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

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

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

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

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

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

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

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

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.