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Algebra & Geometry Seminar

Monday, November 22nd, 2021

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

Speaker: Emily Gunawan (University of Oklahoma)

Title: Cambrian combinatorics on quiver representations

Abstract: First, we will discuss a polygon model of the Auslander--Reiten quiver of a type A quiver. Next, we will introduce a Catalan object which we call a maximal almost rigid representation. Finally, we will define a partial order on the set of maximal almost rigid representations and show that this partial order is a Tamari or Cambrian lattice. This talk is based on joint work with Emily Barnard, Emily Meehan, and Ralf Schiffler.

Website details here:

Statistics Seminar

Wednesday, November 17th, 2021

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

Speaker: Kin Wai Keith Chan (CUHK)

Title: A general and optimal difference-based method for variance estimation in time series

Abstract: Difference-based statistics are asymptotically invariant to arbitrary mean structures. Thus, they are natural building blocks for constructing variance estimators. In this talk, we present a general framework for constructing variance estimators based on observations that are masked by serial dependence structures and time-varying mean structures. The proposed class of estimators is general enough to cover many existing estimators. Necessary and sufficient conditions for consistency are investigated. The first asymptotically optimal estimator is derived. Our proposed estimator is theoretically proven to be invariant to arbitrary mean structures, which may include trends and a possibly divergent number of discontinuities.

Short bio: Kin Wai Chan is an Assistant Professor in the Department of Statistics at the Chinese University of Hong Kong. He completed his B.Sc. and M.Phil. in Risk Management Science in 2013 and 2015, respectively. After that, he did graduate work in Statistics from Harvard University under the supervision of Xiao-Li Meng and received his Ph.D. in 2018. His research interest is statistical inference for dependent data and incomplete data. Many of his research articles are about long-run variance estimation, change point analysis, and multiple imputation. He is particularly keen on developing elegant statistical theories and creating new methodologies that strike a nice balance between statistical and computational properties.

2021 Lorne Campbell Lecture Series

Thursday, November 24th, 2021

Time: 3:30 p.m.  Place: Online via Zoom (contact Abdol-Reza Mansouri for Zoom link)

Speaker: Peter Sarnak (Institute for Advanced Study & Princeton)

Title: Applications of Points on Subvarieties of Tori

Abstract: A basic conjecture of Lang asserts that the intersection of a finitely generated subgroup of a torus with an algebraic subvariety has a simple description. We review the statement and some of the tools in its proof. It has many applications and we highlight some recent ones in Euclidean geometry, group theory, the spectral theory of metric graphs and quasicrystals.

Lecture Video Recording:

Math & Stats Department Colloquium

Friday, November 19th, 2021

Time: 2:30 p.m.  Place: Jeffery Hall 234 & Online (via Zoom)

Speaker: Brad Rodgers (Queen's University)

Title: How random are arithmetic sequences?

Abstract: In this talk I will outline some of the ways that number theoretic sequences -- despite being deterministic -- can be fruitfully studied using the language of probability. Two fundamental number theoretic sequences I hope to discuss are the sequence of primes and the sequence of squarefree numbers (which are numbers whose prime factorization contains no repeated factors). I will focus especially on the statistical distribution of these sequences inside relatively small intervals. In the former case there is a conjectural link to random matrix theory, while in the latter case I will discuss forthcoming work with O. Gorodetsky and A. Mangerel which establishes a link to fractional Brownian motion.

Brad Rodgers is an Assistant Professor in the Department of Mathematics and Statistics at Queen's University. He obtained his Ph.D. in Mathematics from the University of California, Los Angeles in 2013. He held a postdoctoral position at the Institut für Mathematik at University of Zurich from 2013 to 2015, and was a Postdoc Assistant Professor at the University of Michigan from 2015 to 2018. His research interests are in analysis, number theory and probability. He is especially interested in problems at the interface of random matrix theory and analytic number theory.

Math & Stats Department Colloquium

Friday, November 12th, 2021

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

Speaker: Jenya Sapir (Binghamton University)

Title: A projection from geodesic currents to Teichmuller space

Abstract: Given a genus $g$ surface $S$, we consider the space of projective geodesic currents $\mathbb{P}\mathcal{C}(S)$. This space contains many objects of interest in low dimensional topology, such as the set of all closed curves on S up to homotopy, the set of all marked, negatively curved metrics on $S$, as well as some higher Teichmuller spaces. We show that there is a mapping class group invariant, length minimizing projection from the space of filling projective currents onto Teichmuller space, and that this projection is continuous and proper. This is joint work with Sebastian Hensel.

Jenya Sapir is an Assistant Professor at Binghamton University. She got her Ph.D. in Mathematics from Stanford University in 2014, under Maryam Mirzakhani. She was a postdoc at the Max Planck Institute for Mathematics at Bonn, and at the University of Illinois at Urbana-Champaign. Her research areas are low dimensional topology and geometric group theory.

