Mathematics Dept.
  Events
Tue 8 Sep 2015, 10:30am SPECIAL
Math 103
New Instructor Orientation
Math 103
Tue 8 Sep 2015, 10:30am-12:30pm

Details

 Practical information for everyone teaching at UBC for the first time: a quick overview of UBC rules, policies,  key websites one would have to use, and other resources. Pizza lunch will be provided at noon.  

Note for Attendees

 Pizza will be delivered at noon. 
hide
George Bluman
UBC Math
Wed 9 Sep 2015, 3:00pm
Symmetries and Differential Equations Seminar
Math 125
Construction of invariant solutions for differential equations I
Math 125
Wed 9 Sep 2015, 3:00pm-4:00pm

Abstract

The first part of this lecture will be a short discussion on how to find non-invertible mappings of linear PDEs with variable coefficients to linear PDEs with constant coefficient, focusing on parabolic PDEs. The main part of the lecture will focus on how to find invariant solution of DEs.  There will be a presentation of Lie's systematic classical method based on the invariance of PDEs under Lie groups of point transformations--here a symmetry maps every solution of a PDE to another solution of the PDE and one looks for solutions that are themselves invariant.  It will be shown how to generalize Lie's classical method to the systematic nonclassical method based on mapping only a subset of solutions of a PDE to solutions of the same PDE whereas the remaining solutions map into solutions of other PDEs. The aim is to find solutions of such submanifolds of  that are themselves invariant.  Illustrative examples yielding such nonclassical solutions include the Boussinesq equation and the Kompaneets equation.

hide
University of Warwick
Thu 10 Sep 2015, 3:30pm
Number Theory Seminar
IRMACS Theatre (SFU)
Sums of seven cubes
IRMACS Theatre (SFU)
Thu 10 Sep 2015, 3:30pm-4:30pm

Abstract

In 1851, Carl Jacobi made the experimental observation that all integers are sums of seven non negative cubes, with precisely 17 exceptions, the largest of which is 454. Building on previous work by Maillet, Landau, Dickson, Linnik, Watson, Bombieri, Ramare, Elkies and many others, we complete the proof of Jacobi's observation.

Note for Attendees

Coffee/tea/snacks at 3pm, IRMACS 2nd floor
hide
UBC Math
Fri 11 Sep 2015, 3:00pm
Department Colloquium
MATX 1100
Random planar maps
MATX 1100
Fri 11 Sep 2015, 3:00pm-4:00pm

Abstract

Planar maps are of interest to several communities: In Combinatorics Tutte and others have enumerated them.  In Physics they are a model for 2 dimensional quantum gravity. The past 15 years have seen great progress in this area using probabilistic tools.  I will survey the topic, some of the significant progress made so far and big problems we hope to solve.

Note for Attendees

Refreshments will be served in MATH 125 before the colloquium.
hide
Mon 14 Sep 2015, 3:00pm SPECIAL
Institute of Applied Mathematics
LSK 306
IAM Welcome Reception
LSK 306
Mon 14 Sep 2015, 3:00pm-5:00pm

Abstract

Please join us at the Institute of Applied Mathematics to welcome new members at the start of a new academic year. We will have food and drink and will be light on speeches. 
hide
UBC
Mon 14 Sep 2015, 3:00pm
Algebraic Geometry Seminar
ESB 4127
Hochschild cohomology of torus-equivariant D-modules
ESB 4127
Mon 14 Sep 2015, 3:00pm-4:00pm

Abstract

 In this talk I will discuss how to compute the Hochschild cohomology
of the category of D-modules on a quotient stack via a
compactification of the diagonal morphism. I will then apply this
construction to the case of quotients by a torus and describe the
Hochschild cohomology as the cohomology of a D-module on the loop
space of the quotient stack. This work is motivated by a desire to
understand the D-module equivalent of singular support of coherent
sheaves in Geometric Langlands.
 
hide
National Taiwan University
Tue 15 Sep 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Solutions to Poisson-Nernst-Planck type systems with cross-diffusion terms
ESB 2012
Tue 15 Sep 2015, 3:30pm-4:30pm

Abstract

The Poisson-Nernst-Planck (PNP) system is a well-known model of ion transport with many applications in biology, engineering and physics. Cross-diffusion terms may describe the exclusion of steric effects. In this lecture, I shall introduce cross diffusion terms from the Lennard-Jones potential and show the analytical results as follows:

1. Stability of 1D boundary layer solutions to original Poisson-Nernst-Planck (PNP) systems

2. Multiple solutions of PNP systems with steric effects

hide
Tel Aviv University
Tue 15 Sep 2015, 4:00pm
Discrete Math Seminar
ESB 4127
The Elekes-Ronyai-Szabo theory and its applications
ESB 4127
Tue 15 Sep 2015, 4:00pm-5:00pm

Abstract

Let F(x,y,z) be a real trivariate polynomial of constant degree, and let A,B,C be three sets of real numbers, each of size n. 
 
