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Curves Seminar

Monday, February 7th, 2022

Time: 11:00 a.m.  Place: Online via Zoom (contact Deepanshu Prasad for Zoom link)

Speaker: Deepanshu Prasad (Queen's University)

Title: Coxeter arrangements and Shi arrangements.

Abstract: We will look at the Coxeter arrangements, arising from root systems, and Shi arrangements. We will calculate their characteristic polynomial using the finite field method and talk about their "supersolvability" and "freeness".

Curves Seminar

Monday, January 31st, 2022

Time: 11:00 a.m.  Place: Online via Zoom (contact Deepanshu Prasad for Zoom link)

Speaker: Alexandre (Sasha) Zotine (Queen's University)

Title: Graphical Arrangements cont.

Abstract: We will finish our discussion of graphical arrangements by covering orientations of graphs and relating them to the regions of their associated arrangements. Once this is finished, I will start to introduce some finite field methods for computing characteristic polynomials over the rationals.

Curves Seminar

Monday, January 24th, 2022

Time: 11:00 a.m.  Place: Online via Zoom (contact Deepanshu Prasad for Zoom link)

Speaker: Alexandre (Sasha) Zotine (Queen's University)

Title: Graphical Arrangements.

Abstract: Continuing our discussion of hyperplane arrangements, I will give a quick reminder of the talks that Ben gave on matroids, then proceed to introduce arrangements arising from graphs.

Math & Stats Department Colloquium

Friday, February 11th, 2022

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Somnath Pradhan (Queen's University)

Title: Robustness of Stochastic Optimal Control for Controlled Diffusions with Incorrect Models

Abstract: In control theory, typically a nominal model is assumed based on which an optimal control is designed and then applied to an actual (true) system. This gives rise to the problem of performance loss due to the mismatch between the true model and the assumed model. A robustness problem in this context is to show that the error due to the mismatch between true model and assumed model decreases to zero as the assumed model approaches the true model. We study this problem when the state dynamics of the system are governed by controlled diffusion processes. In particular, we will discuss continuity and robustness properties of infinite-horizon $\alpha$-discounted/ergodic optimal control problems for a general class of controlled diffusion processes. Under a general set of assumptions and a convergence criterion on the models, we first establish that the optimal value of the approximate model converges to the optimal value of the true model. We then establish that the error due to mismatch that occurs by application of a control policy, designed for an incorrectly estimated model, to a true model decreases to zero as the incorrect model approaches the true model. We will see that, compared to related results in the discrete-time setup, the continuous-time theory will let us utilize the strong regularity properties of solutions to optimality (HJB) equations, via the theory of uniformly elliptic PDEs, to arrive at strong continuity and robustness properties. A corollary of our analysis is a continuity result on the optimal cost in the control policies under a natural topology which also leads to near-optimality of quantized stationary policies.

Somnath Pradhan is a Coleman postdoctoral fellow in the Department of Mathematics and Statistics at Queen's University. He was a postdoc fellow at the Indian Institute of Science Education and Research, Pune, India from 2019-2021. He obtained his Ph.D.~in Mathematics from Indian Institute of Science in 2019. His research interests include stochastic analysis, applied probability, and controlled Markov processes.

Algebra & Geometry Seminar

Monday, February 7th, 2022

Time: 4:30 p.m.  Place: Online via Zoom (contact Kaveh Mousavand for Zoom link)

Speaker: Dixy Msapato (University of Leeds)

Title: Counting tau-exceptional sequences for Nakayama algebras.

Abstract: The notion of a tau-exceptional sequence was introduced by Buan and Marsh in 2018 as a generalisation of exceptional sequences over finite dimensional algebras. In this talk, I will introduce both this notion, and present counting results of tau-exceptional sequences over some classes of Nakayama algebras (and a general counting strategy). In some of these Nakayama algebra cases we will see obtain closed formulas counting other well-known combinatorial objects, and exceptional sequences over some path algebras of Dynkin quivers.

Website details here:

Math & Stats Department Colloquium

Friday, February 4th, 2022

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Ben Webster (University of Waterloo)

Title: Representation theory and a little bit of quantum field theory

Abstract: One of the central foci of representation theory in the 20th century was the representation theory of Lie algebras, starting with finite dimensional algebras and expanding to a rich, but still mysterious infinite dimensional theory. In this century, we realized that this was only one special case of a bigger theory, with new sources of interesting non-commutative algebras whose representations we'd like to study such as Cherednik algebras. In mathematical terms, we could connect these to symplectic resolutions of singularities, but more intriguing explanation is that they arise 3-d quantum field theories. I'll try to provide an overview about what’s known about this topic, and what we're still confused about.

Ben Webster is an Associate Professor in the Department of Pure Mathematics at the University of Waterloo, and an Associate Faculty at the Perimeter Institute. He got his Ph.D. in Mathematics from the University of California, Berkeley in 2007. His research is on connections between representation theory, mathematical physics, geometry and topology, including knot homology, the geometry of symplectic singularities, and categorification.

Algebra & Geometry Seminar

Monday, January 31st, 2022

Time: 4:30 p.m.  Place: Online via Zoom (contact Kaveh Mousavand for Zoom link)

Speaker: Brett Nasserden (University of Waterloo)

Title: Arithmetic dynamics on projective bundles over elliptic curves.

