Women-in-Math Seminar

Description

Hey, make sure to check out Women in Mathematics @ MIT!

Mission: This seminar series exists for the purpose of encouraging, inspiring, showcasing and celebrating research of women in math, as well as for stimulating mathematical interactions within and around the still minority community. It is also hoped that this seminar will provide a little oasis where the audience are encouraged to try to learn about, judge and celebrate women's work in mathematics without the influence of gender considerations.

Conception: The idea was original suggested by our own graduate student Karene Chu probably as early as in 2009 at one of the Math Women social gatherings. I finally embraced the idea in spring 2011, and a consultation/brainstorming session involving 4-5 female grad students of the department took place in April 2011. The consensus was that women (at least the female grad students) in math would definitely benefit from some women-friendly mathematical activities besides the existing social events, and we should get this thing going!

Speakers: If you are a woman working in mathematics in Toronto or somewhere in Ontario, and have a piece of mathematics you would like to share with us, be it original research or a nice piece of math you learned, we are happy to listen. Don't be shy, student presentations are strongly encouraged! Simply send an email to the organizer (for contact info and format of the seminar, see below), specifying which month you would like to speak in. Please volunteer to give a talk! Also, if you are a man working on a piece of mathematics inspired by some female mathematician, you are also welcome to speak at the seminar. :-)

Audience: Everyone! Regardless of your gender, you are very welcome to join us and get to know us as long as you are willing to strive for equality (complete mastery is not required).

Format and Time: The seminar takes place once a month, usually on a Wednesday or Thursday towards the end of each month (as long as we have a speaker for that month). We try to hold the seminar around lunch time. The talk can be 30 to 50 minutes in length (depending on the speaker's preference), and the level of formality may vary from talk to talk. However, these talks should always be designed for a diverse mathematical audience, that is, no specialized background should be assumed. Special time accommodation may be made for the speaker as long as logistics allow.

Organizer: Lucy Liuxuan Zhang (lzhang *at^ math *dot^ utoronto *dot^ ca).
Faculty Support (organizing some lunches): Yael Karshon

Schedule

Abstracts

May 2011 -- Nevena Francetic, Senior PhD student @ U of T -- Covering arrays with row limit: Covering Arrays with Row Limit (/CARL/s) are a generalization of well studied covering arrays, a family of combinatorial objects modelling software test suites. /CARL/s have one new parameter, weight /w/, representing the number of components tested a once. We are going to discuss the connection of /CARL/s with some existing combinatorial objects, asymptotic bounds on their size, as well as some construction techniques.

June 2011 -- Jessie Yang, Senior PhD student @ U of T -- Tropicalization of Severi varieties: Tropicalization is an operation that turns a complex variety into a polyhedral fan whose maximal cones are assigned with positive integers. It can be used to compute numerical invariants of the complex variety. This approach can be viewed as a generalization of Newton polytope theory which is applicable for the cases of hypersurfaces and complete intersections. It shows a beautiful interplay between algebraic geometry and polyhedral geometry.
I will present both algebraic and geometric definitions of tropicalization and some explicit examples.
For the last 20 minutes, I will focus on Severi varieties, algebraic varieties which parametrize complex plane nodal curves. I will present a description of the tropicalizations of Severi varieties and show how to compute the degrees of Severi varieties using tropical intersection theory which is purely combinatorial.

July 2011 -- Megumi Harada, Faculty @ McMaster University -- Equivariant cohomology, GKM-compatibility and Schubert calculus on Hessenberg varieties: Hessenberg varieties are certain subvarieties of flag varieties G/B which appear in a wide array of mathematical areas, including geometric representation theory, numerical analysis, algebraic geometry, and combinatorics. In this talk I will explain recent joint work with Julianna Tymoczko, in which we develop a `generalized Schubert calculus' in the S^1-equivariant cohomology of Peterson varieties. We view our results as the first steps in the development of an extensive theory of generalized Schubert calculus beyond the realm of the G/P.
The talk will be aimed at beginning graduate students. In particular, at the beginning I will give an impressionistic sketch of Schubert calculus in general. I will then introduce the main characters, which are the flag varieties and the Hessenberg varieties (as well as the torus actions on them), and give an idea of the GKM and Schubert calculus techniques which enter into our proofs. Time permitting, I will conclude with a sampling of open questions.

