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Algebra & Geometry Seminar

Speaker: Monica Garcia (Laboratoire d'Algèbre, de Combinatoire et d’Informatique Mathématique (LACIM))

Title: Infinite super friezes

Abstract: Super friezes were introduced by S. Morier-Genoud, V. Ovsienko, S. Tabachnikov as a supersymmetric analog of classical Coxeter friezes. They show analogous properties of classical friezes: they are determined by the first non-trivial even and odd quiddity rows, they satisfy linear recurrence relations, and exhibit glide symmetry when of finite width. Moreover, as shown by G. Musiker, N. Ovenhouse and S. Zhang, all finite width super friezes arise from a decorated triangulation of a polygon, where even entries correspond to $\lambda$-lengths of arcs, and odd entries to $\mu$-invariants of triangles in the polygon. In this talk, I will report on joint work with A. Burcroff, Ä°. Çanakçı, F. Fedele and V. Klász on how to construct infinite super friezes from decorated skeletal triangulations of annuli.

Department Colloquium

Speaker: Prashant Mehta (University of Illinois, Urbana-Champaign)

Title: What can we learn from signals and systems in a transformer? Insights for probabilistic modeling and inference architecture

Abstract:
Transformer is the name of the core algorithm inside a large language model (LLM). In the so-called decoder-only transformer, a finite sequence of symbols (tokens) is mapped to the conditional probability of the next token.

In this talk, I situate the transformer within the broader history of the prediction theory: In the early 1940s, Wiener introduced a linear predictor, where the conditional expectation of future data is computed by linearly combining the past data. I argue that a decoder-only transformer generalizes this idea and that a transformer is best understood as a causal nonlinear predictor. The technical results for causal nonlinear prediction are described for the special case where the data is discrete-valued and generated from an underlying hidden Markov model (HMM).

The aim of this on-going research is to bridge the classical nonlinear filtering theory with modern inference architectures inspired by transformers. The work is jointly carried out with Heng-Sheng Chang and Jin Won Kim, and the talk is based on the paper:

Department Colloquium

Speaker: Tyler Meadows (Queen's University)

Title: Resource Competition in Sequencing Batch Reactors

Abstract:
Sequencing batch reactors are a type of bioreactor in which bacteria consume resources in order to produce things that we want (pharmaceuticals, biofuels, etc). The bacterial species involved in these reactors are often carefully engineered, and the introduction of competing bacterial strains can lead to unwanted byproducts. In this talk, I will review some of the basic principles behind modeling competition using dynamical systems, including the famous Lotka-Volterra competition model. We will look at how the knowledge of the LV model translates to a simple model of resource competition, the chemostat. Finally, we will look at how some of the fundamental results for the chemostat are altered in the discontinuous environment of sequencing batch reactors.

Curves Seminar

Speaker: Trevor Shillington

Affiliation: Queen's University

Title: Projective Embeddings of Flag Varieties

Abstract: Having wrapped up the representation component, we will see how the Schur modules from Chapter 8 help us study flag varieties. To do this, we will study how Grassmannians embed into projective spaces via the Plucker embedding. We build up to the flag variety by studying the quadratics needed to embed Grassmannians, incidence varieties, and then the flag varieties.

Department Colloquium

Speaker: Graham Denham (University of Western Ontario)

Title: Distances in trees and inequalities for matroids

Abstract:
The distance matrix of a tree appears in a range of contexts, from phylogenetics to physical chemistry. In the context of telephone switching networks, Graham and Pollak (1971) computed the signature of the distance matrix in order to bound length of addresses in Pierce's loop switching model. I will describe a recent refinement of Graham and Pollak's result, together with its use in establishing new inequalities for matroids, including one that Dowling conjectured in 1980. I will highlight the role of Lorentzian polynomials, which were introduced by Brändén and Huh in 2020 and are part of an ongoing close relationship between discrete convexity and polynomial positivity properties. This is joint work with Federico Ardila, Sergio Cristancho, Chris Eur, June Huh, and Botong Wang.

Department Colloquium

Speaker: Brian Ling (Queen's University)

Title: Shape-Constrained Estimation with Incomplete Data

Abstract:
Many problems in statistics involve incomplete or indirect observation of the variable of interest, such as interval-censored data and current duration data. In these settings, likelihood-based estimation is inherently a shape-constrained problem, since the parameter of interest is a distribution or survival function.

In this talk, I will survey several shape-constrained estimation problems and those arising from incomplete-data models. I will then focus on recent work on current duration data and interval-censored data under log-concavity, where additional structural constraints enable faster convergence in a fully nonparametric, tuning-parameter-free framework. I will discuss theoretical properties such as rates of convergence, as well as the associated computational algorithms.