In this talk we link discrete Markov spectrum to geometry of continued fractions. As a result of that we get a natural generalization of classical Markov tree which leads to an efficient computation of Markov minima for all elements in generalized Markov trees.
Convex real projective structures generalize hyperbolic structures in a rich way. We will discuss a class of manifolds introduced by Cooper Long and Tillmann, which include finite-volume cusped hyperbolic manifolds and other manifolds with well-controlled ends. These manifolds have nice deformation theoretic properties, and we will conclude with an existence theorem for novel structures on […]
Title: Geometry of continued fractions Speaker: Oleg Karpenkov, Department of Mathematical Sciences, University of Liverpool Abstract: In this talk we introduce a geometrical model of continued fractions and discuss its appearance in rather different research areas: -- values of quadratic forms (Perron Identity for Markov spectrum) -- the 2nd Kepler law on planetary motion -- Global relation […]
I will explain how to apply presentations of algebras (together with some classical results from non-commutative algebra) to obtain some 5 polynomial invariants telling us when two pairs of 2x2 matrices over a commutative ring are conjugate, assuming that each of these pairs generate the matrix algebra. This talk is based on my joint paper […]
Title: Linear independence, counting, and Hilbert's syzygy theorem Speaker: Youngsu Kim, Department of Mathematics, Cal State San Bernardino Abstract: Linear independence is an essential concept in mathematics and one of the most fundamental notions in linear algebra. Linear algebra studies the solutions of linear equations. Algebraic geometry studies the solutions of polynomial equations (of arbitrary degree). […]
Title: Data science and applications in dynamic topic modeling Abstract: The shockwaves of the big data boom have thrown into sharp relief the critical need for domain-driven, large-scale data analytic techniques across the fields of, among others, finance, political science, economics, psychology, and medicine. It is not simply the size of data sets that contributes […]
As $\lambda$ runs through all integer partitions, the set of Schur functions $\{s_{\lambda}\}_\lambda$ forms a basis in the ring of symmetric functions. Hence the rule $$s_{\lambda}s_{\mu}=\sum c_{\lambda,\mu}^{\gamma} s_{\gamma}$$ makes sense and the coefficients $c_{\lambda,\mu}^{\gamma}$ are called \textit{Littlewood-Richardson (LR) coefficients}. The calculations of Littlewood-Richardson coefficients has been an important problem from the first time they were […]
Title: Contact topology and geometry in high dimensions Speaker: Bahar Acu, Department of Mathematics, Pitzer College Abstract: A very useful strategy in studying topological manifolds is to factor them into ``smaller" pieces. An open book decomposition of an n-manifold (the open book) is a special map (fibration) that helps us study our manifold in terms of its (n-1)-dimensional […]
Title: What is the best shape? Geometric problems arising in aggregation models Abstract: How do pair interactions shape the large-scale behaviour of a cloud of particles (animals, social agents ...) ? In the most basic models, the shape of the cloud is determined by minimizing an attractive-repulsive interaction energy under suitable geometric constraints. When can […]
Suppose you are given a data set that can be viewed as a nonnegative integer-valued function defined on a finite set. A natural question to ask is whether the data can be viewed as a sample from the uniform distribution on the set, in which case you might want to apply some sort of test […]
The non-orientable 4-genus of a knot K is the smallest first Betti number of any non-orientable surface in the 4-ball spanning the knot. It is defined to be zero if the knot is slice. In joint work with Patrick Shanahan and Cornelia Van Cott, we attempt to determine the value of this invariant for double […]
An elliptic curve $ E: y^2 + a_1 \, x \, y + a_3 \, y = x^3 + a_2 \, x^2 + a_1 \, x + a_6 $ is a cubic equation which has two curious properties: (1) the curve is nonsingular, so that we can draw tangent lines to every point $ P […]