How many roots can F have on A x B x C? 
 
This setup arises in many interesting problems in combinatorial geometry, including distinct distances between points on curves, distinct distances from three points, collinear triples of points on curves (the `orchard problem'), unit-area triangles, triple intersection points of families of circles, and more.
 
This question has been studied by Elekes and R\'onyai and then by Elekes and Szab\'o about 15 years ago. One of their striking results is that, for the special case where F(x,y,z) = z-f(x,y), either F vanishes at  only a subquadratic number (o(n^2)) of points of A x B x C, or else f must have one of the special forms f(x,y) = h(p(x)+q(y)) or f(x,y) = h(p(x)q(y)), for univariate polynomials p, q, h.
 
In this talk I will survey recent progress on this problem, in which the analysis is greatly simplified, and the bounds become sharp: If F does not have  a special form, the number of roots is at most O(n^{11/6}). Moreover, the results also hold over the complex field. This yields significantly improved bounds for many geometric problems, as listed above.
 
The proofs use techniques from algebra and algebraic geometry, which are somewhat related to the recent growing body of work on algebraic techniques for incidences and distance problems, inspired by Guth and Katz's seminal papers.
 
Joint work with Orit Raz, Jozsef Solymosi, and Frank de Zeeuw (and others).
hide
University of Cambridge
Wed 16 Sep 2015, 3:00pm
Probability Seminar
ESB 2012
Critical exponents for FK random planar maps
ESB 2012
Wed 16 Sep 2015, 3:00pm-4:00pm

Abstract

We consider random planar maps weighted by the critical Fortuin--Kasteleyn percolation model with parameter q \in (0,4). The study of these surfaces is central to the theory of Liouville quantum gravity.

We obtain rigorously the value of critical exponents associated with the length of cluster interfaces, which is shown to be
$$
 \frac{4}{\pi} \arccos \left( \frac{\sqrt{2 - \sqrt{q}}}{2} \right).
$$
This is consistent with physics predictions; in particular, applying the KPZ formula we recover the dimension of SLE curves.

Joint work with Benoit Laslier and Gourab Ray (Cambridge).
hide
University of Warwick
Thu 17 Sep 2015, 3:30pm
Number Theory Seminar
room MATH 126
Modularity of elliptic curves over totally real fields
room MATH 126
Thu 17 Sep 2015, 3:30pm-4:30pm

Abstract

We combine the latest advances in modularity lifting with a 3-5-7 modularity switching argument to deduce modularity of 'most' elliptic curves over totally real fields. In particular, we show that all elliptic curves over real quadratic fields are modular. This talk is based on joint work with Bao Le Hung (Harvard) and Nuno Freitas (Bonn).
hide
UBC Math
Fri 18 Sep 2015, 3:00pm
Department Colloquium
MATX 1100
The strange appearance of algebras in the combinatorics of polytopes
MATX 1100
Fri 18 Sep 2015, 3:00pm-4:00pm

Abstract

The subject of this talk is the problem of counting faces of various dimensions in a convex n-polytope. When the polytope is simplicial, McMullen's conjecture, proved in 1979 by Billera, Lee and Stanley, gives a complete description of all possible face numbers. One encodes the number of faces of dimension 0,1,...,n in a vector g with n/2 components. The result then is that g comes from a simplicial polytope if and only if the components of g are the dimensions of graded pieces in an algebra generated in degree 1.

In this talk I will explain the conjectural extensions of this theorem to simplicial spheres, non-simplicial polytopes and barycentric subdivisions. In every problem that involves counting faces, it is conjectured that there exists an algebra generated in degree 1 whose graded pieces give the face numbers.

