Geometry and Topology

Speaker: Hugh Thomas (UNB)

"Monodromy for the quintic mirror"

Time: 15:30

Room: MC 107

The mirror to the quintic in P^4 is a family X_p of Calabi-Yau 3-folds over a thrice-punctured sphere. As p moves in a loop around each of the three punctures, we can parallel transport classes in H^3(X_p), and observe the monodromy. H^3(X_p) is four-dimensional, and the monodromy can be expressed by matrices in Sp(4,Z). These matrices generate a subgroup which is dense in Sp(4,Z), but it was not known whether or not it was of finite index. We showed that the subgroup is isomorphic to the free product Z/5 * Z, from which it follows that it cannot be of finite index. The mirror quintic family is one of 14 similar families of CY 3-folds; our methods establish similar results for 7 of the 14 families. For the other 7, it has recently been shown that the monodromy is of finite index, so our result is best possible. This talk is based on joint work with Chris Brav, arXiv:1210.0523.

Homotopy Theory

Speaker: Marcy Robertson (Western)

"DG-categories and motives"

Time: 14:30

Room: MC 107

Noncommutative Geometry

Speaker: Latham Boyle (Perimeter Institute)

"Non-Commutative Geometry, Non-Associative Geometry, and the Standard Model of Particle Physics"

Time: 16:30

Room: MC 107

Connes and others have developed a notion of non-commutative geometry (NCG) that generalizes Riemannian geometry, and provides a framework in which the standard model of particle physics, coupled to Einstein gravity, may be concisely and elegantly recast. I will explain how this formalism may be reformulated in a way that naturally generalizes from non-commutative to non-associative geometry. In the process, several of the standard axioms of NCG are conceptually reinterpreted. This reformulation also suggests a new constraint on the class of NCGs used to describe the standard model of particle physics. Remarkably, this new condition resolves a long-standing puzzle about the embedding of the standard model in NCG, by precisely eliminating from the action formula the collection of seven unwanted terms that previously had to be removed by an extra (empirically-motivated, ad hoc) assumption.

Analysis Seminar

Speaker: Myrto Manolaki (Western)

"Harmonic functions with universal expansions"

Time: 11:30

Room: MC 108

Let $G$ be a domain in $\mathbb{R}^N$ and let $w$ be a point in $G$. This talk is concerned with harmonic functions on $G$ with the property that their homogeneous polynomial expansion about $w$ are "universal" in the sense that they can approximate all plausible functions in the complement of $G$. We will discuss topological conditions under which such functions exist, and the role played by the choice of the point $w$. These results can be generalised for the corresponding class of universal holomorphic functions on certain domains of $\mathbb{C}^N$.

Index Theory Seminar

Speaker: Sean Fitzpatrick (Western)

"Atiyah's "pushed symbol" construction and index theory on noncompact manifolds"

Time: 12:00

Room: MC 107

On any manifold with Spin$^c$ structure, one can construct a corresponding Dirac operator, which is a first-order elliptic differential operator whose principal symbol can be expressed in terms of Clifford multiplication. Dirac operators are Fredholm on compact manifolds, but not on noncompact manifolds.

I'll give a construction due to Atiyah that deforms the symbol of a $G$-invariant Dirac operator into a transversally elliptic symbol whose equivariant index is well-defined, by "pushing" the characteristic set of the symbol off the zero section using an invariant vector field. This construction is essentially topological in nature, and has been used by Paradan, Ma and Zhang, and others in the study of the "quantization commutes with reduction" problem in symplectic geometry.

I will say a few words about this problem, and will end with a discussion of a construction due to Maxim Braverman of a generalized Dirac operator whose analytic index coincides with the topological index of the "pushed symbol" of Atiyah.

Colloquium

Speaker: Aravind Asok (Univ. of Southern California)

"Projective modules and $\mathbb{A}^{1}$-homotopy theory"

Time: 15:30

Room: MC 107

The theory of projective modules has, from its inception, taken as inspiration for theorems and techniques ideas from the topological theory of vector bundles on (nice) topological spaces. I will explain another chapter in this story: ideas from classical homotopy theory can transplanted to algebraic geometry via the Morel-Voevodsky $\mathbb{A}^{1}$-homotopy category to deduce new results about classification and splitting problems for projective modules over smooth affine algebras. Some of the results I discuss are the product of joint work with Jean Fasel.

Noncommutative Geometry

Speaker: Mohammad Hassanzadeh (University of Windsor)

"On representation theory of integrals for Hopf algebras"

Time: 14:30

Room: MC 108

The study of integrals is an important topic in the theory of Hopf algebras. This notion was introduced by Sweedler in 1969 motivated by the uniquenness of the Haar integral on locally compact groups. In this talk we focus on the representation theory of integrals and explain how special types of integrals can produce (Anti)-Yetter-Drinfeld modules. This can be used to classify total integrals and (cleft) Hopf Galois extensions.

