Geometry and Topology

Speaker: Bernard Badzioch (University of Buffalo)

"Higher torsion invariants of smooth bundles"

Time: 15:30 - 16:30

Room: MC 107

Higher torsion invariants generalize the notion of the classical Reidemeister torsion. While the Reidemeister torsion is a tool for distinguishing between finite CW-complexes that are homotopy equivalent but not homeomorphic, higher torsion can detect bundles of smooth compact manifolds that are fiberwise homotopy equivalent (or even fiberwise homeomorphic), but have different smooth structures. In recent years various constructions of higher torsion invariants appeared including the higher analytical torsion of Bismut and Lott and Morse-theoretical construction of Igusa and Klein. The talk will present a construction of higher torsion that is based on the machinery of homotopy theory and some of its applications. The talk is based on joint work with W. Dorabiala, J. Klein and B. Williams.

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"Rational Convexity of Lagrangian inclusions (Part II)"

Time: 14:30 - 15:30

Room: MC 107

In the second part I will outline the proof of rational convexity of singular Lagrangian inclusions with a Whitney umbrella. The proof consists of finding a pair of functions that satisfy certain properties. These functions are constructed as deformation of the standard symplectic structure on $C^2$.

Noncommutative Geometry

Speaker: Boris Ugurcan (Western University (Assistant Professor and Postdoctoral Fellow))

"Non-commutative Stochastic Processes, Semi-groups and Dilation Theory"

Time: 15:00 - 16:30

Room: MC 107

In the first part of the talk, we recall the well-known correspondence between semi-groups and stochastic processes both in the commutative and non-commutative settings. We explain how this correspondence can be used to develop analysis on singular spaces such as fractals. Then, we proceed to a survey (including our results) of dilation theorems in operator algebras and how these theorems appear in the study of non-commutative stochastic processes.

Graduate Seminar

Speaker: Dinesh Valluri (Western)

"canceled"

Time: 13:00 - 14:00

Room: MC 106

Homotopy Theory

Speaker: (Western)

"No meeting today"

Time: 14:00 - 15:30

Room: MC 107

We resume next week.

Colloquium

Speaker: Tony Pantev (University of Pennsylvania)

"*moved to Fall 2015*"

Time: 15:30 - 16:30

Room: MC 107

TBA

Noncommutative Geometry

Speaker: Mitsuru Wilson (Western University (PhD Candidate))

"NCG Learning Seminar: The Local index formula III: Examples"

Time: 11:00 - 12:30

Room: MC 106

This is the third part of my talk on the local index formula. In this talk, I give two classical examples, the circle $\mathbb{T}$ and the torus $\mathbb{T}^2$ as spectral triples to demonstrate the use of the local index formula in the odd case and the even case, respectively.

Geometry and Topology

Speaker: Martin Helmer (Computer Science, Western)

"Algorithms to Compute Chern-Schwartz-Macpherson and Segre Classes and the Euler Characteristic"

Time: 15:30 - 16:30

Room: MC 107

Let V be a closed subscheme of a n dimensional projective space. We give algorithms to compute the Chern-Schwartz-MacPherson class, Euler characteristic and Segre class of V. These algorithms can be implemented using either symbolic or numerical methods. The basis for these algorithms is a method for calculating the projective degrees of a rational map defined by a homogeneous ideal. When combined with formula for the Chern-Schwartz-MacPherson class of a projective hypersurface and the Segre class of a projective variety in terms of the projective degrees of certain rational maps this gives us algorithms to compute the Chern-Schwartz-MacPherson class and Segre class of a projective variety. Since the Euler characteristic of V is the degree of the zero dimensional component of the Chern-Schwartz-MacPherson class of V our algorithm also computes the Euler characteristic of V. The algorithms are tested on several examples and are found to perform favourably compared to other algorithms for computing Chern-Schwartz-MacPherson classes, Segre classes and Euler characteristics. For the special case where V is a global complete intersection we develop a additional algorithm to compute the Chern-Schwartz-MacPherson class. This algorithm complements existing algorithms by providing performance improvements in the computation Chern-Schwartz-MacPherson class for some complete intersection schemes, particularly those which correspond to ideals which have few singular generators. These algorithms are implemented in Macaulay2.

Analysis Seminar

Speaker: Myrto Manolaki (Western)

"A result on harmonic measure with applications to Taylor series (Part I)"

Time: 14:30 - 15:30

Room: MC 107

Let $f$ be a holomorphic function on the unit disc, and let $(S_{n_k})$ be a subsequence of its Taylor polynomials about 0. In this talk we will see that the nontangential limit of $f$ and $\lim_{k\rightarrow \infty} S_{n_{k}}$ agree at a.e. point of the unit circle where they simultaneously exist. In the first part of this talk we will focus on this result and its applications. In the second part of the talk we will discuss a convergence theorem of harmonic measures on domains in $\mathbb{R}^{N}$ which played a key role in the proof of the above result and it is of independent interest. (Joint work with Stephen Gardiner)

Noncommutative Geometry

Speaker: Joakim Arnlind (Linkoping University, Sweden)

"Naive Riemannian geometry of the noncommutative 4-sphere"

Time: 11:00 - 12:00

Room: MC 106

I will present a pedestrian way of introducing the concepts of Riemannian geometry for the noncommutative 4-sphere. This is done in analogy with the classical view of the 4-sphere as being embedded in five dimensional Euclidean space. By closely mimicking the construction of the tangent space for an embedded manifold in classical geometry, a particular module over the noncommutative 4-sphere is found and compared with the tangent bundle. Together with a set of corresponding derivations, one may introduce a connection on this module and show that it shares several properties with the Levi-Civita connection on the classical tangent bundle.