Note for Attendees

Refreshments will be served in MATH 125 before the colloquium.
hide
UBC
Mon 21 Sep 2015, 3:00pm
Algebraic Geometry Seminar
MATH 126
Curve counting on Abelian varieties, modular forms, and the combinatorics of box counting
MATH 126
Mon 21 Sep 2015, 3:00pm-4:00pm

Abstract

An Abelian variety (of complex dimension g) is an algebraic geometer's version of a torus — it is a variety which is topologically equivalent to a (real) 2g-dimensional torus. Geometers consider the problem of counting the number of curves on an Abelian variety subject to some set of constraints. In dimensions g=1,2, and 3, these geometric numbers have a surprising connections to number theory and combinatorics. They occur as the coefficients of Fourier expansions of various modular forms and they can also be determined in terms of combinatorics of 2D and 3D partitions (a.k.a. box counting). We illustrate this using only elementary ideas from topology and combinatorics in the case of g=1. For g=2 and g=3, we describe recent theorems and conjectures which complete determine the enumerative geometry of Abelian surfaces and threefolds in terms of Jacobi forms and in the process we indicate how Jacobi forms arise from the combinatorics of box counting.
hide
UBC
Tue 22 Sep 2015, 12:30pm
Scientific Computation and Applied & Industrial Mathematics
ESB 4133 (PIMS Lounge)
A new misfit function for the full waveform inversion
ESB 4133 (PIMS Lounge)
Tue 22 Sep 2015, 12:30pm-1:30pm

Abstract


hide
Liangming Shen
UBC Math
Tue 22 Sep 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Unnormalized conical Kahler-Ricci flow
ESB 2012
Tue 22 Sep 2015, 3:30pm-4:30pm

Abstract

Conical Kahler metrics have become an interesting topic in Kahler geometry, and played an important role in the solution of Yau-Tian-Donaldson conjecture. In this talk, we make use of approximation method of Guenancia-Paun to extend Tian-Zhang's maximal existence result of Kahler-Ricci flow to conic case. Finally if possible, we can talk a little about C^{2,\alpha}-estimate for conical Kahler-Ricci flow based on Tian's master thesis.
hide
Jair Taylor
University of Washington
Tue 22 Sep 2015, 4:00pm
Discrete Math Seminar
ESB 4127
Chromatic symmetric functions of hypertrees
ESB 4127
Tue 22 Sep 2015, 4:00pm-5:00pm

Abstract

The chromatic symmetric function X_H of a hypergraph H is the generating function for all colorings of H so that no edge is monochromatic. When H is an ordinary graph, it is known that X_H is positive in the fundamental quasisymmetric functions F_S, but this is not the case for general hypergraphs. We exhibit a class of hypergraphs H --- hypertrees with prime-sized edges --- for which X_H is F-positive, and give an explicit combinatorial interpretation for the F-coefficients of X_H.  We also present a conjecture that certain chromatic symmetric functions of hypergraphs are Schur-positive.
hide
University of Washington
Wed 23 Sep 2015, 3:00pm
Probability Seminar
ESB 2012
Random dessins d'enfants
ESB 2012
Wed 23 Sep 2015, 3:00pm-4:00pm

Abstract

Circle packings provide a conformally natural way to draw triangulations of the sphere. Similarly and more generally, dessins d'enfants provide natural drawings of plane graphs. I will explain how this works, with an emphasis on trees. Then I will discuss a few results and a few questions in the uniformly random case.
hide
Ric Wade
UBC Math
Wed 23 Sep 2015, 3:15pm
Topology and related seminars
ESB 4133
Subspace arrangements and BNS invariants.
ESB 4133
Wed 23 Sep 2015, 3:15pm-4:15pm

Abstract

 We introduce a natural chain complex associated to a collection of subspaces of a vector spaces, and discuss the associated homology. We will give some background on Bieri-Neumann-Strebel invariants of groups, and show how the BNS invariant of a group leads to a nice subspace arrangement, whose associated homology is (yet) another invariant of the group. This can give a useful way of distinguishing between finitely presented groups - we will give some examples involving right-angled Artin groups.
hide
Thu 24 Sep 2015, 10:30am
Math Education Research Reading
Math 126
A Comparison of Web-based and Paper-and-Pencil Homework on Student Performance in College Algebra
Math 126
Thu 24 Sep 2015, 10:30am-11:30am

Abstract

 
hide
Thu 24 Sep 2015, 4:00pm
Fluids Lab Meeting
ESB 2012
Transition to asymmetry in pipe flow of shear-thinning fluids
ESB 2012
Thu 24 Sep 2015, 4:00pm-5:00pm

Abstract

Previous studies of shear-thinning fluids in pipe flow have shown that, although the time-averaged velocity profile is – as expected – axisymmetric when the flow is laminar or fully turbulent, contrary to expectation it exhibits marked asymmetry in the laminar-turbulent transition regime.  Despite this strange effect being observed in different experimental facilities no satisfactory explanation yet exists.  The general consensus of these previous experiments is that the location of the peak velocity remains at a fixed point in space (at least during one experimental realisation). 

