In the 19th century, physicists were interested in determining the conditions under which the equations of motion for a classical mechanical system could be found by integrating a finite number of times. Such a system was said to be completely integrable. Using symplectic geometry, we can generalize the notion of an integrable system beyond the realm of physics and into Lie theory and representation theory. Such "abstract" integrable systems can be used to geometrically construct infinite dimensional representations of Lie algebras.

In this talk, I will discuss a family of integrable systems, the Gelfand-Zeitlin systems, that arise from purely linear algebraic data. For an \(n\times n\) complex matrix \(X\), we consider the eigenvalues of all the \(i\times i\) submatrices in the top left hand corner of \(X\). These are known as Ritz values and play an important role in numerical linear algebra. We will see how questions about Ritz values naturally lead to the construction of the Gelfand-Zeitlin integrable systems. I will explain results about the geometric properties of these systems and indicate how they answer questions of Parlett and Strang about Ritz values. I will also show how this research provides the foundation for the geometric construction of a category of infinite dimensional representations of certain classical Lie algberas using the theory of quantization.

Linkage is a classical topic in algebraic geometry and commutative algebra. Fix an affine space A. We say two subschemes X, Y of A are (directly) linked if their union is a complete intersection in A while X and Y having no common component. Two linked subschemes share several properties in common. Linkage has been studied by various people, Artin-Nagata, Peskine-Szprio, Huneke-Ulrich, to name a few. In 2014, Niu showed that if Y is a generic link of a variety X, then \(LCT (A, X) \le LCT (A, Y)\), where LCT stands for the log canonical threshold. In this talk, we show that if Y is a generic link of a determinantal variety X, then X and Y have the same log canonical threshold. This is joint work with Lance E. Miller and Wenbo Niu.

Let X be a smooth projective curve over an algebraically closed field and let G be a finite group acting on X. The space of holomorphic m-polydifferentials is defined to be the space of global sections of the m-fold tensor product of the sheaf of relative differentials with itself. This space provides a representation of G and a classical problem is to determine the decomposition of the space of holomorphic polydifferentials as a direct sum of indecomposable representations. We discuss work on this problem in the case when k has prime characteristic p and G has cyclic Sylow p-subgroups.

In 1980, David Eisenbud introduced a gadget called a matrix factorization as a tool for studying the homological properties of modules over hypersurface rings (i.e. rings of the form R/(f), where R is a regular local ring and f is a nonzero element of R). Decades later, it was discovered that matrix factorizations also play a key role in algebraic geometry, for instance in homological mirror symmetry and noncommutative Hodge theory. In this talk, I will give an overview of Eisenbud's classical theory of matrix factorizations, and I will give a glimpse of some modern applications of matrix factorizations beyond commutative algebra.

I'll explain the words in the title and their relation to representation theory of the general linear Lie algebra.

I’ll discuss some fundamental objects in commutative algebra, in particular, Gorenstein and Koszul algebras. One way to construct such objects is via idealization, but to guarantee that the idealization has the desired properties, we need certain condtions to hold. I’ll discuss how one can use the Lefschetz property to guarantee this.

Consider a projective nonsingular algebraic curve embedded in a projective space by a very ample line bundle. The (k + 1)-secant k-plans to the curve span the k-th secant variety of the curve. There has been a great deal of work in the last three decades to understand properties of such secant varieties, including their local properties, defining equations, and syzygies. In this talk, I will present a recent study on secant varieties of curves by showing the interaction between singularities and syzygies. The main idea in the talk is that if the degree of the embedding line bundle increases, then the properties of secant varieties become better. This is a joint work with L. Ein and J. Park.

A Moufang loop is a set under binary multiplication which possesses a unit element and two-sided inverses; the multiplication also obeys the ``nearly associative" law \((zx)(yz)=z((xy)z)\). In the spirit of Lie theory, Klim and Majid defined Hopf quasigroups and Hopf coquasigroups --extending Moufang loops to categories of vector spaces-- in order to capture algebraic aspects of \(S^7\), the unit octonions. We will show how to construct, out of a finite Moufang loop \(Q\) and a subgroup \(H\leq \text{Aut}(Q)\), a Hopf coquasigroup that is ``genuinely quantum" (neither commutative nor cocommutative) and ``genuinely quasi" (noncoassociative). Through this example, we'll illustrate some features of Hopf (co)quasigroup representation theory.

Morse theory provides an elegant and powerful method for studying the topology and geometry or manifolds by means of smooth functions. In this talk, we’ll start from the beginnings of Morse theory and show its utility through a variety of examples. We’ll survey how Morse theory has been used in the past, and how it can be used to understand persistent homology, a new area of data science. The first half is especially intended for graduate students, and we won’t assume much beyond multivariable calculus.

In this talk, we begin by presenting the crystal structure of finite-dimensional irreducible representations of the Lie algebra \(\mathfrak{sl}_n\) in terms of Gelfand-Zeitlin patterns. We then define a crystal structure using the set of symplectic Zhelobenko patterns, parametrizing bases for finite-dimensional irreducible representations of \(\mathfrak{sp}_4\). This is obtained by a bijectionwith Kashiwara-Nakashima tableaux and the symplectic jeu de taquin of Sheats and Lecouvey. We offer some conjectures on the generalization of this structure to rank \(n\).

The universal enveloping algebra of a Lie algebra \(\mathfrak{g}\) is of utmost importance when studying representations of \(\mathfrak{g}\). In 2010, V. Futorny and S. Ovsienko gave a realization of \(U(\mathfrak{gl}_n)\) as a subalgebra of the ring of invariants of a certain noncommutative ring with respect to the action of \(S_1\times S_2\times\cdots\times S_n\) where \(S_j\) is the symmetric group on \(j\) variables. With some connections to Galois Theory, an interesting question is what would a similar object be in the invariant ring with respect to a product of Alternating groups? We will discuss such an object and some results about its representations. In particular, we will focus on the case of \(n=2\).

Rational Cherednik algebras have a representation due to Dunkl-Opdam by differential operators making them look similar to a recently introduced class of principal flag orders due to Webster. We show that there is a common generalization using bialgebras. Via Morita equivalence this unifies the Gelfand-Tsetlin representation theory of type A enveloping algebras (and Yangians, finite W-algebras, Coulomb branches) with that of Cherednik algebras and conjecturally other Hecke algebras. We state some open problems regarding these bialgebra Galois orders.