Department of Mathematics

# Analysis Seminar 2017-18

Tuesdays, 3:30–4:30 PM, MC 107*

Date Speaker Talk
Sep 12 Lizhen Ji
University of Michigan
Riemann's moduli space and metric Schottky problem

I will discuss the history and some results on understanding the right complex structure on the moduli space of Riemann surfaces and metric geometric perspective on the Schottky problem of determining the Jacobian locus in the moduli space of principally polarized abelian varieties.
Sep 19 Tatyana Barron
Western University
On some vector-valued functions associated with submanifolds of the unit ball

The talk will be about vector-valued holomorphic automorphic forms on the unit ball (or, more generally, on a bounded domain in $$\mathbb{C}^n$$) and related analytic questions.
Sep 26 Graham Denham
Western University
Cohomological vanishing for complex hypersurface arrangements

Cohomology jump loci are secondary cohomological invariants of discrete groups and topological spaces. I will describe some recent work on the cohomology jump loci of complements of unions of complex hypersurfaces. The study of such hypersurface arrangements involves a mix of combinatorics and complex geometry.
Oct 03
Oct 10
Oct 17 Robert Haslhofer
University of Toronto
Minimal two-spheres in three-spheres

We prove that any manifold diffeomorphic to $$S^3$$ and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal two-sphere was obtained by Simon-Smith in 1983. Our approach combines ideas from min-max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in three-manifolds. Finally, we apply our methods to solve a problem posed by S. T. Yau in 1987, and to show that the assumptions in the multiplicity one conjecture and the equidistribution of widths conjecture are in a certain sense sharp. This is joint work with Dan Ketover.
Oct 24 Ihor Stasyuk
Nipissing University
Extension of functions and metrics

Let $$(X,d)$$ be a complete, bounded, metric space. For a nonempty, closed subset $$A$$ of $$X$$ denote by $$C^*(A \times A)$$ the set of all continuous, bounded, real-valued functions on $$A \times A$$. Denote by $C^\dagger= \bigcup \{C^*(A \times A) \mid A \text{ is a nonempty closed subset of } X\}$ the set of all partial, continuous and bounded functions. We discuss a recent result obtained in collaboration with T. Banakh, M. Zarichnyi and E. D. Tymchatyn on existence of linear, regular extension operators from $$C^\dagger$$ endowed with the topology of convergence in the Hausdorff distance of graphs of partial functions to the space $$C^*(X \times X)$$ with the topology of uniform convergence on compact sets. The extension operator we construct preserves constant functions, pseudometrics, metrics, and admissible metrics. For a fixed, nonempty, closed subset $$A$$ of $$X$$ the restricted extension operator from $$C^*(A \times A)$$ to $$C^*(X \times X)$$ is continuous with respect to the topologies of pointwise convergence, uniform convergence on compact sets, and uniform convergence considered on both $$C^*(A \times A)$$ and $$C^*(X \times X)$$.
Oct 31 Rasul Shafikov
Western University
Polynomial density on compact real manifolds

I will discuss a recent result joint with P. Gupta concerning approximation of continuous complex-valued functions on abstract compact real manifolds by polynomial combinations of a certain number of smooth functions on them.
Nov 07 Mircea Voda
University of Toronto
On the spectrum of multi-frequency quasiperiodic Schrödinger operators with large coupling

The spectrum of single-frequency quasiperiodic Schrödinger operators with analytic potentials is known to be a Cantor set. Chulaevsky and Sinai conjectured that the spectrum of multi-frequency quasiperiodic Schrödinger operators is an interval for generic large potentials. I will discuss a proof of the Chulaevsky-Sinai conjecture based on joint work with M. Goldstein and W. Schlag.
Nov 14 Ron Kerman
Brock University
Consequence of Hermite series in Orlicz spaces

