Week of September 21, 2025

September 22, 2025
Geometric Analysis and Application Quantitative unique continuation on asymptotically conic manifolds Ruoyu Wang - Yale University 3:45pm -
KT 906
Geometry, Symmetry and Physics The microlocal theory of constructible sheaves Tong Zhou - MIT 4:30pm -
KT 801
September 23, 2025
Geometry & Topology Distinguishing filling curves and designer metrics Sayantika Mondal - City University of New York 4:30pm -
KT 203
Algebraic Geometry Quantizations and their representations, lecture 6 Ivan Loseu - Yale University 10:25pm -
KT 801
September 25, 2025
Algebraic Geometry Quantizations and their representations, lecture 7 Ivan Loseu - Yale University 10:25am -
KT 801
Analysis Available Seminar Slot 4:00pm -
Walter Feit Memorial Lecture Periodic pencils of flat connections and their p-curvature Pavel Etingof - MIT 4:15pm -
KT 205
September 26, 2025
Friday Morning Seminar Bruhat interval polytopes and posets arising as 1-skeleta of (directionally) simple polytopes Patricia Hersch - University of Oregon 10:00am -
KT 801

Abstracts

Week of September 21, 2025

September 22, 2025
Geometric Analysis and Application Quantitative unique continuation on asymptotically conic manifolds 3:45pm -
KT 906

Abstract: Quantitative unique continuation states that eigenfunctions of the Laplacian on smooth closed manifolds cannot vanish faster than exponentially in its eigenvalue in any open set. It is interpreted physically that the probability of a quantum particle to appear in the classically forbidden region (total energy < potential) is at least exponentially small, also known as quantum tunnelling. In this informal talk, we will discuss the strategy to prove it on both closed and open manifolds with specified end structures (like cones or cylinders) and discuss open problems.
Geometry, Symmetry and Physics The microlocal theory of constructible sheaves 4:30pm -
KT 801

The microlocal point of view was introduced by M. Sato in the 1960s for studying partial differential equations. It was then adopted by M. Kashiwara and P. Schapira and developed into a systematic theory in the context of sheaves on manifolds. The theory has since had applications in many fields, including partial differential equations, symplectic geometry, geometric Langlands, and exponential sums. In this talk, I will explain the basic ingredients of this theory, and discuss recent development of its analogues in the contexts of étale sheaves on algebraic varieties and rigid analytic varieties.
September 23, 2025
Geometry & Topology Distinguishing filling curves and designer metrics 4:30pm -
KT 203

There are many topological invariants one can associate with homotopy classes of closed curves. These include algebraic and geometric self-intersection number, intersection with curves in a class of curves (for example, simple ones), complementary component types of a curve, mapping class group stabilizers of a curve, and many others. How these invariants interact and determine the curve type (mapping class group orbit) is an active area of research today. In this talk, we focus on the so called inf invariant (shortest length metric) associated to a filling curve, its relationship with the geometric self-intersection number, and its relation to the optimal metric that is tailored to produce the minimum length. While clearly the geometric self-intersection number is a type invariant, we address whether the inf invariant can distinguish between curves that have the same self-intersection. This is joint work with Ara Basmajian.
Algebraic Geometry Quantizations and their representations, lecture 6 10:25pm -
KT 801

I will talk about modules over quantizations, their good filtrations and associated varieties. We’ll also discuss holonomic modules and distinguished good filtrations.
September 25, 2025
Algebraic Geometry Quantizations and their representations, lecture 7 10:25am -
KT 801

This is the last lecture in the series. I will discuss the restriction functor for Harish-Chandra modules.
Analysis Available Seminar Slot 4:00pm -
Walter Feit Memorial Lecture Periodic pencils of flat connections and their p-curvature 4:15pm -
KT 205

Abstract: A periodic pencil of flat connections on a smooth algebraic variety X is a linear family of flat connections ∇(s1, …, sn) = d − Xr i=1 Xn j=1 sjBijdxi , where {xi} are local coordinates on X and Bij : X → MatN are matrixvalued regular functions. A pencil is periodic if it is generically invariant under the shifts sj 7→ sj + 1 up to isomorphism. I will explain that periodic pencils have many remarkable properties, and there are many interesting examples of them, e.g. Knizhnik-Zamolodchikov, Dunkl, Casimir connections and equivariant quantum connections for conical symplectic resolutions with finitely many torus fixed points. I will also explain that in characteristic p, the p-curvature operators {Ci , 1 ≤ i ≤ r} of a periodic pencil ∇ are isospectral to the commuting endomorphisms C ∗ i := Pn j=1(sj − s p j )B (1) ij , where B (1) ij is the Frobenius twist of Bij . This allows us to compute the eigenvalues of the p-curvature for the above examples, and also to show that a periodic pencil of connections always has regular singularites. This is joint work with Alexander Varchenko.
September 26, 2025
Friday Morning Seminar Bruhat interval polytopes and posets arising as 1-skeleta of (directionally) simple polytopes 10:00am -
KT 801

Abstract:  Questions regarding the complexity of the simplex method in linear programming  turn out to be related in special cases to questions about partially ordered sets.  Exploring this connection led us to study lattices arising as 1-skeleta of simple polytopes, obtaining the homotopy type of the intervals in these posets as well as a geometric way of constructing lattice-theoretic joins.  In recent joint work with Christian Gaetz, we generalized this to the setting of directionally simple polytopes, motivated by questions about Bruhat interval polytopes.  This allowed us to  generalize a result of Athanasiadis, Edelman and Reiner that the reduced expressions for any fixed permutation are highly connected under braid moves and commutation moves, obtaining an analogous statement for the BCFW bridge decompositions of reduced plabic graphs (a family of graphs arising in the study of the totally nonnegative real part of the Grassmannian); we also determined the homotopy type for each interval in posets arising as 1-skeleta of Bruhat interval polytopes.   This talk will not assume background in this area and will mention several open questions.