Abstract

 We will discuss how to modify an approach of Solymosi and Tao, based on the so-called relative triangle removal lemma, to show that subsets of positive relative density of P^2 contain infinitely many corners, that is sets of the form {(a,b), (a+d,b),(a,b+d)}. This is ongoing joint work with T. Titichetrakun.

Abstract

We characterize Schubert varieties (for GLn) which are local complete intersections (lci) by the combinatorial notion of pattern avoidance.  For the Schubert varieties which are local complete intersections, we give an explicit minimal set of equations cutting out their neighborhoods at the identity.  Although the statement only requires ordinary pattern avoidance, showing the other Schubert varieties are not lci appears to require more complicated combinatorial ideas which have their own geometric underpinnings.  The Schubert varieties defined by inclusions, originally introduced by Reiner and Gasharov, turn out to be an important subclass of lci Schubert varieties.  Using the explicit equations at the identity for the lci Schubert varieties, we can recover formulas for some of their local singularity invariants at the identity as well as explicit presentations for their cohomology rings.  

This is joint work with Henning Ulfarsson (Reykjavik U.).

Abstract

T cells of the immune system, upon maturation, differentiate into either Th1 or Th2 cells that have different functions. The decision to which cell type to differentiate depends on the concentrations of transcription factors T-bet (x_1) and GATA-3 (x_2). These factors are translated by the mRNA whose levels of expression, y_1 and y_2, depend, respectively, on x_1 and x_2 in a nonlinear nonlocal way. The population density of T cells, \phi(t,x_1,x_2, y_1, y_2), satisfies a hyperbolic conservation law with coefficients depending nonlinearly and nonlocally on (t, x_1,x_2, y_1, y_2), while the x_i, y_i satisfy a system of ordinary differential equations. We study the long time behavior of \phi and show, under some conditions on the parameters of the system of differential equations, that the gene expressions in the T-cell population aggregate at one, two or four points, which connect to various cell differentiation scenarios.

Abstract

Over the past few years there has been considerable activity in exploiting the power of algebraic topology to investigate areas outside of mathematics. The phrase 'applied algebraic topology'
is no longer an oxymoron!

Even more recently the intrinsically random nature of the world is beginning to bring statistical and probabilistic tools to these problems, leading to the birth 
of a new area of 'random algebraic topology'.

In this talk I will discuss some of the few results in random algebraic topology, including the persistence homology of the sub-level sets of Gaussian processes 
over manifolds, and limit theorems for the Betti numbers of random complexes built over random point processes.

Since this is to be a joint Probability/Topology seminar, I shall assume no prior knowledge in either area.

Abstract

Feynman's famous quote is of particular relevance for research at the interface of mathematics and physics in recent decades. A striking example is the impressive number of fascinating results (many of them still conjectural) in various areas of mathematics, such as geometry, topology and number theory, that have been obtained via string theory. In this vein, mirror symmetry and topological string theory have been particularly fruitful. In this talk I will focus on a number of mathematical conjectures and theorems that we have obtained through careful study of mirror symmetry. I will discuss what string theory tells us about (quasi-)modularity of the generating functions of Gromov-Witten invariants of Calabi-Yau threefolds, and what it implies for the crepant resolution conjecture relating Gromov-Witten invariants of an orbifold to the invariants of its crepant resolution. I will also talk about a new mysterious recursive structure conjecturally satisfied by the Gromov-Witten generating functions for toric Calabi-Yau threefolds, with far-reaching and still mostly unexplored consequences. By the end of the talk, you should hopefully be convinced of "the unreasonable effectiveness of string theory in mathematics"!

Abstract

In this talk, I will give a survey on Harnack inequalities for solutions of second-order elliptic equations on Riemannian manifolds.

Abstract

Let S be an ergodic measure-preserving automorphism on a non-atomic
probability space, and let T be the time-one map of a topologically weak
mixing suspension flow over an irreducible subshift of finite type under a
Holder ceiling function. We show that if the measure-theoretic entropy of
the S is strictly less than the topological entropy of T, then there
exists an embedding from the measure-preserving automorphism into the
suspension flow. As a corollary of this result and the symbolic dynamics
for geodesic flows on compact surfaces of negative curvature developed by
Bowen and Ratner, we also obtain an embedding from the measure-preserving
automorphism into a geodesic flow, whenever the measure-theoretic entropy
of S is strictly less than the topological entropy of the time-one map of
the geodesic flow.

Details

In recent years, a unifying theme has been found for a surprising number of counting problems. It appears that in many seemingly unrelated contexts, generating functions for enumerative invariants satisfy a particular topological recursion, based on the geometry of a complex curve. As examples, Hurwitz theory,
Gromov-Witten theory of the complex line, and open/closed Gromov-Witten theory of the three-dimensional complex plane and other toric Calabi-Yau threefolds are all encoded by the same topological recursion on certain complex curves. In this talk, I will first review applications of the topological recursion on genus zero curves, and then report on work in progress on studying the topological recursion for families of elliptic curves. In this context, the recursion
produces an infinite tower of quasi-modular forms. The question is: to what counting problem should these quasi-modular forms be related?

Abstract

Abstract

This week we will study the section titled "Foundations for beginning calculus". This section is composed of four papers: 1) On developing a rich conception of variable 2) Rethinking change 3) Foundation reasoning abilities that promote coherence in students' function understanding and 4) The concept of accumulation in calculus. Please see the Seminar's wiki page available at: http://wiki.ubc.ca/Sandbox:Math_Teaching_Seminar

You will find access to the pdf version of these papers.

Abstract

Abstract

This talks introduces a novel framework for phase retrieval, a problem which arises in X-ray crystallography, diffraction imaging, astronomical imaging and many other applications. Our approach combines multiple structured illuminations together with ideas from convex programming to recover the phase from intensity measurements, typically from the modulus of the diffracted wave. We demonstrate empirically that any complex-valued object can be recovered from the knowledge of the magnitude of just a few diffracted patterns by solving a simple convex optimization problem inspired by the recent literature on matrix completion. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. Finally, we present some novel theory showing that our entire approach may be provably surprisingly effective.