The kth Hermite function, $$h_k$$, is given at $$x \in \mathbb{R}$$ by $h_k(x)=(-1)^k \gamma_k e^{x^2/2} \frac{\mathrm{d}^k e^{-x^2}}{\mathrm{d}x^k}, \gamma_k=\pi^{-1/4} 2^{-k/2} (k!)^{-1/2}, k=0,1,\dots.$ Given a suitable function $$f$$ on $$\mathbb{R}$$, its Hermite series is $$\sum_{k=0}^\infty c_k(f)h_k$$, where $$c(k)=\int_{\mathbb{R}} f(y)h_k(y)\mathrm{d}y, k=0,1,\dots$$. In 1965, R. Ashey and S. Wainger proved that $$\tag{1}\label{eq:1} \qquad \lim_{n \to \infty} \int_{\mathbb{R}} |f(x)-\sum_{k=0}^n c_k(f) h_k(x)|^p\, \mathrm{d}x=0$$ for all $$f$$ with $$\int_{\mathbb{R}} |f(y)|^p\, \mathrm{d}y <\infty$$ if and only if $$4/3<p<4$$. One consequence of our general results asserts that, given the index $$p$$, $$0< p<\infty$$, and $$f$$ satisfying $$\int_{\mathbb{R}} |f(y)|^p\, \mathrm{d}y < \infty$$, one has $$\tag{2}\label{eq:2}\qquad \lim_{n \to \infty} \int_{|x|< T_n} |g(x)-\sum_{k=0}^n c_k(g) h_k(x)|^p\,\mathrm{d}x=0,$$ where $$T_n=n^{1/34+\epsilon}$$, $$0<\epsilon\ll 1$$, and $$g(y)=\frac{f(y)}{1+y^{36}}$$, and so $$f$$ can be recovered from a related Hermite series. The numbers $$T_n$$ and the weight $$(1+y^{36})^{-1}$$ come out of certain estimates of the Dirichlet kernel of the partial sum operator derived by G. Sansone from earlier work of J. Uspensky. We will consider when, in \eqref{eq:1} and \eqref{eq:2}, the modular $$t^p$$ can be replaced by modulars of the form $$\Phi(t)=\int_0^t \phi(s)\,\mathrm{d}s$$, where $$0=\phi(0) \leq \phi(t) \uparrow \infty$$.
Nov 21 Nikolay Shcherbina
University of Wuppertal
Squeezing functions and Cantor sets

We construct "large" Cantor sets whose complements resemble the unit disk arbitrarily well from the point of view of the squeezing function, and we construct "large" Cantor sets whose complements do not resemble the unit disk from the point of view of the squeezing function.
Nov 28 Regina Rotman
University of Toronto
Lengths of periodic geodesics and related questions

Thirty five years ago M. Gromov asked if it is true that the length of a shortest periodic geodesic on a closed Riemannian manifold does not exceed $$c(n)\mathrm{Vol}^{1/n}$$, where $$\mathrm{Vol}$$ denotes the volume of the manifold, and $$c(n)$$ is a constant that depends only on its dimension $$n$$. This question and a similar question with the diameter of the manifold instead of $$\mathrm{Vol}^{1/n}$$ are still open.
Dec 05 Almaz Butaev
Concordia University
Some refinements of the embedding of critical Sobolev spaces into BMO

In 2004, Van Schaftinen showed that the inequalities established by Bourgain and Brezis give rise to some new function spaces that refine the classical embedding $$\mathring{W}^{1,n}(\mathbb{R}^n)\subset \rm{BMO}(\mathbb{R}^n)$$. In this talk, we discuss the non-homogeneous analogs of these function spaces and their properties.
Jan 09
Happy New Year!
Masoud Khalkhali
Western University
Curvature of the determinant line bundle and Quillen's conformal anomaly (I)

In this series of three lectures I shall recall the classical theory of determinant line bundle on the space of Fredholm operators with emphasis on del bar operators acting on spaces of smooth sections of holomorphic vector bundles on Riemann surfaces. In the very last part I shall indicate how to generalize these results to a noncommutative setting (based on joint work with A. Ghorbanpour and A. Fathi). The following topics will be discussed:
• Zeta regularized determinant of Cauchy-Riemann operators and Quillen's metric;
• Curvature of the determinant line bundle and obstruction to a conformal gauge invariant determinant;
• Extension to a noncommutative setting.
Jan 16 André Belotto
Toulouse Mathematics Institute
The Sard conjecture on Martinet surfaces