Analysis Seminar

Speaker: Ilia Binder (University of Toronto)

"The rate of convergence of Cardy-Smirnov observable"

Time: 15:30

Room: MC 108

Convergence of the Cardy-Smirnov observables is the crucial part of the famous proof of existence of the scaling limit of critical percolation on hexagonal lattice. I will discuss a proof of the power law convergence of Cardy-Smirnov observables on arbitrary simply-connected planar domains. The proof works for the usual critical percolation on hexagonal lattice, as well as for some modified versions. In the heart of the proof lies a careful study of the fine boundary properties of arbitrary planar domains. I will also explain the relevance of this result for the investigation of the rate of convergence of the critical percolation to its scaling limit. This is a joint work with L. Chayes and H. K. Lei.

Index Theory Seminar

Speaker: Sean Fitzpatrick (Western)

"Index theory for twisted Dirac operators on Spin$^c$ manifolds"

Time: 12:00

Room: MC 107

After a brief introduction to the "quantization commutes with reduction" credo in symplectic geometry (and what index theory has to do with quantization), I will report on recent results of Paradan and Vergne on the multiplicities of the equivariant index for twisted Spin$^c$-Dirac operators. (In this case, "recent" means that the results appeared on the arXiv the day before the talk.)

Algebra Seminar

Speaker: Kirsten Wickelgren (Georgia Tech)

"Splitting varieties for triple Massey products in Galois cohomology"

Time: 15:00

Room: MC 107

The Brauer-Severi variety a $x^2 + b y^2 = z^2$ has a rational point if and only if the cup product of cohomology classes with $F_2$ coefficients associated to $a$ and $b$ vanish. The cup product is the order-$2$ Massey product. Higher Massey products give further structure to Galois cohomology, and more generally, they measure information carried in a differential graded algebra which can be lost on passing to the associated cohomology ring. For example, the cohomology of the Borromean rings is isomorphic to that of three unlinked circles, but non-trivial Massey products of elements of $H^1$ detect the more complicated structure of the Borromean rings. Analogues of this example exist in Galois cohomology due to work of Morishita, Vogel, and others. This talk will first introduce Massey products and some relationships with non-abelian cohomology. We will then show that $b x^2 =$ $(y_1^2 - a y_2^2$ $+ c y_3^2 - ac y_4^2)^2 - c(2 y_1 y_3 - 2 a y_2 y_4)^2$ is a splitting variety for the triple Massey product $\langle a,b,c \rangle$ with $F_2$ coefficients, and that this variety satisfies the Hasse principle. It follows that all triple Massey products over global fields vanish when they are defined. Jan Minac and Nguyen Duy Tan have extended this result to all $\mathbb{F}_p$ and with $p=2$ to all fields of characteristic different from $2$. The method discussed in the talk could produce splitting varieties for higher order Massey products. This is joint work with Michael Hopkins.

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"Singular Levi-flat hypersurfaces and holomorphic webs (Part I)"

Time: 15:30

Room: MC 107

Levi flat hypersurfaces are characterized by vanishing of the Levi form on them. Their regular part is foliated by complex hypersurfaces, this is called the Levi foliation. It is an open question whether one can extend this foliation to the ambient space. As example of Brunella shows, near singular points the extension may exist in general only as a singular holomorphic web. A smooth holomorphic web is simply the union of several foliations. A singular web is a more general object which loosely can be thought of as a foliation with branching. In this talk I will give a detailed background concerning holomorphic webs, and will discuss some recent progress on extension of the Levi foliation. This is joint work with A. Sukhov.

Index Theory Seminar

Speaker: Matthias Franz (Western)

"Representations of compact connected Lie groups"

Time: 12:00

Room: MC 107

We review basic results of the representation theory of compact connected Lie groups.

Comprehensive Exam Presentation

Speaker: Nadia Alluhaibi (Western)

"the geometry of complex hyperbolic and automorphic forms of the unit ball quotients"

Time: 10:00

Room: MC 107

I will describe the ball model and the Siegel space model of the n-dimensional complex hyperbolic space H^n_C. The matrix group SU(n,1) acts on H^n_C. The group of holomorphic isometries of H^n_C is PU(n,1).

Let \Gamma be a discrete subgroup of SU(n,1) which acts freely and properly discontinuously on H^n_C. I will give the definition of an automorphic form for \Gamma. I will talk about constructing automorphic forms associated to certain submanifolds of H^n_C / \Gamma .

