We present a theorem of Li and Yorke, that if a continuous function from the reals to the reals has an orbit with period 3, then it has orbits of all sizes.

The Markov numbers are the positive integers that appear in the solutions of the equation x^2 + y^2+ z^2 =3xyz. These numbers are a classical subject in number theory and have important ramifications in hyperbolic geometry, algebraic geometry and combinatorics. It is known that the Markov numbers can be labeled by the lattice points in the first quadrant and below the diagonal whose coordinates are coprime. In this paper, we consider the following question. Given two lattice points, can we say which of the associated Markov numbers is larger? A complete answer to this question would solve the uniqueness conjecture formulated by Frobenius in 1913. Using tools from cluster algebras, we give a partial answer in terms of the slope of the line segment that connects the two lattice points. As a special case, namely when the slope is equal to 0 or 1, we obtain a proof of two conjectures from Aigner's book "Markov's theorem and 100 years of the uniqueness conjecture". This is joint work with Li Li, Michelle Rabideau, and Ralf Schiffler.

Abstract: Least Squares is the standard method for approximating the solution to overdetermined systems of linear equations. It is known to converge quickly to the optimal solution. A solved linear system can be seen as a diagonalized system. For many applications data are multilinear, and we want to use that structure. A multilinear analogue to a diagonalized system is a rank decomposition. Alternating Least Squares is one standard method for decomposing tensors into a rank decomposition by attempting to reduce this problem into a sequence of least squares optimizations. While this method can be effective in some situations, it is limited because it doesn’t always converge, and it can be computationally expensive. However, when tensor data arrive sample by sample, we can use stochastic methods to attempt to decompose a model tensor from its samples. We show that under mild regularity and boundedness assumptions, the Stochastic Alternating Least Squares (SALS) method converges. Even though tensor problems often have high complexity, the tradeoff for using sampling in place of exactness can lead to large savings in time and resources. I’ll describe the SALS algorithm and its advantages, and I’ll give some hints as to why (and when) it converges.

Abstract: The Witten-Reshetikhin-Turaev (WRT) invariants for SU(2) are defined from the Jones polynomial and assign complex numbers to roots of unity. It is a long-standing problem to relate these numbers to suitable functions like power series defined on the unit disk. These extensions are supposed to lead the way to a categorification of quantum 3-manifold invariants and possibly also a proof for the volume conjecture in knot theory. I will formulate a corresponding recent conjecture by Gukov-Manolsecu, which refines a previous conjecture by Gukov-Putrov-Vafa. The work of Gukov and Manolescu sheds new light on quantum link invariants like the colored Jones polynomial, and it also discusses the various examples for which the conjecture is presently known.

Often in Algebraic Geometry linear systems we have an expectation on how many conditions vanishing on certain subvarieties should impose. For instance, for a nonempty linear system we expect that vanishing at a random point imposes a single condition.  In this talk we discuss a version of this problem. In particular we introduce the concept of a "very unexpected hypersurface" passing through a fixed set of points Z. These occur when Z imposes less than the "expected" number of conditions on certain ideal sheaves coming from generic linear subspace. A key feature is the use of projective duality and certain combinatorial tools from the theory of matroids. We close by discussing relationships between this problem and certain motivating problems in matroid theory and hyperplane arrangements.

Thesis defense.

Abstract:  Knot theory is the study of closed loops embedded in three space.  A knotoid is a knotted arc with fixed endpoints.  A knotoid can be closed to obtain an arc in a variety of ways.  This gives a possible relationship between knots.  That is, two knots are related if there is a knotoid that can be closed in different ways to obtain the two knots.  In this talk I will introduce both knots and knotoids, and share our surprising result on knots related by knotoids.

Cyclically presented groups are a class of finitely presented groups with a balanced presentation that is equipped with cyclic symmetry. An example of such groups is the generalized Fibonacci group H(r,n,s). A conjecture of Williams states that "if H(r,n,s) is perfect then either r or s is congruent to zero modulo n". In my talk, I will describe a proof of this conjecture, give a topological application, and mention a more general problem.

The Hilbert scheme of n points on an algebraic surface parametrizes finite, length n subschemes of the surface. When the surface is smooth and projective, its Hilbert schemes of points are smooth, projective varieties. In favorable situations, one can construct 'tautological' projective embeddings of these Hilbert schemes out of embeddings of the original surface. In this talk, I will explain what is known about the degrees of these tautological embeddings, including structural results on their generating functions, and some special cases in which explicit results are known.

Classifying the irreducible characters of the groups of unipotent upper-triangular matrices over a finite field is known to be a "wild" problem. By instead studying related objects known as "supercharacters" one obtains a more manageable structure with nice combinatorial properties. I will talk about the supercharacters of unipotent matrix groups in symplectic, orthogonal, and unitary types. No knowledge of characters or representation theory will be assumed.

We consider skew group algebras that are built from a group action on a polynomial ring.  Their deformations in polynomial degree zero include numerous algebras with interesting representation theory and important connections to noncommutative geometry and physics, especially when the group involved is a reflection group.  We introduce parametrized families of polynomial degree-one deformations of skew group algebras for the symmetric group.  These generalize rational Cherednik algebras and provide numerous avenues for further exploration.  Along the way we describe some related algebraic varieties that may be of independent geometric or combinatorial interest.  This is joint work with Briana Foster-Greenwood.

Abstract: Branched covers provide a natural setting to better understand contact manifolds and the transverse knots they contain. In this talk, we explore how branched covers can generally be used to derive effective and computable invariants of transverse knots. We will present a specific example of one such construction using grid diagrams. This is joint work with C.-M. Mike Wong.