August (17) 2011 -- Bei Zeng, Faculty @ University of Guelph -- A simple construction for quantum error-correcting codes: The discovery of quantum error-correcting codes (QECCs) has greatly improved the long-term prospects for quantum communication and computation technology. Stabilizer codes, a quantum analogue of classical additive codes, are the most important class of QECCs. These codes have dominated the study of quantum error correction ever since their discovery in the mid-1990s. Here we present a simple unifying approach to quantum error-correcting code design that encompasses stabilizer (additive) codes, as well as all known examples of nonadditive codes with good parameters. We use this framework to generate new codes with superior parameters to any previously known.

August (23) 2011 -- Iva Halacheva & Emily Cliff, Master students @ U of T -- Classifying Lie bialgebras: Analogously to Lie algebras and Lie groups, Lie bialgebras are the infinitesimal objects associated to Poisson-Lie groups. One reason for studying them is in the context of understanding finite-type invariants of virtual knots. We will give a survey of the theory of Lie bialgebras and discuss the Belavin-Drinfel'd classification theorem of Lie bialgebra structures on simple Lie algebras.

September (23) 2011 -- Tanya Yarmola, Postdoc @ U of T -- Discrete-time degenerate random perturbations of dynamical systems: An important first step in studying open systems, i.e. systems exchanging energy and/or matter with an outside environment, is understanding their steady states and the rates at which various initial conditions converge to them. In many situations open systems can be thought of as random perturbations of isolated systems and, for a variety of particle systems, such random perturbations occur discretely in time and only in a few directions of a many dimensional and possibly non-compact phase space. We will discuss basic properties and examples of randomly perturbed dynamical systems with this degenerate discrete-time noise. Aside from the situations when the phase space is compact or the steady states can be written down explicitly, their existence is a nontrivial and frequently open question. In this talk we choose to focus on the properties that would guarantee uniqueness. The degeneracy of the noise may lead to the coexistence of two or more steady states; worse yet, they may be supported on lower codimension manifolds and statistically attract almost all points of the phase space. Clearly, this is not the situation we expect in the case of particle systems for which steady states should be 'volume-like'. We will provide relatively general conditions that guarantee uniqueness of the steady states for randomly perturbed dynamical systems with discrete-time degenerate noise and see how those conditions apply to certain examples.

October (14) 2011 -- Tatyana Barron, Faculty @ University of Wester Ontario -- Berezin-Toeplitz quantization: This will be a mostly introductory talk. The word "quantization" means, in general, a way to pass from classical mechanics to quantum mechanics. Mathematically it can be formulated as a question about existence of a representation of a certain Lie algebra. Geometric quantization is a part of symplectic geometry (very closely related to representation theory), and Berezin-Toeplitz quantization can be viewed as a version of geometric quantization. I will explain what kind of mathematics is involved and will comment on the physics meaning as well.

November (25) 2011 -- Sarah Croke, Postdoc @ Perimeter Institute -- Introducing "coherent measurement" in quantum mechanics: I will discuss measurements in quantum mechanics, with particular reference to the task of quantum state discrimination. Much of the talk will be introductory and I will not assume prior familiarity with quantum mechanics. For the remainder of the talk, I will introduce the concept of "coherent measurement", in which a quantum device, rather than a classical one, learns something about the system being measured. I will discuss the advantages of this approach for the task of state discrimination, showing that quantum information processors (QIPs) with O(1) qubits can substantially reduce measurement complexity -- i.e., the number of samples required to learn something about an unknown quantum state.
The resulting protocols demonstrate useful applications for the 2-14 qubit QIPs that exist today. Such QIPs are computationally trivial (their dynamics can be easily simulated on classical computers), but our results suggest nontrivial applications in sensing, detection, metrology, and characterization of larger prototype QIPs.
Joint work with Robin Blume-Kohout and Michael Zwolak.

