Upcoming Events

We will discuss a class of ideals in a polynomial ring studied by Mark Haiman in his work on the Hilbert scheme of points, and ask some purely algebraic questions about them. It turns out that these questions are very closely tied to homology of affine Springer fibers, Khovanov-Rozansky homology of links, and to the ORS conjecture. We will discuss which cases are known and unknown, and compute some simple examples.

Ramsey's theorem roughly states that "complete disorder is impossible": any sufficiently large system, no matter how complex and unstructured, must contain large, structured pieces. Since its proof almost 100 years ago, Ramsey's theorem and related statements have become fundamental results in combinatorics, functional analysis, geometry, number theory, theoretical computer science, and many other fields of mathematics.

A natural question arising from Ramsey's theorem is "how large is sufficiently large?", which leads to the notion of Ramsey numbers. The problem of finding good upper and lower bounds on Ramsey numbers has proved to be surprisingly difficult, but has seen some remarkable breakthroughs in recent years. In this talk, I will discuss the background and motivation for this problem, and highlight some of the new ideas that have been introduced in its study.

The combinatorics of (q,t)-Catalan polynomials and related objects is known to have deep connections to geometry, representation theory, and knot theory. In particular, the (q,t)-Catalan polynomials arise as the hook components in the celebrated Shuffle theorem of Carlsson-Mellit which combinatorially computes the Frobenius character of the space of diagonal harmonics. At the same time, the (q,t)-Catalan polynomials arise as the lowest degree part of the Khovanov-Rozanksy homology of certain torus knots. Khovanov-Rozanksy homology is a distinguished knot-homology theory that is notoriously difficult to compute. In this talk I will describe how both the combinatorial and topological components in this story are related and introduce two new families of polynomials that generalize the (q,t)-Catalan polynomials in different directions. Along the way, these polynomials will not only compute the KR-homology for larger previously unknown families of knots and prove some open conjectures, they will also pose new unexplored questions by changing the representation theory that underlies them. 

In the 19th century, German Mathematician Hermann Schubert was interested in questions in enumerative geometry, like "given 4 generic lines in 3-space, how many lines intersect all four lines?" The modern treatment of such questions centers around the study of the cohomology ring of the complete flag variety, with a basis given by the classes of Schubert varieties. While this topic and its variations have been extensively studied, a central question, which seeks a combinatorial interpretation of the structure constants, remains open. However, the same question has been fully solved using many different methods in the special case when the classes are pullbacks from the Schubert classes in Grassmannians, in which case the Schubert structure constants are known as the Littlewood-Richardson coefficients. The "lifting dream" of Schubert calculus hopes to find ways to lift the Littlewood-Richardson rules to the general case. I will talk about the mathematical pipeline that turns the geometric questions into combinatorics, potential venues to lift the combinatorial and geometric techniques for the Grassmannian case using bumpless pipe dreams and positroid varieties, and some success stories so far.

Our lab is focused on developing tumor forecasting methods by integrating advanced imaging technologies with mathematical models to predict tumor growth and treatment response.  In this presentation, we will focus on how quantitative magnetic resonance imaging (MRI) data can be employed to calibrate mathematical models built on first-order effects related to well-established “hallmarks” of cancer including proliferation, migration/invasion, vascular status, and drug-related tumor growth inhibition and cell death.  In particular, we will present some of our recent results on using these models to build personalized digital twins that provide a rigorous, but practical, methodology for optimizing therapeutic interventions on a patient-specific basis.