Graduate Seminar

Speaker: Armin Jamshidpey (Western)

"Rationality problem for algebraic tori"

Time: 13:00 - 14:00

Room: MC 106

In this session we will talk about rationality of algebraic tori. We will first define the notion of rational algebraic variety and then some relaxed notions of rationality. Algebraic tori are important objects in studying algebraic groups. In fact the role which they play is similar to the role of tori in the theory of Lie groups. In order to talk about the results about rationality of algebraic tori we will take a look at duality between the category of algebraic tori and category of G-lattices. We will end the session with the main results about birational classification of tori in small dimensions (up to 5).

Homotopy Theory

Speaker: Martin Frankland (Western)

"Secondary chain complexes and derived functors"

Time: 14:00 - 15:30

Room: MC 107

The $E_2$ term of the Adams spectral sequence is given by Ext groups over the Steenrod algebra, namely the algebra of primary (stable) cohomology operations. In this talk, we will present work of Baues and Jibladze on secondary chain complexes and secondary derived functors, which generalize the usual chain complexes and derived functors in homological algebra. With this machinery, the $E_3$ term can be expressed as a secondary Ext group over the algebra of secondary cohomology operations.

Colloquium

Speaker: Farzad Fatizadeh (Caltech)

"Local computations in noncommutative geometry"

Time: 15:30 - 16:30

Room: MC 107

Index theory on noncommutative algebras that arise from far more complicated spaces than manifolds, such as the space of leaves of a foliation, and properties of a noncommutative Chern character led to the discovery of cyclic cohomology and Connes' index formula. It states the coincidence between an analytic and a topological index for noncommutative algebras. The local index formula of Connes and Moscovici is an effective tool for computing the index pairings in noncommutative geometry by local formulas. This talk will be a review of these ideas and an indication of my joint works with Masoud Khalkhali on the computation of local geometric invariants of noncommutative tori, such as scalar curvature, when their flat geometry is conformally perturbed by a Weyl factor.

Noncommutative Geometry

Speaker: Piotr M. Hajac ((IM PAN Warszawa/University of New Brunswick))

"There and back again: from the Borsuk-Ulam theorem to quantum spaces"

Time: 11:00 - 12:30

Room: MC 106

Assuming that both temperature and pressure are continuous functions, we can conclude that there are always two antipodal points on Earth with exactly the same pressure and temperature. This is the two-dimensional version of the celebrated Borsuk-Ulam Theorem which states that for any continuous map from the n-dimensional sphere to n-dimensional real Euclidean space there is always a pair of antipodal points on the sphere that are identified by the map. Our quest to unravel topological mysteries in the Middle Earth of quantum spaces will begin with gentle preparations in the Shire of elementary topology. Then, after riding swiftly through the Rohan of C*-algebras and Gelfand-Naimark Theorems and carefully avoiding the Mordor of incomprehensible technicalities, we shall arrive in the Gondor of compact quantum groups acting on unital C*-algebras. It is therein that the generalized Borsuk-Ulam-type statements dwell waiting to be proven or disproven. Time permitting, we shall pay tribute to the ancient quantum group SUq(2), and show how the proven non-trivializability of the SUq(2)-principal instanton bundle is a special case of two different noncommutative Borsuk-Ulam-type conjectures. (Based on joint work with Paul F. Baum and Ludwik Dabrowski.)

Algebra Seminar

Speaker: Michael Bush (Washington and Lee University)

"Non-abelian generalizations of the Cohen-Lenstra Heuristics"

Time: 14:30 - 15:30

Room: MC 107

The class group of a number field is a finite abelian group which measures the failure of unique factorization in the ring of integers of the field. In the context of quadratic fields (both real and imaginary), the Cohen-Lenstra Heuristics make precise predictions about the statistical behavior of the class group if one orders fields by discriminant. Over the last several years, Nigel Boston, Farshid Hajir and I have formulated analogous non-abelian heuristics for such fields in which we replace the p-class group (p an odd prime) with the Galois group of the maximal unramified p-extension of the field. I'll discuss both the formulation of our conjectures and the evidence for them. No prior knowledge of the Cohen-Lenstra Heuristics will be assumed.

Geometry and Topology

Speaker: Ergun Yalcin (Bilkent University, Ankara)

"Finite group actions on homotopy spheres"

Time: 15:30 - 16:30

Room: MC 107

We are interested in classifying all finite groups which can act on a finite $CW$-complex homotopy equivalent to a sphere, such that all isotropy subgroups are rank one groups, meaning that they do not include $Z/p\times Z/p$ for any prime $p$. The similar question for free actions (all isotropy subgroups are trivial) has been answered completely by the works of P.A. Smith and R. Swan. There is a complete list of such groups (they are finite groups with periodic group cohomology) and we would like to obtain a similar list for actions with rank one isotropy. This is joint work with Ian Hambleton.

