Abstracts

Week of February 8, 2026

February 9, 2026
Geometric Analysis and Application Widths, Index, Intersection, and Isospectrality 3:45pm -
KT 906

In this talk, I will discuss a series of works on Gromov’s p-widths, $\{\omega_p\}$, on surfaces. For ambient dimensions larger than $2$, $\omega_p$ morally realizes the area of an embedded minimal surface of index p. This characterization was historically used to prove the existence of infinitely minimal hypersurfaces in closed Riemannian manifolds. In ambient dimension $2$, $\omega_p$ realizes the length of a union of (potentially immersed) geodesics, and heuristically, $p$ is equal to the sum of the indices of the geodesics plus the number of points of self-intersection. Joint with Lorenzo Sarnataro and Douglas Stryker, we prove upper bounds on the index and vertices, making progress towards this heuristic. Along the way, we prove a generic regularity statement for immersed geodesics. If time allows, we will also discuss the isospectral problem for the p-widths and how surfaces provide a convenient setting to investigate this. 
Geometry, Symmetry and Physics Stabilization of Kac polynomials along root strings 4:30pm -
KT801

In this talk, I will present the stabilization phenomenon of cohomology groups and Kac polynomials associated with moduli spaces of quiver representations. Specifically, for Q and chosen dimension vectors d and e satisfying reasonable conditions, the cohomologies of various types of quiver varieties associated with dimension vectors d+ne stabilize as e tends to infinity. I will provide explicit generating functions for these stabilized dimensions and explain the implications for root multiplicities of Kac-Moody Lie algebras.
February 10, 2026
Geometric Analysis Learning Seminar Geometric Analysis Learning Seminar 10:30am -
KT 801

TBA
Group Actions, Geometry and Dynamics The pentagram zoo 4:30pm -
KT 203

Schwartz’s pentagram map is a dynamical system defined on moduli spaces of polygons by intersecting diagonals. It is an integrable system, meaning that in appropriate coordinates, the map becomes a family of translations on complex tori. Some natural generalizations of the pentagram map produce integrable systems, but numerical experiments by Khesin-Soloviev suggest that others do not. In this talk, we use tools from dynamical systems to prove that the “skew” pentagram map is non-integrable.
February 12, 2026
Analysis Support of semiclassical measures in higher dimensions 4:00pm -
KT 201

A central question in quantum chaos is how classical chaotic dynamics influence quantum behavior. On compact Riemannian manifolds, pure quantum states correspond to Laplacian eigenfunctions. The quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak predicts that on hyperbolic manifolds, all high-energy eigenfunctions become uniformly distributed. The asymptotic behavior of eigenfunctions can be formulated in terms of semiclassical measures, which describe the microlocal distribution of eigenfunction mass. One approach towards the QUE conjecture applies microlocal analysis and uncertainty principles to characterize the support of semiclassical measures. I will discuss recent work that uses the breakthrough higher-dimensional fractal uncertainty principle of Cohen. Using this uncertainty principle, we prove the first result on the support of semiclassical measures on real hyperbolic n-manifolds. To explain some of the main proof ideas, we will discuss work on the toy model of quantum cat maps. This is joint work with Nicholas Miller.
Quantum Topology and Field Theory Relationships between skein algebras 4:30pm -
KT 801

We will examine the multiplicative structure of two skein algebras—the usual Kauffman bracket skein algebra of a surface (generated by loops) and a generalization of it due to Roger-Yang (generated by loops and arcs).  In joint work with Chloe Marple, we found an unexpected homomorphism between the usual skein algebra for a closed torus and the Roger-Yang skein algebra for a twice-punctured annulus.  In this talk, I’ll discuss how we  used the homomorphism to help compute representations and structural constants of the Roger-Yang skein algebra for a twice-punctured annulus, and  whether there might be similar relationships between skein algebras for other surfaces. 
February 13, 2026
Learning seminar on Matroids and Algebraic Cycles Learning seminar on Matroids and Algebraic Cycles 2:15pm -
KT 801

TBA
Learning seminar on Groups, Geometry and Dynamics Coding of hyperbolic diffeomorphisms. 4:00pm -
KT 217

I’ll discuss how to leverage symbolic dynamics to understand Anosov diffeomorphisms of surfaces, starting with the most basic case of linear total automorphisms. The key idea is the construction of so-called Markov partitions that code the dynamics of the diffeomorphism. 

More recently, the same ideas have been extended to prove new results about general diffeomorphisms of manifolds.