalgebra of all fibrewise bounded holomorphic functions (arising in corona problem for $H^\infty$) (3) subalgebra of holomorphic functions having fibrewise limits.

Our proofs are based on the analogues of Cartan theorems A and B for coherent type sheaves on the maximal ideal spaces of these subalgebras.

This is joint work with Alexander Brudnyi.

Colloquium

Speaker: Mircea Mustata (University of Michigan)

"Invariants of singularities in positive characteristic"

Time: 15:30

Room: MC 107

Given a polynomial with integer coefficients, one can define invariants of singularities by either considering the polynomial over the complex numbers, or by taking reduction modulo prime integers, and using the Frobenius morphism. I will describe these invariants, and I will discuss some known and conjectural relations between them.

Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"Einstein Manifolds and Distinct 7-Manifolds Admitting Positively Curved Riemannian Structures (Part 2)"

Time: 10:30

Room: MC 108

An Einstein manifold is a smooth manifold whose Ricci tensor is proportional to the metric. Many homogeneous spaces can be realized as Einstein manifolds, and have been widely studied for general existence and nonexistence of Einstein metrics. In this talk we will give examples of homogeneous and Einstein manifolds and discuss some of the general underlying theory related to these spaces. We will also briefly discuss how this can be extended to the noncommutative case. Finally, we will show that if we are given a closed, connected, one-dimensional subgroup H of SU(3) that has no nonzero fixed points, then SU(3)/H admits an SU(3)-invariant Riemannian structure of strictly positive curvature. This result was first proven in 1975 by Aloff and Wallach, and it was here that the famous Aloff-Wallach spaces were introduced.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Curvature in Noncommutative Geormetry"

Time: 13:30

Room: MC 108

Algebra Seminar

Speaker: Mehdi Garrousian (Western)

"A random walk around Koszul algebras"

Time: 14:30

Room: MC 107

A connected graded algebra is called Koszul if the ground field has a linear resolution, i.e. differentials are defined by matrices that only have linear entries. This condition has less than a million equivalent descriptions. In this survey talk, I will mention a few of these characterizations and examine the resulting homological behavior. As a motivation, I start off by showing the LCS formula for the pure braid group. This is an instance of a more general result about the cohomology ring of a nice class of hyperplane arrangements. I am also planning to describe more examples with origins in quantum groups and show a quick proof for the classical PBW theorem. If there is time left, I will say a few words about the interaction of the Koszul property with the Bloch-Kato conjecture. At last but not least, I will mention the biggest open problem of this area which asks for the correct pronunciation of the word Koszul.

Geometry and Topology

Speaker: Timo Schurg

"Perfect obstruction theories and quasi-smooth derived schemes"

Time: 15:30

Room: MC 107

We discuss the equivalence of perfect obstruction theories (extensively used in formation of Gromov-Witten invariants) and quasi-smooth derived schemes. The latter can be thought of as a derived zero locus. When the zero locus is locally given by a complete intersection, one gets a classical scheme.

Analysis Seminar

Speaker: Seyed Mehdi Mousavi (Western)

"Maximal Tori in Symplectomorphism Groups and Convexity"

Time: 14:40

Room: MC 107

Symplectomorphism groups are one of the classical infinite-dimensional Lie groups that have been studied. Arnold's paper in 1966, where he used methods of infinite-dimensional Lie theory to study the hydrodynamics of a perfect incompressible fluid, has motivated intensive research in infinite-dimensional Lie theory. He showed that the geodesics on the group of volume preserving diffeomorphisms are essentially solutions of the Euler's equations. In a fundamental paper in 1970 Marsden and Ebin studied some infinite-dimensional groups in more details which included symplectomorphism groups. In this talk we study a special class of symplectomorphism groups that resemble compact Lie groups in a particular way. We see there is a similar notion of the so-called maximal tori in the symplectomorphism groups of toric manifolds. As a consequence we see there is an analogue of the Schur-Horn-Kostant convexity theorem in this infinite-dimensional setting. It also should be mentioned that these results are a generalization of results that were obtained by Bao-Ratiu 1997, Bloch-Flaschka-Ratiu 1993 and El-hadrami 1996 for special cases of toric manifolds.

