Halloween 2018

Dr. Octopus won this year’s Halloween contest!

Drop courses deadline – October 29, 2018

Drop courses absolute deadline: Monday, October 29, 2018

Students dropping courses on ACORN must also fill out a drop courses
form and submit to the Graduate Office.

http://www.sgs.utoronto.ca/Documents/Add+Drop+Courses.pdf

Forms are also available on the counter in the math mailroom (BA 6290A).

Departmental PhD Thesis Exam – Chia-Cheng Liu

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, December 5, 2018
11:10 a.m.
BA1170

PhD Candidate:  Chia-Cheng Liu
Co-Supervisors:   Joel Kamnitzer/Alexander Braverman
Thesis title:  Semi-innite Cohomology, Quantum Group Cohomology, and the Kazhdan-Lusztig
Equivalence

The Kazhdan-Lusztig tensor equivalence is a monoidal functor which sends modules over ane
Lie algebras at a negative level to modules over quantum groups at a root of unity. A positive
level Kazhdan-Lusztig functor is dened using Arkhipov-Gaitsgory’s duality between ane Lie
algebras of positive and negative levels. We prove that the semi-innite cohomology functor
for positive level modules factors through the positive level Kazhdan-Lusztig functor and the
quantum group cohomology functor with respect to the positive part of Lusztig’s quantum
group. This is the main result of the thesis.

Monoidal structure of a category can be interpreted as factorization data on the associated
global category. We describe a conjectural reformulation of the Kazhdan-Lusztig tensor equivalence
in factorization terms. In this reformulation, the semi-innite cohomology functor at
positive level is naturally factorizable, and it is conjectured that the factorizable semi-innite
cohomology functor is essentially the positive level Kazhdan-Lusztig tensor functor modulo the
Riemann-Hilbert correspondence. Our main result provides an important technical tool in a
proposed approach to a proof of this conjecture.

A copy of the thesis can be found here: thesis_chiachengliu-1

Halloween Tea and Costume Party

There will be a special tea time on Halloween (Wednesday October 31) in the Department lounge at 2:00 pm.

Costumes are encouraged, we will have prizes available for the best costumes.

Halloween 2018

Departmental PhD Thesis Exam – Krishan Rajaratnam

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Friday, October 19, 2018
2:10 p.m.
BA6183

PhD Candidate:  Krishan Rajaratnam
Supervisor:   Michael Sigal
Thesis title: Abrikosov lattice solutions of the ZHK Chern-Simons equations

****

In this thesis we study the ZHK Chern-Simons equations which occur in the study of the fractional quantum hall effect of condensed matter physics. After stating basic properties of these equations, we first prove the existence of Abrikosov lattice solutions of them.  Among these solutions, we find the physically interesting one whose lattice shape minimizes the average energy per lattice cell. In addition to the Abrikosov lattice solutions, we find solutions of the ZHK Chern-Simons equations on Riemann surfaces of higher genus $g$, by utilizing similar results for the Ginzburg-Landau equations.

Finally, we study the orbital stability of the Abrikosov lattice solutions under perturbations which preserve the lattice.

A copy of the thesis can be found here:

Departmental PhD Thesis Exam – Steven Amelotte

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Tuesday, June 26 2018
1:10 p.m.
BA6183

PhD Candidate:  Steven Amelotte
Supervisor:   Paul Selick
Thesis title: Unstable Homotopy Theory Surrounding the Fibre of the $p^\text{th}$ Power Map on Loop Spaces of Spheres