January (20) 2012 -- Artemis Dude (female form of Artem Dudko), Senior PhD student @ University of Toronto -- Asymptotic expansions and dynamics of parabolic germs: An asymptotic expansion is a (divergent) formal series with the property that truncating the series after a finite number of steps provides an approximation to a given function as the argument of the function tends to a particular point. In the talk we will discuss Borel-Laplace method of summation of an asymptotic expansion. We will briefly consider examples of applications of asymptotic expansions in Differential Equations, Mathematical Physics and Dynamics with the focus on dynamics of simple parabolic germs. We will present recent results on Ecalle's analytic (conjugacy) invariants for simple parabolic germs.

February (17) 2012 -- Marina Tvalavadze, Postdoc @ Fields Institute -- Enveloping algebras of non-associative structures: In the classical theory of Lie algebras, the Poincar-Birkhoff-Witt (PBW) theorem (Poincar (1900), G. D. Birkhoff (1937), Witt (1937)) establishes a fundamental connection between Lie and associative algebras by saying that any Lie algebra L can be viewed as a subalgebra of U(L)^{-} where U(L) is its universal (associative) enveloping algebra. Since then the analogue results for other non-associative structures such as ε-Lie algebras, Leibnitz n-algebras, Malcev algebras,etc. have been discovered. This allowed to describe their representations by means of universal enveloping algebras and to make calculations in a similar way as in a ring of polynomials F[x1,...,xn]. The aim of this talk is to discuss some classical as well as recent (including the speaker's) results on the universal enveloping algebras for different types of algebras.

March (23) 2012 -- Kristin Shaw, Postdoc @ U of T -- Tropical intersection theory and lines in surfaces: Tropical geometry studies polyhedral complexes with integral affine structures often obtained from degenerations of algebraic subvarieties of toric varieties. Tropical spaces are often more manageable and more combinatorial then their complex counterparts. One of the most celebrated results of tropical geometry is Mikhalkin's correspondence theorem. This relates tropical and complex (or real) curves in toric surfaces, and provides a combinatorial way of computing Gromov-Witten (or Welschinger) invariants for toric surfaces. However, this correspondence theorem cannot always be generalised to other spaces. To illustrate these points I will focus on the very classical example of studying lines in surfaces. It is a well known fact that in CP^3 a non-singular degree 3 surface contains exactly 27 lines, and a generic non-singular surface of degree larger than 3 contains no lines. Tropically, however, Vigeland discovered the existence of infinite families of lines in generic non-singular tropical surfaces of degrees larger than or equal to 3, and also isolated lines in generic non-singular surfaces of degree larger than or equal to 4. In joint work with Erwan Brugall, we apply intersection theory to study Vigeland's lines (and others) in surfaces.

April (20) 2011 -- Feride Tiglay, University of Western Ontario -- Integrable evolution equations on spaces of tensor densities: In a pioneering paper V. Arnold presented a general framework within which it is possible to employ geometric and Lie theoretic techniques to study the equations of motion of a rigid body in R^3 and the equations of ideal hydrodynamics. I will describe how to extend his formalism and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. These two PDE possess all the hallmarks of integrability: the existence of a Lax pair formulation, a bi-Hamiltonian structure, the presence of an infinite family of conserved quantities and the ability to write down explicitly some of its solutions. I will also talk about local well-posedness of the corresponding Cauchy problem and global existence of solutions.

July (26) 2012 -- Zsuzsanna Dancso, Postdoc @ Fields Institute -- Odd Khovanov homology via hyperplane arrangements: I will give an introduction to hyperplane arrangements, a combinatorial structure that includes both graphs and (alternating) links as sub-classes. I will also explain what categorification of a polynomial invariant means, the most famous example of this is Khovanov's categorification of the Jones polynomial of links. I will conclude by describing a new construction of Odd Khovanov homology via hyperplane arrangements.