Given a totally nonholonomic distribution of rank two $$\Delta$$ on a three-dimensional manifold $$M,$$ it is natural to investigate the size of the set of points $$\mathcal{X}^x$$ that can be reached by singular horizontal paths starting from a same point $$x \in M$$. In this setting, the Sard conjecture states that $$\mathcal{X}^x$$ should be a subset of the so-called Martinet surface of $$2$$-dimensional Hausdorff measure zero. In this seminar, I present a reformulation of the conjecture in terms of the singular behavior of a vector field. Next, I present a recent work in collaboration with Ludovic Rifford where we show that the conjecture holds whenever the Martinet surface is smooth. Moreover, we address the case of singular real-analytic Martinet surfaces and show that the result holds true under an assumption of non-transversality of the distribution on the singular set of the Martinet surface. Our methods rely on the control of the divergence of vector fields generating the trace of the distribution on the Martinet surface and some techniques of resolution of singularities.
Jan 23 Masoud Khalkhali
Western University
Curvature of the determinant line bundle and Quillen's conformal anomaly (II)

In this series of three lectures I shall recall the classical theory of determinant line bundle on the space of Fredholm operators with emphasis on del bar operators acting on spaces of smooth sections of holomorphic vector bundles on Riemann surfaces. In the very last part I shall indicate how to generalize these results to a noncommutative setting (based on joint work with A. Ghorbanpour and A. Fathi). The following topics will be discussed:
• Zeta regularized determinant of Cauchy-Riemann operators and Quillen's metric;
• Curvature of the determinant line bundle and obstruction to a conformal gauge invariant determinant;
• Extension to a noncommutative setting.
Jan 30

Notice:
This talk is moved to the Noncommutative Geometry Seminar on Wed, Jan 31, to be held from 2:00–3:00 PM in MC 108.
Masoud Khalkhali
Western University
Curvature of the determinant line bundle and Quillen's conformal anomaly (III)

In this series of three lectures I shall recall the classical theory of determinant line bundle on the space of Fredholm operators with emphasis on del bar operators acting on spaces of smooth sections of holomorphic vector bundles on Riemann surfaces. In the very last part I shall indicate how to generalize these results to a noncommutative setting (based on joint work with A. Ghorbanpour and A. Fathi). The following topics will be discussed:
• Zeta regularized determinant of Cauchy-Riemann operators and Quillen's metric;
• Curvature of the determinant line bundle and obstruction to a conformal gauge invariant determinant;
• Extension to a noncommutative setting.
Feb 06 Gord Sinnamon
Western University
Interpolation of Banach Spaces Related to Decreasing Functions

Two scales of Banach spaces of functions on the half line will be defined using the least decreasing majorant construction and the level function construction. The two scales are shown to be dual to one another and the interpolation structure within each scale is shown to closely parallel that of the rearrangement-invariant spaces. In particular it is proved that a couple of these spaces is a Calderon-Mityagin couple if and only if the corresponding couple of rearrangement-invariant spaces is a Calderon-Mityagin couple. Consequently, the interpolation spaces for a couple of these spaces admits a complete description by the K-method if and only if same is true for the corresponding couple of rearrangement-invariant spaces.
Feb 13 Ruxandra Moraru
University of Waterloo
Hermitian-Einstein equations on generalized Kähler manifolds

In this talk, we discuss an analogue of the Hermitian-Einstein equations for generalized Kähler manifolds. We explain in particular how these equations are equivalent to a notion of stability for generalized holomorphic bundles, and that there is a Kobayahsi-Hitchin-type correspondence between solutions of these equations and stable bundles. Time permitting, we will also describe moduli spaces of these stable generalized holomorphic bundles on some speciﬁc examples of generalized Kähler manifolds. This is joint work with Mohamed El Alami and Shengda Hu.
Feb 20
Feb 27 Dmitry Faifman
University of Toronto
Curvature in contact manifolds and integral geometry

Valuations are finitely additive measures on nice subsets; for example, the Euler characteristic, volume, and surface area are valuations. During the 20th century, valuations have been studied predominantly on convex bodies and polytopes, in linear spaces and lattices. Valuations on manifolds were introduced about 15 years ago by S. Alesker, with contributions by A. Bernig, J. Fu and others, and immediately brought under one umbrella a range of classical results in Riemannian geometry, notably Weyl's tube formula and the Chern-Gauss-Bonnet theorem. These results circle around the real orthogonal group. In the talk, the real symplectic group will be the central player. Drawing inspiration from the Lipschitz-Killing curvatures in the Riemannian setting, we will construct some natural valuations on contact and dual Heisenberg manifolds, which generalize the Gaussian curvature. We will also construct symplectic-invariant distributions on the grassmannian, leading to Crofton-type formulas on the contact sphere and symplectic space.
Mar 06