Here we present new experimental data obtained using 2D 3C stereo particle image velocimetry which demonstrates that, in fact, the asymmetry does not stay fixed in space and that it may be related to a linear instability occurring prior to classical transition to turbulence (i.e. the appearance of puffs/slugs). The experiments are performed using aqueous solutions of a xanthan gum (0.15wt %), which exhibits shear-thinning of the shear viscosity by about three-orders of magnitude and is only very weakly elastic in the range of shear rates probed in our pipe.   To quantitatively describe the degree of azimuthal flow asymmetry we define an “asymmetry factor”. Our results once again confirm significant departures from axisymmetry in transitional flows of shear-thinning fluids but indicate that the asymmetry may occur slightly before transition.  Furthermore at higher flowrates within transition, it can be seen that the asymmetry is not fixed in space but that, although it preferentially arises at certain azimuthal locations, there are short durations when the flow switches back to a quasi-axisymmetric state.  Associated with some of these events, the flow can also briefly probe an asymmetric state of a different orientation to the preferred state.  

With Chaofan Wen & David J.C. Dennis.

hide
UBC Math
Fri 25 Sep 2015, 3:00pm
Department Colloquium
MATX 1100
Differentiation, lacunarity and Kakeya-type sets
MATX 1100
Fri 25 Sep 2015, 3:00pm-4:00pm

Abstract

Pretend that your car is a unit line segment. How do you perform a three point turn using zero area on the road? It turns out that this seemingly impossible driving stunt is related to the fundamental theorem of calculus! We will explore this connection and see how these ideas have been useful in many problems in mathematics.
hide
Princeton University
Mon 28 Sep 2015, 3:00pm
Institute of Applied Mathematics
ESB 2012
From two-dimensional sheets to three-dimensional structures in developing tissues
ESB 2012
Mon 28 Sep 2015, 3:00pm-4:00pm

Abstract

 

I will present the results of our recent work on epithelial morphogenesis, a highly conserved set of processes that transform two-dimensional sheets of cells into complex three-dimensional structures. Such transformations play key roles during embryogenesis and their understanding is important both from a purely scientific standpoint and for the design of man-made tissues and organs. Our laboratory is using the eggshell morphogenesis in the fruit fly Drosophila melanogaster as a model for studying epithelial morphogenesis in a relatively simple setting, with a constant number of cells. Based on the combination of time lapse imaging studies and three-dimensional image reconstructions we found that epithelial morphogenesis in this system can be viewed as a two-stage process, in which buckling of a group of cells out of a curved sheet is followed by ordered cell rearrangements, generating a tube-like structure. A small number of cells involved in this transformation (<100) and their reproducible dynamics enable its computational modeling. I will present models based on vertex description of cell sheets and discuss some applied mathematical problems that arise in the analysis of these models.

References:

Osterfield M, Du X, Schüpbach T, Wieschaus E, Shvartsman SY. Three-dimensional epithelial morphogenesis in the developing Drosophila egg. Dev Cell. 2013 Feb 25;24(4):400-10.

Fletcher AG, Osterfield M, Baker RE, Shvartsman SY. Vertex models of epithelial morphogenesis. Biophys J. 2014 Jun 3;106(11):2291-304.

Osterfield M, Schüpbach T, Wieschaus E, Shvartsman SY. Diversity of epithelial morphogenesis during eggshell formation in drosophilids. Development. 2015 Jun 1;142(11):1971-7. 

Note for Attendees

Note the change from our usual venue this term. 
hide
SFU
Mon 28 Sep 2015, 3:00pm
Algebraic Geometry Seminar
MATH 126
K-Stability for Fano Varieties with Torus Action
MATH 126
Mon 28 Sep 2015, 3:00pm-4:00pm

Abstract

It has been recently shown by Chen-Donaldson-Sun that the existence of a Kähler-Einstein metric on a Fano manifold is equivalent to the property of K-stability. In general, however, this does not lead to an effective criterion for deciding whether such a metric exists, since verifying the property of K-stability requires one to consider infinitely many special degenerations called test configurations. I will discuss recent joint work with H. Süß in which we show that for Fano manifolds with complexity-one torus actions, there are only finitely many test configurations one needs to consider. This leads to an effective method for verifying K-stability, and hence the existence of a Kähler-Einstein metric. As an application, we provide new examples of Kähler-Einstein Fano threefolds.
hide
Fernuniversitat Hagen, Germany
Tue 29 Sep 2015, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Area law for the entanglement entropy of the free Fermi gas at nonzero temperature
ESB 2012
Tue 29 Sep 2015, 3:30pm-4:30pm

Abstract

The leading asymptotic large-scale behavior of the spatially bipartite entanglement entropy (EE) of the free Fermi gas at temperature T=0 is by now well understood. Here, we present and discuss the first rigorous results for the corresponding EE of thermal equilibrium states at T>0. The leading large-scale term of this thermal EE turns out to be twice the first finite-size correction to the infinite-volume thermal entropy (density). However, it is given by a rather complicated integral derived from semiclassical trace formulas and differs, at least at high temperature, from simpler expressions previously obtained by arguments based on a conformal field theory. In the zero-temperature limit, the leading large-scale term of the thermal EE considerably simplifies and displays a \ln(1/T)-singularity which one may identify with the known logarithmic correction at T=0 to the so-called area-law scaling. This is joint work with Hajo Leschke and Alexander Sobolev.
hide
University of Georgia
Tue 29 Sep 2015, 4:00pm
Discrete Math Seminar
ESB 4127
Asymptotic laws for knot diagrams
ESB 4127
Tue 29 Sep 2015, 4:00pm-5:00pm

Abstract

We consider a model of random knots akin to the one proposed by Dunfield et. al.; a random knot diagram is a random immersion of the circle into the sphere with randomly assigned crossing signs. By studying diagrams as annotated planar maps, we are able to show that any given ``tangle diagram'' substructure almost certainly occurs many times in a random knot diagram with sufficiently many crossings. Thus, in this model, it is exponentially unlikely for a diagram with n crossings to represent an unknot as n \rightarrow \infty. This asymptotic behavior is similar to that seen in other models of random knots such as random lattice walks and random polygons.
hide
Konstantin Tikhomirov
University of Alberta
Wed 30 Sep 2015, 3:00pm
Probability Seminar
ESB 2012
The smallest singular value of random matrices with independent entries
ESB 2012
Wed 30 Sep 2015, 3:00pm-4:00pm

Abstract

 
We consider a classical problem of estimating the smallest singular value of random rectangular and square matrices with independent identically distributed entries. The novelty of our results lies in very weak, or nonexisting, moment assumptions on the distribution of the entries. We prove that, given a sufficiently “tall” N \times n rectangular matrix  A = (a_{ij} ) with i.i.d. entries satisfying the condition \sup_{\lambda \in \mathbb{R}} \mathbb{P} \left\{ \lvert a_{ij} − \lambda \rvert \le 1 \right\} \le 1/2, the smallest singular value s_n (A) satisfies s_n(A) \gtrsim \sqrt{N} with probability very close to one.

Our second theorem is an extension of the fundamental result of Bai and Yin from the early 1990’s. Let \{a_{ij} \}^\infty_{i,j=1} be an infinite double array of i.i.d. random variables with zero mean and unit variance, and let (N_m )_{m=1}^\infty be an integer sequence satisfying \lim_{m \to \infty} \frac{N_m}{m} = r\) for some \(r \in (1, \infty). Then, denoting by A_m the N_m \times m top-left corner of the array \{a_{ij}\}, we have

\[ \lim_{m \to \infty} \frac{s_m(A_m)}{\sqrt{N_m}} = \sqrt{r}-1 \hspace{3mm}\mbox{ almost surely}.\]

This result does not require boundedness of any moments of a_{ij}'s higher than the 2-nd and resolves a long standing question regarding the weakest moment assumptions on the distribution of the entries sufficient for the convergence to hold.
hide
Ric Wade
UBC Math
Wed 30 Sep 2015, 3:15pm
Topology and related seminars
ESB 4113
Subspace arrangements and BNS invariants, continued.
ESB 4113
Wed 30 Sep 2015, 3:15pm-4:15pm

Abstract

(This is a continuation of the talk of 23 Septemer) We introduce a natural chain complex associated to a collection of subspaces of a vector spaces, and discuss the associated homology. We will give some background on Bieri-Neumann-Strebel invariants of groups, and show how the BNS invariant of a group leads to a nice subspace arrangement, whose associated homology is (yet) another invariant of the group. This can give a useful way of distinguishing between finitely presented groups - we will give some examples involving right-angled Artin groups.
hide