***

In this thesis, we study the fibre of the $p^\text{th}$ power map on loop spaces of spheres with a view toward obtaining homotopy decompositions. For the prime $p=2$, we give an explicit decomposition of the fibre $\Omega^3S^{17}\{2\}$ of the squaring map on the triple loop space of the $17$-sphere, building on work of Campbell, Cohen, Peterson and Selick, who showed that such decompositions are only possible for $S^5$, $S^9$ and $S^{17}$. This induces a splitting of the mod $2$ homotopy groups $\pi_\ast(S^{17}; \mathbb{Z}/2\mathbb{Z})$ in terms of the integral homotopy groups of the fibre of the double suspension $E^2 \colon S^{2n-1} \to \Omega^2S^{2n+1}$ and refines a result of Cohen and Selick, who gave similar decompositions for $S^5$ and $S^9$. For odd primes, we find that the decomposition problem for $\Omega S^{2n+1}\{p\}$ is equivalent to the strong odd primary Kervaire invariant problem. Using this unstable homotopy theoretic interpretation of the Kervaire invariant problem together with what is presently known about the $3$-primary stable homotopy groups of spheres, we give a new decomposition of $\Omega S^{55}\{3\}$.

We relate the $2$-primary decompositions above to various Whitehead products in the homotopy groups of mod $2$ Moore spaces and Stiefel manifolds to show that the Whitehead square $[i_{2n},i_{2n}]$ of the inclusion of the bottom cell of the Moore space $P^{2n+1}(2)$ is divisible by $2$ if and only if $2n=2,4,8$ or $16$. As an application of our $3$-primary decomposition, we prove two new cases of a longstanding conjecture which states that the fibre of the double suspension is a double loop space.

A copy of the thesis can be found here:  ut-thesis

Departmental PhD Thesis Exam – Vincent Gelinas

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Monday, June 18,  2018
11:10 a.m.
BA6183

PhD Candidate:  Vincent Gelinas
Co-Supervisors:   Joel Kamnitzer, Colin Ingalls
Thesis title:  Contributions to the Stable Derived Categories of Gorenstein Rings

***

Abstract:

The stable derived category ${\rm D}_{sg}(R)$ of a Gorenstein ring $R$ is defined as the Verdier quotient of the bounded derived category $ {\rm D}^b(\modsf R) $ by the thick subcategory of perfect complexes, that is, those with finite projective resolutions, and was introduced by Ragnar-Olaf Buchweitz as a homological measure of the singularities of $R$. This thesis contributes to its study, centered around representation theoretic, homological and Koszul duality aspects.

In Part I, we first complete (over $\C$) the classification of homogeneous complete intersection isolated singularities $R$ for which the graded stable derived category ${\rm D}^{\Z}_{sg}(R)$ (respectively, $ {\rm D}^b(\coh X) $ for $X = \proj R$) contains a tilting object. This is done by proving the existence of a full strong exceptional collection of vector bundles on a $2n$-dimensional smooth complete intersection of two quadrics $X = V(Q_1, Q_2) \subseteq \mathbb{P}^{2n+2}$, building on work of Kuznetsov. We then use recent results of Buchweitz-Iyama-Yamaura to classify the indecomposable objects in ${\rm D}_{sg}^{\Z}(R_Y)$ and the Betti tables of their complete resolutions, over $R_Y$ the homogeneous coordinate rings of $4$ points on $\mathbb{P}^1$ and $4$ points on $\mathbb{P}^2$ in general position.

In Part II, for $R$ a Koszul Gorenstein algebra, we study a natural pair of full subcategories whose intersection $\mathcal{H}^{\mathsf{lin}}(R) \subseteq {\rm D}_{sg}^{\Z}(R)$ consists of modules with eventually linear projective resolutions. We prove that such a pair forms a bounded t-structure if and only if $R$ is absolutely Koszul in the sense of Herzog-Iyengar, in which case there is an equivalence of triangulated categories ${\rm D}^b(\mathcal{H}^{\mathsf{lin}}(R)) \cong {\rm D}_{sg}^{\Z}(R)$. We then relate the heart to modules over the Koszul dual algebra $R^!$. As first application, we extend the Bernstein-Gel’fand-Gel’fand correspondence beyond the case of exterior and symmetric algebras, or more generally complete intersections of quadrics and homogeneous Clifford algebras, to any pair of Koszul dual algebras $(R, R^!)$ with $R$ absolutely Koszul Gorenstein. In particular the correspondence holds for the coordinate ring of elliptic normal curves of degree $\geq 4$ and for the anticanonical model of del Pezzo surfaces of degree $\geq 4$. We then relate our results to conjectures of Bondal and Minamoto on the graded coherence of Artin-Schelter regular algebras and higher preprojective algebras; we characterise when these conjectures hold in a restricted setting, and give counterexamples to both in all dimension $\geq 4$.