Abstract: Let $X$ be a smooth projective variety defined over a number field $K$. Suppose that $X$ is endowed with a surjective endomorphism $f: X\rightarrow X$. A numerical measure of the complexity of the morphism $f$ is its dynamical degree, which can be defined as the spectral radius of the pullback morphism $f^*:N^1(X)\rightarrow N^1(X)$, where $N^1(X)$ is the Neron-Severi group of $X$. On the other hand, given a point $P$ in X defined over $K$, we have the following arithmetic measure of complexity of $f$ at $P:$ The arithmetic degree of $P$ with respect to $f$ is defined to be the limit, as $n\to\infty$, $h(f^n(P))^{1/n}$ where $h(x)$ is the height of a point $x$ in $X$. The Kawaguchi-Silverman conjecture predicts that if the forward orbit of $P$, $\lbrace P,f(P), f^2(P),\dotsc\rbrace$, is Zariski dense, then the arithmetic degree of $P$ with respect to $f$ equals the dynamical degree of $f$.\par In this talk, we will discuss how to prove the Kawaguchi-Silverman conjecture when $X$ is the projectivization of certain vector bundles on an elliptic curve $\mathcal{C}$. Specifically, Atiyah proved that for each integer $r>0$, there is a unique indecomposable rank $r$ degree zero vector bundle $F_r$ on $\mathcal{C}$ with a non-zero global section. We will discuss how one may prove the Kawaguchi-Silverman conjecture for the projectivizations of these bundles. Along the way, we will extend some results of Atiyah in the following way: Atiyah showed that the Iitaka dimension of the line bundle $\mathcal{O}(1)$ on $P(F_2)$ is zero. We prove that the Iitaka dimension of the line bundle $\mathcal{O}(1)$ on $P(F_r)$ is strictly positive whenever $r>2$ and relate this to the Kawaguchi-Silverman conjecture.

Website details here:

Math & Stats Department Colloquium

Friday, January 28th, 2022

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Elizabeth Ultee (Middlebury College)

Title: Mathematical avenues toward climate science

Abstract: Global climate change is one of the most pressing challenges facing humankind. From a wide variety of mathematical starting points, we have the opportunity to address important open questions in climate science. I will highlight some of those open questions and sketch emerging approaches. I will then give a detailed example from my own work: how to find a "speed limit" on glacier retreat and the resulting global mean sea-level rise. Whether climate intersects your research, teaching, or simply your human interests, I encourage you to join the conversation.

Elizabeth Ultee is an Assistant Professor of Geology at Middlebury College, Vermont. She held Postdoctoral positions at Georgia Institute of Technology in 2021, and at Massachusetts Institute of Technology from 2018-2021. She got her Ph.D.~in Climate and Space Science from the University of Michigan in 2018, and B.Sc.~in Mathematical Physics from ºÚÁϳԹÏ×ÊÔ´ in 2013. She is a glaciologist focused on describing the processes and societal impacts of glacier and ice sheet change. Her current projects include glacial water supply, ice fracture, and global sea level rise. She received the Early Career Scientist Medal from the International Glaciological Society in 2021.

Algebra & Geometry Seminar

Monday, January 24th, 2022

Time: 4:30 p.m.  Place: Online via Zoom (contact Kaveh Mousavand for Zoom link)

Speaker: Ana Garcia-Elsener (University of Glasgow)

Title: Grassmannian cluster categories

Abstract: The category of maximal Cohen-Macaulay modules over a certain quotient of a boundary algebra provides a categorification of Scott's cluster algebra structure of the Grassmannian $Gr(k,n)$, by work of Jensen, King and Su. This category is of infinite type in general, with finite types corresponding to the ADE Dynkin diagrams. We study this category in the infinite types. It is known to be tau-periodic, and we show that it is a tubular category. This makes it a very interesting family of categories of infinite types and allows us to characterize small rank modules.

Website details here:

Algebra & Geometry Seminar

Monday, January 17th, 2022

Time: 4:30 p.m.  Place: Online via Zoom (contact Kaveh Mousavand for Zoom link)

Speaker: Andrew P. Staal (University of Waterloo)

Title: Small Elementary Components of Hilbert Schemes of Points

Abstract: Hilbert schemes of points are moduli spaces of fundamental importance in algebraic geometry, commutative algebra, and algebraic combinatorics. Since their construction by Grothendieck, they have seen broad-ranging applications from the McKay correspondence to Haiman's proof of the Macdonald positivity conjecture.\par I will present some recent progress in the study of Hilbert schemes $\mathrm{Hilb}^d(\mathbb{A}^n)$ of $d$ points in affine space, and the related (local) punctual Hilbert schemes $\mathrm{Hilb}^d(\mathcal{O}_{\mathbb{A}^n,p})$ at fixed $p \in \mathbb{A}^n$. Specifically, I will discuss some results on \emph{elementary} components of Hilbert schemes of points and tie these to a question posed by Iarrobino in the 80's: does there exist an irreducible component of the punctual Hilbert scheme $\mathrm{Hilb}^d(\mathcal{O}_{\mathbb{A}^n,p})$ of dimension less than $(n-1)(d-1)$? I will answer this question by describing a new infinite family of irreducible components satisfying this bound, when $n=4$. A secondary family of elementary components also arises, providing further new examples of elementary components of Hilbert schemes of points, and improving our knowledge surrounding a folklore question on the existence of certain Gorenstein local Artinian rings.\par This is joint work with Matt Satriano (U Waterloo).

Website details here: