Mathematics Dept.
  Events
Courant Institute, NYU
Mon 23 Oct 2017, 3:00pm
Institute of Applied Mathematics / PIMS Seminars and PDF Colloquiums
ESB 2012
Numerical Investigation of Crouzeix’s Conjecture
ESB 2012
Mon 23 Oct 2017, 3:00pm-4:00pm

Abstract

Crouzeix's conjecture is among the most intriguing developments in matrix theory in recent years. Made in 2004 by Michel Crouzeix, it postulates that, for any polynomial p and any matrix A, ||p(A)|| <= 2 max(|p(z)|: z in W(A)), where the norm is the 2-norm and W(A) is the field of values (numerical range) of A, that is the set of points attained by v*Av for some vector v of unit length. Crouzeix proved in 2007 that the inequality above holds if 2 is replaced by 11.08, and very recently this was greatly improved by Palencia, replacing 2 by 1+sqrt(2). Furthermore, it is known that the conjecture holds in a number of special cases, including n=2. We use nonsmooth optimization to investigate the conjecture numerically by attempting to minimize the “Crouzeix ratio”, defined as the quotient with numerator the right-hand side and denominator the left-hand side of the conjectured inequality. We present numerical results that lead to some theorems and further conjectures, including variational analysis of the Crouzeix ratio at conjectured global minimizers. All the computations strongly support the truth of Crouzeix’s conjecture. This is joint work with Anne Greenbaum and Adrian Lewis.

Note for Attendees

Refreshments before talk in room ESB 4133 (the PIMS Lounge).
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SNS Pisa
Mon 23 Oct 2017, 4:00pm
Algebraic Geometry Seminar
MATH 126
Chow rings of some stacks of smooth curves
MATH 126
Mon 23 Oct 2017, 4:00pm-5:00pm

Abstract

 There is by now an extensive theory of rational Chow rings of stacks of smooth curves. The integral version of these Chow rings is not as well understood. I will survey what is known, including some recent developments.
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McGill
Tue 24 Oct 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
An isometric embedding problem and related geometric inequalities
ESB 2012
Tue 24 Oct 2017, 3:30pm-4:30pm

Abstract

Solutions to the classical Weyl problem by Nirenberg and Pogorelov
play fundamental role in the notion of quasi local masses and positive quasi
local mass theorems in general relativity. An interesting question in
differential geometry is whether one can isometrically embed compact surfaces
with positive Gauss curvature to a general 3 dimensional ambient space. Of
particular importance is the anti de Sitter Schwarzchild space in general
relativity.  We discuss some recent progress in this direction, the a priori
estimates for embedded surfaces in a joint work with Lu, the openness and
non-rigidity results of Li -Wang, and a new quasi local type inequality of
Lu-Miao. We will also discuss open problem related to isometric embeddings to
ambient spaces with horizons.
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UBC
Tue 24 Oct 2017, 4:00pm
Discrete Math Seminar
ESB 4127
Disjoint Cycles and Equitable Colorings in Graphs
ESB 4127
Tue 24 Oct 2017, 4:00pm-5:00pm

Abstract

The Corradi-Hajnal Theorem gives a minimum-degree condition for the existence of a given number of vertex-disjoint cycles in a simple graph. We discuss a number of variations on the Corradi-Hajnal Theorem, changing both the nature of the necessary condition (for example, minimum degree sum instead of minimum degree) and the kind of subgraph whose existence is desired. We also briefly discuss the connections between these types of theorems and equitable graph colourings.

 

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Krishanu Sankar
UBC
Wed 25 Oct 2017, 3:15pm
Topology and related seminars
ESB 4133
Symmetric Powers and the Dual Steenrod Algebra - Part 1
ESB 4133
Wed 25 Oct 2017, 3:15pm-4:15am

Abstract

The mod $p$ Steenrod algebra is the (Hopf) algebra of

stable operations on mod $p$ cohomology, and in part

measures the subtle behavior of $p$-local homotopy theory

(as opposed rational homotopy theory, which is much simpler).

A classical theorem of Dold-Thom tells us that the infinite

symmetric power of the $n$-dimensional sphere is the Eilenberg-Maclane

space K(Z, n),and one can use an appropriate modification of this

construction to compute the dual Steenrod algebra. The infinite

symmetric power of the sphere spectrum has a filtration whose

$k$-th cofiber miraculously turns out to be the Steinberg summand

(from modular representation theory of GL_k(F_p)) of the

classifying space of (Z/p)^k. This opens the door for slick

computations - for example, the Milnor indecomposables can

be picked out as explicit cells.

 

In this talk, I will introduce the concepts and results

chronologically. I will also include hands-on homotopy

theory computations as time permits.

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Alain Prat
UBC Math
Thu 26 Oct 2017, 12:30pm
Lunch Series on Teaching & Learning
Math 126
Working against the WeBWork clock: What are the behaviour patterns of students who struggle to complete online calculus assignments?
Math 126
Thu 26 Oct 2017, 12:30pm-1:30pm

Abstract

Since 2010, the math department at UBC has been gradually adopting the WeBWork online homework system in most first and second year courses. Instructors typically give students several days to complete their WeBWork assignments, and allow students several attempts at each problem. Despite this, many students struggle to complete their online assignments. In this talk, I'll discuss how the timing of answer submissions recorded in WeBWork log files can reveal the behaviour patterns of students who struggle with WeBWork. In particular, students who don't complete the WeBWork start the assignments closer to the deadline, have shorter login sessions and don't persist for as long once they encounter a problem they can't solve. I'll discuss what these observations reveal about the mindset of struggling students, and how assignments could be restructured to help increase their completion rate.
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Nikhil Kher
Fri 27 Oct 2017, 12:00pm
Graduate Student Seminar
MATH 203
Operator Semigroups and Hille Yosida Theorem
MATH 203
Fri 27 Oct 2017, 12:00pm-1:00pm

Abstract

Semigroups are useful in solving a large class of problems known as evolution equations. These kind of equations often appear in many disciplines including physics, chemistry, biology and engineering. I will be discussing an introduction to theory of C0-semigroups on Banach spaces. We then discuss uniformly continuous semigroups and prove some important results and characterizations related to them. Following which, we present examples of how new semigroups can be constructed out of a given semigroup. We associate a generator to a C0-semigroup and illustrate their properties by means of some theorems. Subsequently we prove Hille-Yosida theorem, which provides a necessary and sufficient condition for an unbounded operator on a Banach space to be a generator of a contraction C0-semigroup. This was proved in 1948 by mathematicians E. Hille and K. Yosida. We also prove Feller-Miyadera-Phillips theorem which generalizes Hille-Yosida theorem. This was proved around 1952. If time permits I'll do some other interesting stuff related to theory of C_0-semigroups.
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