MONDAY - 4/29/24

Combinatorics Seminar

Location: 182 George Street, Room 110

Time: 12:00-1:00pm

Speaker: Ethan Partida, Brown University

Title: The combinatorial topology of lagrangian complexes and posets of matroids

Abstract: In this talk, I will introduce the conormal complex of a matroid and discuss its combinatorial topology. The conormal complex is a simplicial complex which keeps track of the flats and dual flats of a matroid in an intricate manner. The conormal complex was implicitly introduced by recent work of Ardila, Denham and Huh. In their work, they use a simplicial fan associated to the conormal complex to prove the log concavity of the h-vector of the broken circuit complex of a matroid. To better understand the conormal complex, I will also introduce the poset of biflats. The poset of biflats will also keep track of the flats and dual flats of a matroid but in a less delicate manner than the conormal complex. I will show that, despite this reduction in complexity, the poset of biflats and the conormal complex are simple homotopy equivalent. I will also compare and contrast the combinatorial topology of the conormal complex and poset of biflats with that of the Bergman complex, a well understood simplicial complex related to the conormal complex. This is joint work with Anastasia Nathanson.

Analysis Seminar

Location: Kassar 105

Time: 3:00-4:00pm

Speaker: Xiumin Du, Northwestern

Title: Weighted Fourier Extension Estimates

Abstract: In this talk, we will survey recent results on weighted Fourier extension estimates and its variants. Such estimates ask for $L^2$ bounds of the Fourier extension operator on $\alpha$-dimensional sets, and they have applications to several problems in PDEs and geometric measure theory, including size of divergence set of Schrodinger solutions, spherical average Fourier decay rates of fractal measures, and Falconer’s distance set problem.

Number Theory & Algebra Seminar

Location: Kassar 105

Time: 4:00-5:00pm

Speaker: Dmitrii Pavlov, MPI MiS

Title: Santaló Geometry of Convex Polytopes

Abstract: The Santaló point of a convex polytope is the interior point which leads to a polar dual of minimal volume. This minimization problem is relevant in interior point methods for convex optimization, where the logarithm of the dual volume is known as the universal barrier function. When translating the facet hyperplanes, the Santaló point traces out a semi-algebraic set. In this talk, we will describe this geometry using algebraic and numerical techniques. We will also explore connections to algebraic statistics and positive geometries.

TUESDAY - 4/30/24

Geometric Analysis Seminar

Location: GEO155 106 (155 George Street, side entrance)

Time: 3:00-4:00pm

Speaker: Ruobing Zhang, Princeton

Title: Metric geometric aspects of Einstein manifolds

Abstract: This lecture concerns the metric Riemannian geometry of Einstein manifolds. We will exhibit the rich geometric/topological structures of Einstein manifolds and specifically focus on the structure theory of moduli spaces of Einstein metrics.  My recent works center around the intriguing problems regarding the compactification of the moduli space of Einstein metrics, which tells us how Einstein manifolds can degenerate. The metric geometry of Einstein manifolds is a central theme in modern differential geometry, and it is deeply connected to fundamental problems in algebraic geometry, analysis of nonlinear PDEs, and mathematical physics. We will introduce recent major progress and propose several open questions in the field.

WEDNESDAY - 5/1/24

Geometry & Topology Seminar

Location: Kassar 105

Time: 3:00-4:00pm

Speaker: Dan Cristofaro-Gardiner, Maryland

Title: The symplectic packing numbers

Abstract: The k-th "symplectic packing number" of a finite volume symplectic manifold is the proportion of the volume that can be filled by k disjoint symplectically embedded balls.  In the first part of my talk, I will explain what we know about these numbers, and give some examples of what we'd like to know.  In particular, I will explain the following remarkable stability property, which sharply contrasts the Riemannian case: for many symplectic manifolds, the packing number is 1 for sufficiently large k.  This "packing stability" phenomenon was discovered by Biran in the 90s.  In the second part, I will explain some recent work aimed at understanding how characteristic packing stability actually is.  In fact, it has been conjectured that it holds for all symplectic manifolds, but we will show that it does not.  A symplectic "fractal Weyl law" plays a major role in our proof.

Colloquium

Location: Foxboro Auditorium

Time: 4:30-5:30pm

Speaker: Dan Cristofaro-Gardiner, Maryland

Title: Periodic orbits of Hamiltonian systems and Hofer-Wysocki-Zehdner's two or infinity conjecture 

Abstract: The evolution of a system in classical mechanics is encoded in Hamilton's equations of motion.  The dynamics of Hamiltonian systems are therefore of considerable interest.  My talk will be about the problem of finding periodic orbits, when the Hamiltonian is independent of time.  In the first part of my talk, I will survey some of the main results about this topic and tell you about some of the main open questions.  In the second part, I will focus on a 2001 conjecture of Hofer, Wysocki and Zehnder, asserting that in the special case where the state space is R^4, there are either two or infinitely many simple periodic orbits on any compact star-shaped energy level.  We recently proved this conjecture and I want to explain a few ideas in the proof.  A key point is to construct a "global surface of section" for the flow: this is done by appealing to Gromov's theory of pseudoholomorphic curves, and so my talk will include a very gentle introduction to this concept.  My aim is to make all of this quite accessible.

FRIDAY - 5/3/24

Algebraic Geometry Seminar

Location: Kassar 105

Time: 3:00-4:00pm

Speaker: Samir Canning, ETH

Title: Tautological non-relations

Abstract: There is a conjecture due to Pixton describing all the relations in the tautological ring of the moduli space of curves. A central difficulty in the study is classes that the conjecture predicts to be nonzero, yet intersect trivially with all tautological classes of complementary codimension. I will explain some recent progress on studying these classes by intersecting with classes outside of the tautological ring.

PDE Seminar

Location: 170 Hope Street, Room 108

Time: 3:00-4:00pm

Speaker: Sanchit Chaturvedi, NYU

Title: Phase mixing in astrophysical plasmas with an external Kepler potentia

Abstract: In Newtonian gravity, a self-gravitating gas around a massive object such as a star or a planet is modeled via Vlasov Poisson equation with an external Kepler potential. The presence of this attractive potential allows for bounded trajectories along which the gas neither falls in towards the object or escape to infinity. We focus on this regime and prove first a linear phase mixing result in 3D outside symmetry with exact Kepler potential. Then we also prove a long-time nonlinear phase mixing result in spherical symmetry. The mechanism is phenomenologically similar to Landau damping on a torus but mathematically the situation is quite a lot more complex.