Pizza Seminar

Speaker: Marcy Robertson (Western)

"What is Algebraic Topology?"

Time: 16:30

Room: MC 107

The goal of this talk is to introduce some of the most basic notions in the field of topology. We focus on the concept of a surface or 2-dimensional manifold. A surface is a mathematical abstraction of the familiar concept of a surface made of paper - like the surface of a sphere, the Mobius strip, and so on. We will spend time constructing these surfaces and then I will demonstrate the tools an algebraic topologist would use to classify all possible surfaces.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Curvature in Noncommutative Geormetry"

Time: 10:30

Room: MC 108

Noncommutative Geometry

Speaker: Ali Motadelro (Western)

"Spectral Aspects of Non-commutative Geometry"

Time: 13:30

Room: MC 108

Algebra Seminar

Speaker: Janusz Adamus (Western)

"Effective flatness criterion - from Auslander to Vasconcelos' conjecture"

Time: 14:30

Room: MC 107

In his seminal 1961 paper, Auslander gave a beautiful characterization of flatness of a finite module over a regular local ring $R$ in terms of torsion in tensor powers of the module. Almost 40 years later, Vasconcelos conjectured a generalization of this criterion to the category of finite type $R$-algebras. I will survey a recent development in this area, leading to the establishing of Vasconcelos' conjecture (and then some) in characteristic zero, by local analytic methods. These are joint works with E. Bierstone and P.D. Milman, and with Hadi Seyedinejad.

Colloquium

Speaker: Yongbin Ruan (University of Michigan)

"Gromov-Witten theory and quasi-modular forms"

Time: 15:30

Room: MC 107

For years, much effort were spent to connect geometry-physics to number theory. Recently, a deep relation between Gromov-Witten theory and number theory were discovered on the topic of quasi-modular forms. It has already generated a great deal of interest among Gromov-Witten theory community. Moreover, it appears to be the beginning of an exciting area, which could potentially benefit both subjects. In the talk, I will explain these interesting discoveries.

Geometry and Topology

Speaker: Teena Gerhardt (Michigan State University)

"Cyclotomic spectra and computations in algebraic K-theory"

Time: 15:30

Room: MC 107

In this talk I will describe joint work with Vigleik Angeltveit, Mike Hill, and Ayelet Lindenstrauss, yielding new computations of algebraic K-theory groups. Techniques from equivariant stable homotopy theory are often key to algebraic K-theory computations. In this case we use n-cubes of cyclotomic spectra to compute the topological cyclic homology, and hence K-theory, of truncated polynomial algebras in several variables.

Ph.D. Presentation

Speaker: S. Pal (Western)

"Segre Varieties and the Reflection Principle"

Time: 14:40

Room: MC 107

Colloquium

Speaker: Alexandre Sukhov (Universite de Lille)

"Hartogs Lemma and Symplectic Non-Squeezing"

Time: 15:30

Room: MC 107

We discuss a result on filling real 2-tori by Levi-flat hypersurfaces in almost complex manifolds and some applications to global rigidity of symplectic structures.

Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"Ricci Flow in Differential and Noncommutative Geometry"

Time: 10:30

Room: MC 108

Studying the Ricci flow of a smooth, closed manifold M equipped with a Riemannian metric g involves the process of allowing the metric g to evolve over time under the PDE g_{t} = -2Ric(g). Ricci flow was, in fact, the main tool used by Perelman to prove the Poincare conjecture. The purpose of this talk will be to discuss what is Ricci flow, to present where it comes from and to provide examples of Ricci flow of certain manifolds. Our discussion will then lead into an analysis of a paper written by Bhuyain and Marcolli, who constructed a version of Ricci flow for noncommutative two-tori. The Ricci flow is a fundamental tool used to understand the geometry and topology of manifolds, and understanding it well will help us understand how we can classify other noncommutative spaces such as noncommutative tori in higher dimensions.