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"Singular Levi-flat hypersurfaces and holomorphic webs (Part II)"

Time: 15:30

Room: MC 107

Levi flat hypersurfaces are characterized by vanishing of the Levi form on them. Their regular part is foliated by complex hypersurfaces, this is called the Levi foliation. It is an open question whether one can extend this foliation to the ambient space. As example of Brunella shows, near singular points the extension may exist in general only as a singular holomorphic web. A smooth holomorphic web is simply the union of several foliations. A singular web is a more general object which loosely can be thought of as a foliation with branching. In this talk I will give a detailed background concerning holomorphic webs, and will discuss some recent progress on extension of the Levi foliation. This is joint work with A. Sukhov.

Index Theory Seminar

Speaker: Matthias Franz (Western)

"Representations of compact connected Lie groups II"

Time: 11:00

Room: MC 108

This time we discuss highest weights, various constructions of irreducible representations and the Weyl character formula.

Comprehensive Exam Presentation

Speaker: Sajad Sadeghi (Western)

"Dirac Operators and Geodesic Metric on the Sierpinski Gasket and the Harmonic Gasket"

Time: 11:30

Room: MC 107

This talk is based on the paper `` Dirac operators and Geodesic metric on Harmonic Sierpienski gasket and other fractals" by Lapidus and Sarhad. First, the Sierpinski gasket will be introduced as the unique fixed point of a certain contraction on the set of compact subsets of the Euclidean plane. Then, by defining the graph approximation of the Sierpinski gasket, I will define the energy form on that space. I will talk about Kusuoka's measurable Riemannian geometry on the Sierpinski gasket and introduce counterparts of the Riemannian volume, the Riemannian metric and the Riemannian energy in that setting. Thereafter harmonic functions on the Sierpinski gasket will be introduced as energy minimizing functions. Using those functions we can define the harmonic gasket. I will also talk about Kigami's geodesic metric on the harmonic gasket. Using a spectral triple on the unit circle, a Dirac operator and a spectral triple for the Sierpinski gasket and the harmonic gasket will be constructed. Next, we will see that Connes' distance formula of noncommutative geometry which provides a natural metric on these fractals, is the same as the geodesic metric on the Sierpinski gasket and the kigami's geodesic metric on the harmonic gasket. It will be shown also that the spectral dimension of the Sierpinski gasket is the same as its Hausdorff dimension. Finally some conjectures about the harmonic gasket will be stated.

Algebra Seminar

Speaker: Franz-Viktor Kuhlmann (Saskatchewan)

"Notes on tame and extremal valued fields"

Time: 14:30

Room: MC 107

In the year 2003 I first heard of the notion of extremal valued fields when Yuri Ershov gave a talk at a conference in Tehran. He proved that algebraically complete discretely valued fields are extremal. However, the proof contained a mistake, and it turned out in 2009 through an observation by Sergej Starchenko that Ershov's original definition leads to all extremal fields being algebraically closed. In joint work with Salih Durhan (formerly Azgin) and Florian Pop, we chose a more appropriate definition and then characterized extremal valued fields in several important cases.

We call a valued field $(K,v)$ extremal if for all natural numbers $n$ and all polynomials $f$ in $K[X_1,...,X_n]$, the set {$f(a_1,...,a_n) | a_1,...,a_n$ in the valuation ring} has a maximum (which is allowed to be infinity, which is the case if $f$ has a zero in the valuation ring). This is such a natural property of valued fields that it is in fact surprising that it has apparently not been studied much earlier. It is also an important property because Ershov's original statement is true under the revised definition, which implies that in particular all Laurent Series Fields over finite fields are extremal. As it is a deep open problem whether these fields have a decidable elementary theory and as we are therefore looking for complete recursive axiomatizations, it is important to know the elementary properties of them well. That these fields are extremal seems to be an important ingredient in the determination of their structure theory, which in turn is an essential tool in the proof of model theoretic properties.

Further, it came to us as a surprise that extremality is closely connected with Pop's notion of "large fields". Also the notion of tame valued fields plays a crucial role in the characterization of extremal fields; both large and tame valued fields have the right model theoretic properties. A valued field $K$ with algebraic closure $K^{ac}$ is tame if it is henselian and the ramification field of the extension $K^{ac}|K$ coincides with the algebraic closure.

In my talk I will introduce the above notions, try to explain their meaning and importance also to the non-expert, and discuss in detail what is known about extremal fields and how the properties of large and of tame fields appear in the proofs of the characterizations we give. I will also discuss the connections with additive polynomials in the case of positive characteristic. Finally, I will present some challenging open problems whose solution may have an impact on the above mentioned problem for Laurent Series Fields over finite fields.