Notice:
This talk might be is cancelled.
Greg Reid
Western University
Mar 13 Joey van der Leer Duran
University of Toronto
Hodge theory for Lie algebroids on manifolds with boundary

The behaviour of the delbar operator on a complex manifold with boundary depends heavily on the geometry of the boundary. In this talk we will review this relation and discuss how it generalizes to the setting of complex Lie algebroids.
Mar 20
Happy Spring!
Hristo Sendov
Western University
Stronger Rolle's Theorem for Complex Polynomials

Every Calculus student is familiar with the classical Rolle's theorem stating that if a real polynomial $$p$$ satisfies $$p(-1) = p(1)$$, then it has a critical point in $$(-1, 1)$$. In 1934, L. Tschakaloff strengthened this result by finding a minimal interval, contained in $$(-1,1)$$, that holds a critical point of every real polynomial with $$p(-1) = p(1)$$, up to a fixed degree. In 1936, he expressed a desire to find an analogue of his result for complex polynomials. This talk will present the following Rolle's theorem for complex polynomials. If $$p(z)$$ is a complex polynomial of degree $$n\geq 5$$, satisfying $$p(-i)=p(i)$$, then there is at least one critical point of $$p$$ in the union $$D[-c;r] \cup D[c;r]$$ of two closed disks with centres $$-c, c$$ and radius $$r$$, where $c= \cot (2\pi/n),\;\;\; r=1/ \sin (2\pi/n).$ If $$n=3$$, then the closed disk $$D[0; 1/\sqrt{3}]$$ has this property; and if $$n=4$$ then the union of the closed disks $$D[-1/3; 2/3] \cup D[1/3; 2/3]$$ has this property. In the last two cases, the domains are minimal, with respect to inclusion, having this property. This theorem is stronger than any other known Rolle's Theorem for complex polynomials of any degree. A minimal Rolle's domain are found for polynomials of degree $$3$$ and $$4$$, answering Tschakaloff's question. This is a joint work with Blagovest Sendov from the Bulgarian Academy of Sciences.
Mar 27 Keewatin Dewdney
Western University
Romancing the J-Curve: steps toward a working theory of biodiversity

Computer simulations have paved the way to a working hypothesis of how the individuals in most living species appear to behave over time. The Stochastic Species Hypothesis (SSH) declares that, over suitable lengths of time, each individual is as likely to die as to reproduce. This results in a normal distribution of such events and, in turn, becomes the statistical driver for an equilibrium process in communities, an equilibrium process that results in a hyperbolic distribution of species vs abundances. The resulting density function is shown to be a pure hyperbola ($$y = c/x$$) translated by small amounts of epsilon and delta to intersect the axes and produce, in consequence, a truncated density function. The resulting proposal has its work cut out for it in confronting some dozen proposals that have emerged since the 1930s from a cottage industry of (essentially) guesswork by population biologists. The resulting J Distribution is deployed, via the Pielou transform into a method for making accurate estimates of the species richness in a living community—the holy grail of population biology. But does it actually work? You be the judge by reviewing the results of a massive meta-study that appears to strongly support that claim.
Apr 03

Notice:
This talk is cancelled.
Aftab Patel
Western University
Apr 10

Notice:
This talk is cancelled.
Western University
Extra talk:
May 16

Notice:
To be held in MC 108. Also note the unusual day (Wed).
Minghua Lin
University of Waterloo & Shanghai University
A norm inequality for positive semidefinite block matrices

Any positive semidefinite matrix $$M=(M_{i,j})_{i,j=1}^m$$ with each block $$M_{i,j}$$ square satisfies the symmetric norm inequality $\|M\|\le \|\sum_{i=1}^mM_{i,i}+\sum_{i=1}^{m-1}\omega_iI\|,$ where $$\omega_i$$ ($$i=1, \ldots, m-1$$) are quantities involving the width of numerical ranges. This extends the main theorem of [J.-C. Bourin, A. Mhanna, C. R. Acad. Sci. Paris, Ser. I 355 (2017), 1077–1081] to higher number of blocks.