Printer-friendly versionAbstracts
Week of April 19, 2026
| Geometry, Symmetry and Physics | Thesis defense: "On certain Lagrangian subvarieties in minimal resolutions of Kleinian singularities" |
4:30pm -
KT 801
|
Abstract: Kleinian singularities are quotients of C^2 by finite subgroups of SL_2(C). They are in bijection with the ADE Dynkin diagrams via the McKay correspondence. In this talk, I will introduce certain singular Lagrangian subvarieties in the minimal resolutions of Kleinian singularities, motivated by the geometric classification of unipotent Harish-Chandra (g,K)-modules. The irreducible components of these singular Lagrangian subvarieties are P^1’s and A^1’s. I will describe how they intersect with each other through the realization of Kleinian singularities as Nakajima quiver varieties. I will also discuss their connections with nilpotent K-orbits and symmetric pairs in semisimple Lie algebras. The defense will be preceeded by special tea at 3.30 in the lounge. |
| Geometric Analysis Learning Seminar | Geometric Analysis Learning Seminar |
10:30am -
KT 801
|
TBA |
| Geometry, Symmetry and Physics | Positivity of Canonical Bases |
4:30pm -
KT 801
|
Lusztig’s theory of canonical bases reveals a remarkably rigid and positive algebraic structure on quantum groups and their modules. In symmetric types, it is known that the structure constants for multiplication in the negative part U −, as well as for the action of Chevalley generators Ei and Fi on a single simple module, all belong to N[v, v−1 ]. Lusztig conjectured that this strong positivity holds for the multiplication within the modified quantum group and the action on the tensor product of modules. In this talk, I will present recent joint work with Jiepeng Fang towards this conjecture. A key innovation in our approach is the ”thickening philosophy”, an algebraic technique inspired by geometric ideas from total positivity, building on my earlier work with Huanchen Bao. This method embeds a suitable approximation of the tensor product into the negative part U˜ − of a larger quantum group, constructed via a framed quiver. This allows us to inherit the desired positivity directly from the well-established positivity of the canonical basis of U˜ −. This approach demonstrates how the large Kac-Moody groups can provide a powerful framework for elucidating the structure and representations of quantum groups even for the finite and affine types. |
| Geometry & Topology | On a conjecture of Ayala–Francis–Rozenblyum |
4:30pm -
KT 207
|
I will present some finite stratified homotopy types constructed from smooth data which are not the homotopy types of conically smooth manifolds, disproving a 2015 conjecture of Ayala, Francis and Rozenblyum. The main tool is a new combinatorial and fully faithful exit path infinity-category construction. Time permitting, I will discuss a future direction for such conjectures. |
| Analysis | On the exact failure of the hot spots conjecture |
4:00pm -
KT 201
|
The hot spots conjecture asserts that as time goes to infinity, the hottest and coldest points in an insulated domain will migrate towards the boundary of the domain. In this talk, I will describe joint work with Jaume de Dios Pont and Alex Hsu where we find the exact failure of the hot spots conjecture in every dimension. |
| Quantum Topology and Field Theory | Triangulations and volume conjectures |
4:30pm -
KT 801
|
A volume conjecture relates a certain asymptotical growth of a given quantum topological invariant of a hyperbolic 3-manifold to the hyperbolic volume of this manifold. In this talk I will mention several of these volume conjectures, their common points and differences, notably those associated to the Baseilhac-Benedetti invariants and to the Andersen-Kashaev TQFT. A general strategy to prove a volume conjecture is to use the combinatorial properties of a given triangulation of the manifold to simplify the expression of the quantum invariant, and hopefully to successfully apply the saddle point method in the desired asymptotics. I will use the figure-eight knot complement as a recurring example, as it is the simplest member of two infinite families, the hyperbolic twist knots and the once-punctured torus bundles over the circle. No prerequisite in quantum topology or hyperbolic geometry will be needed. (This talk will cover joint works with François Guéritaud, Stéphane Baseilhac and Ka Ho Wong) |
| 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 | Learning seminar on Groups Geometry and Dynamics |
4:00pm -
KT 801
|
TBA |