Algebra Seminar

Speaker: Lila Kari (Western)

"DNA Computing: Implications for Theoretical Computer Science"

Time: 14:30

Room: MC 107

We are now witnessing exciting interactions between computer science and mathematics on one side, and the natural sciences on the other. While the natural sciences are rapidly absorbing notions, techniques and methodologies intrinsic to computer science and mathematics, theoretical computer science is adapting and extending its traditional notion of computation and computational techniques, to account for computation taking part in nature around us.

This talk will outline several of the fruitful directions of theoretical computer science research that originated from the study of DNA. I will describe comma-free codes inspired by the studies into the genetic code, splicing systems, optimal encodings for DNA Computing, sticker systems, Watson-Crick automata, combinatorics on DNA words, cellular computing, and computing by DNA self-assembly.

Langlands seminar

Speaker: Zack Wolske (Western)

"Classical Themes in Number Theory"

Time: 16:00

Room: MC 108

We introduce fundamental topics involving number fields, including ideal splitting, ramification, and the Frobenius element, along with many motivating questions and examples. We conclude with a discussion of the local global principle, and ask some number theoretic questions which can be easily understood, but require automorphic forms to resolve.

Geometry and Topology

Speaker: Marcy Robertson (Western)

"Operads, multicategories, and higher dimensional deformations"

Time: 15:30

Room: MC 107

Operads, and the more general multicategories, are combinatorial devices originally used in algebraic topology as a ``bookkeeping'' devices that described the internal operations of iterated loop spaces. The basic idea of an operad, however, is quite flexible and can be adapted to problems in algebra, mathematical physics, and computer science. The goal of this talk is to give a quick introduction to the Grothendieck-Teichm\"{u}ller group, as introduced by Drinfeld and Ihara, describe some of the conjectures relating this group to quantized deformations, and explain how this conjecture is being understood through the machinery of operads (up to homotopy).

Colloquium

Speaker: Boris Khesin (University of Toronto)

"Symplectic fluids and point vortices"

Time: 15:30

Room: MC 107

We describe the motion of symplectic fluids as an Euler-Arnold equation for the group of symplectic diffeomorphisms. We relate it to the Lagrangian study of symplectic fluids by D.Ebin, describe a symplectic analog of vorticity and the finite-dimensional Hamiltonian systems of symplectic point vortices.

Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"Ricci Flow in Differential and Noncommutative Geometry (2)"

Time: 10:30

Room: MC 108

Studying the Ricci flow of a smooth, closed manifold M equipped with a Riemannian metric g involves the process of allowing the metric g to evolve over time under the PDE g_{t} = -2Ric(g). Ricci flow was, in fact, the main tool used by Perelman to prove the Poincare conjecture. The purpose of this talk will be to discuss what is Ricci flow, to present where it comes from and to provide examples of Ricci flow of certain manifolds. Our discussion will then lead into an analysis of a paper written by Bhuyain and Marcolli, who constructed a version of Ricci flow for noncommutative two-tori. The Ricci flow is a fundamental tool used to understand the geometry and topology of manifolds, and understanding it well will help us understand how we can classify other noncommutative spaces such as noncommutative tori in higher dimensions.

Algebra Seminar

Speaker: Ali Moatadelro (Western)

"Spectral geometry of noncommutative two torus"

Time: 14:30

Room: MC 107

Recently an analogue of the Gauss-Bonnet theorem has been proved by Connes-Tretkoff and Fathizadeh-Khalkhali for noncommutative two torus. The idea is based on the direct computation of the value at origin of the zeta function associated to the corresponding Laplacian. In this talk we will briefly discuss the above theorem and explain a related problem.