Algebra & Geometry Seminar

Monday, November 15th, 2021

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

Speaker: Lauren Cranton Heller (University of California, Berkley)

Title: Characterizing multigraded regularity on products of projective spaces

Abstract: Eisenbud and Goto described the Castelnuovo-Mumford regularity of a sheaf on projective space in terms of three different properties of the corresponding graded module: its betti numbers, its local cohomology, and its truncations. For the multigraded generalization of regularity defined by Maclagan and Smith, these three conditions are no longer equivalent. I will characterize each of them for sheaves on products of projective spaces.

Website details here:

Algebra & Geometry Seminar

Monday, November 8th, 2021

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

Speaker: Jacob Matherne (University of Bonn & Max Planck Institute)

Title: Singular Hodge theory for combinatorial geometries

Abstract: If you take a collection of planes in R^3, then the number of lines you get by intersecting the planes is at least the number of planes. This is an example of a more general statement, called the "Top-Heavy Conjecture", that Dowling and Wilson conjectured in 1974. On the other hand, given a hyperplane arrangement, I will explain how to uniquely associate to it a certain polynomial, called its Kazhdan–Lusztig (KL) polynomial.  I will spend some portion of the talk comparing and contrasting these KL polynomials with the classical ones in Lie theory.

The problems of proving the "Top-Heavy Conjecture" and the non-negativity of the coefficients of these KL polynomials are related, and they are controlled by the Hodge theory of a certain singular projective variety. The "Top-Heavy Conjecture" was proven for hyperplane arrangements by Huh and Wang in 2017, and the non-negativity was proven by Elias, Proudfoot, and Wakefield in 2016. I will discuss joint work with Tom Braden, June Huh, Nicholas Proudfoot, and Botong Wang which resolves these two problems for arbitrary matroids.

Website details here:

Math & Stats Department Colloquium

Thursday, November 4th, 2021

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

Speaker: Masoud Khalkhali (University of Western Ontario)

Title: Bootstrapping Dirac ensembles

Abstract: In this talk, I shall explain certain techniques we have employed so far in the study of spectral statistics of certain classes of random matrices suggested by noncommutative geometry. In some cases, one can apply the Coulomb gas method to find the empirical spectral distribution and rigorously prove existence of phase transition. More recently, we applied the newly developed bootstrap technique to find the moments of certain multi-trace and multi-matrix random matrix models. Using bootstrapping, we are able to find the relationships between the coupling constants of these models and their second moments. Using the Schwinger-Dyson equations, all other moments can be expressed in terms of the coupling constant and the second moment. Explicit relations for higher moments are obtained. The talk will be mostly an overview of techniques we have used so far. (Based on joint works with H. Hessam and N. Pagliaroli).

Masoud Khalkhali is a Professor in the Department of Mathematics at the University of Western Ontario. His research interests include noncommutative geometry, cyclic cohomology, operator algebras, quantum groups, and Hopf algebras.

Algebra & Geometry Seminar

Monday, November 1st, 2021

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

Speaker: Fatemeh Mohammadi (Ghent University)

Title: Toric degenerations of Grassmannians and combinatorial mutations of their associated polytopes

Abstract: Many toric degenerations of the Grassmannians Gr(2, n) are described by trees, or equivalently subdivisions of polygons. These degenerations can be also seen to arise from the cones of the tropicalization of the Grassmannian. In this talk, I focus on particular combinatorial types of cones in tropical Grassmannians indexed by matching fields, whose corresponding degenerations are toric. Moreover, I will show how their associated polytopes are connected by combinatorial mutations. I will present several combinatorial conjectures and computational challenges around this problem.

Website details here:

Math & Stats Department Colloquium

Friday, October 29th, 2021

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

Speaker: Kaveh Mousavand (Queen's University)

Title: An elementary approach to an open conjecture in $\tau$-tilting theory

Abstract: In 2014, Adachi-Iyama-Jasso introduced the concept of $\tau$-tilting theory in the setting of finite dimensional algebras. This modern approach to the notion of mutation of tilting modules soon received a lot of attention and established fruitful connections to other areas, including homological algebras, cluster algebras, lattice theory, etc. Motivated by the algebro-geometric aspects of representation theory, in 2019 I proposed a conjectural equivalence for $\tau$-tilting finiteness of algebras in terms of irreducible components of module varieties. The conjecture has proved to be novel and can result in new linkages between the aforementioned domains. In this talk, I adopt an elementary approach to the study of this open conjecture and present it as a problem in linear algebra. Namely, I fully avoid the technicality of the subject and only use linear algebra and assume familiarity with rudiments of module theory. I begin with some background to motivate the conjecture, then outline our general methodology in the treatment of the problem, and end with some new results from my recent joint work with Charles Paquette, where we verify the conjecture for some important cases.

Kaveh Mousavand is a Coleman postdoctoral fellow in the Department of Mathematics and Statistics at Queen's University. He obtained his Ph.D.~from Universite du Quebec Montreal in 2020. His research interests include representation theory of algebras, $\tau$-tilting theory, cluster algebras, and algebraic combinatorics.