A copy of the thesis can be found here:  thesis

Departmental PhD Thesis Exam – Huan Vo

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, June 20 2018
11:10 a.m.
BA6183

PhD Candidate:  Huan Vo
Supervisor:   Dror Bar-Natan
Thesis title:  Alexander Invariants of Tangles via Expansions

****

Abstract:

In this thesis we describe a method to extend the Alexander polynomial to tangles. It is based on a
technology known as expansions, which is inspired by the Taylor expansion and the Kontsevich integral.
Our main object of study is the space of w-tangles, which contains usual tangles, but has a much simpler
expansion. To study w-tangles, we introduce an algebraic structure called meta-monoids. An expansion
of w-tangles together with a particular Lie algebra, namely the non-abelian two-dimensional Lie algebra,
gives us a meta-monoid called Γ-calculus that recovers the Alexander polynomial. Using the language
of Γ-calculus, we rederive certain important properties of the Alexander polynomial, most notably the
Fox-Milnor condition on the Alexander polynomials of ribbon knots [Lic97, FM66]. We argue that our
proof has some potential for generalization which may help tackle the slice-ribbon conjecture. In a sense
this thesis is an extension of [BNS13].

A copy of the thesis can be found here:  Thesis_HuanVo_V1

2018 Graduate Scholarship Recipients

Over the last few years, the generosity of faculty, alumni and friends of the Department have allowed us to create a number of significant scholarships to support graduate students. This year’s winners are listed below.

1) Ida Bulat Memorial Graduate Fellowship:

Lennart Döppenschmitt (student of Marco Gualtieri)

2) Vivekananda Graduate Scholarship for International students:

Debanjana Kundu (student of Kumar Murty)

3) Canadian Mathematical Society Graduate Scholarship:

Saied Sorkhou (student of Joe Repka)

4) Coxeter Graduate Scholarship:

Mateusz Olechnowicz (student of Jacob Tsimerman, Patrick Ingram)

5) International Graduate Student Scholarship:

Abhishek Oswal (student of Jacob Tsimerman)

6) Margaret Isobel Elliott Graduate Scholarship:

Keegan Da Silva Barbosa (student of Stevo Todorcevic)

7) Irving Kaplansky Scholarship:

Jamal Kawach (student of Stevo Todorcevic)

Congratulations to all!

2018 Award Winners

We will celebrate the significant contributions of this year’s award winners during our reception in the Math lounge on Wednesday, June 13 starting at 3:10 p.m. Graduation-Awards-Invitation-2018

F. V. Atkinson Teaching Award for Postdoctoral Fellows

  • Bhishan Jacelon (working with George Elliott)
  • Mihai Nica (working with Jeremy Quastel)

Daniel B. DeLury Teaching Assistant Awards

  • Anne Dranovski, student of Joel Kamnitzer
  • Jeffrey Im, student of George Elliott
  • Larissa Richards, student of Ilia Binder

Ida Bulat Teaching Awards for Graduate Students

  • Thaddeus Janisse, student of Joe Repka
  • Zackary Wolske, student of Henry Kim

We thank the faculty members, undergraduate students and course instructors who took the time to submit their nominations for the various awards.

The awards committee received positive comments about the work of our TAs and CIs.  We can take pride in their performance.

Congratulations to Anne, Jeffrey, Larissa, Thaddeus and Zackary!