Analysis Seminar

Speaker: Myrto Manolaki (Western)

"A result on harmonic measure with applications to Taylor series (Part II)"

Time: 14:30 - 15:30

Room: MC 107

Let $f$ be a holomorphic function on the unit disc, and let $(S_{n_k})$ be a subsequence of its Taylor polynomials about 0. In this talk we will see that the nontangential limit of $f$ and $\lim_{k\rightarrow \infty} S_{n_{k}}$ agree at a.e. point of the unit circle where they simultaneously exist. In the first part of this talk we will focus on this result and its applications. In the second part of the talk we will discuss a convergence theorem of harmonic measures on domains in $\mathbb{R}^{N}$ which played a key role in the proof of the above result and it is of independent interest. (Joint work with Stephen Gardiner)

Noncommutative Geometry

Speaker: Luuk Verhoeven (Radboud University, Nijmegen)

"NCG Learning Seminar: Can one hear the shape of a drum?"

Time: 11:00 - 12:30

Room: MC 106

In this talk we will explore the current state of this famous question, first formulated this way by Marc Kac in 1966. The answer to this question is known to be "no, you can not," but that is not the end of it. We will discuss the mathematical formulation of the problem and the counter examples, followed by some, fairly recent, positive results and the tools, specifically the heat- and wave-trace, that can be used.

Graduate Seminar

Speaker: Ahmed Ashraf (Western)

"Homological Sylow Theorem"

Time: 13:00 - 14:00

Room: MC 106

We'll prove the Homological Sylow theorem using the elementary approach of Surowski. This involves poset theory, simplicial complexes and character theory of finite groups. The talk is quite elementary. The main highlights are Order complexes with a G-action, Quillen's fibre lemma and Lefschetz character. All of these interact in an interesting way to prove the required result.

Homotopy Theory

Speaker: Gaohong Wang (Western)

"A-infinity structure on Ext-algebras"

Time: 14:00 - 15:30

Room: MC 107

We give an introduction to A-infinity algebras in this talk, which is a generalisation of differential graded algebras. We show that for a graded algebra A, the Ext-algebra has an A-infinity structure that contains sufficient information to recover A. On the other hand, we will present an example where the usual associative algebra structure on the Ext-algebra cannot recover A. We also show that the A-infinity structure is closely related to Massey products.

Colloquium

Speaker: Paul Balmer (UCLA)

"Prime ideals in the equivariant stable homotopy category"

Time: 15:30 - 16:30

Room: MC 107

Tensor triangulated categories appear in algebraic geometry, in homotopy theory and in representation theory, and beyond. Once the naive idea of classifying all objects is abandoned, the natural question becomes to classify the so-called ``thick tensor-ideals". The latter classification can always be achieved, via the spectrum of prime ideals. We shall review these ideas and see what new results we can obtain in the equivariant stable homotopy category. -- This is joint work with Beren Sanders.

Noncommutative Geometry

Speaker: Shahab Azarfar (Western University)

"An example for the local index formula on the two torus"

Time: 11:00 - 12:00

Room: MC 106

We consider the canonical spectral triple associated to a two-torus as a compact Riemannian spin manifold. As an example for the local index formula in the even case, we compute the index of the corresponding Dirac operator with coefficients in a rank one vector bundle given by a special class of projections.

Algebra Seminar

Speaker: Adam Topaz (UC Berkeley)

"On mod-$\ell$ birational anabelian geometry"

Time: 15:30 - 16:30

Room: MC 107

In the early 90's, Bogomolov introduced a program whose ultimate goal is to reconstruct function fields of dimension $> 1$ over algebraically closed fields from their pro-$\ell$ 2-step nilpotent Galois groups. Although it is far from being resolved in full generality, this program has since been carried through for function fields over the algebraic closure of a prime field. After an introduction to birational anabelian geometry and Bogomolov's program, in this talk I will describe the possibility of a $Z/\ell$-analogue, including the inherent problems/difficulties, as well as some partial results in this direction.

Geometry and Topology

Speaker: Dan Grayson (UIUC)

"Homotopy Type Theory and Univalent Foundations"

Time: 15:30 - 16:30

Room: MC 107

Homotopy type theory with the univalence axiom of Voevodsky provides both a new logical foundation for mathematics (Univalent Foundations) and a formal language usable with computers for checking the proofs mathematicians make daily. As a foundation, it replaces set theory with a framework where sets are defined in terms of a more primitive notion called "type". As a formal language, it encodes the axioms of mathematics and the rules of logic simultaneously, and promises to make the extraction of algorithms and values from constructive proofs easy. With a semantic interpretation in homotopy theory, it offers an alternative world where the proofs of basic theorems of mathematics can be formalized with minimal verbosity and verified by computer.

As a relative newcomer to the field, I will survey these recent developments and sketch the basic concepts for a general mathematical audience.