Boston Graduate Math Colloquium

please click here so we can order enough food!

Brian Choi - BU

Title: Would Schroedinger's Cat die under nonlinearity?

Abstract: In 1925, Schroedinger published his famous equation that eventually obtained recognition as the quantum analogue of the Newton's law F=ma. To motivate further discussion, I briefly explain some nice properties generated by the Schroedinger group; since this is a linear problem at heart, a bit of functional analysis is helpful for understanding. In the 1970s, with the rise of nonlinear optics, it was suggested that the mathematical model for the propagation of laser should be Schroedinger's equation with nonlinearity. I describe some classical results regarding existence and uniqueness. In particular, the existence issue on a periodic domain, via Fourier transform, allows us to borrow ideas from analytic number theory. I finish by describing some current projects of mine regarding wellposedness of quintic NLS and achieving either wellposedness or illposedness, via randomisation of initial data, at the regime of supercriticality.

Roderic Guigo - BU

Title: Topological recursion of spectral curves

Abstract: Topological Recursion is as a machinery that inputs a spectral curve and outputs certain numerical invariants that can be computed recursively. Although first developed in the context of random matrix theory it has proven to be related to different areas ranging from hyperbolic geometry to Gromov-Witten theory. I’ll give an overview and will comment on known applications and further questions that are still unanswered.

Rahul Singh - NEU

Title: The conormal variety of a Schubert subvariety of the Grassmannian

Abstract: Let $Gr(d,n)$ be the variety of $d$-dimensional subspaces of $\mathbb C^n$. Recall that a Schubert variety $X\subset Gr(d,n)$ is the closure of a $B$-orbit in $Gr(d,n)$, where $B$ is the group of upper triangular matrices in $GL_n(\mathbb C)$. The conormal variety $N^*X$ is defined to be the closure (in $T^*Gr(d,n)$) of the conormal bundle of $X_{sm}$, the smooth locus of $X$. We construct a resolution of singularities for $N^*X$, and compute the ideal sheaf cutting out $N^*X$ as a closed subvariety in $T^*Gr(d,n)$. Time permitting, we discuss a conjecture regarding the equations of $N^*X$, for $X$ a Schubert variety in the flag variety.

Ben Thompson - BU

Title: Generalizations of Poncelet's Theorem

Abstract: Jean-Victor Poncelet, a member of Napolian's army, wrote his seminal work while being held as a prisoner of war in Russia. In the work he outlined the foundations of projective geometry and introduced Poncelet's Theorem. There are several ways to state the theorem - the simplest being than any billiard reflection inside an ellipse will envelop another ellipse (or hyperbola). The theorem has be proven using synthetic geometry (Poncelet), analytic geometry (Jacobi), and algebraic geometry (Griffiths and Harris). I'll briefly sketch a proof for the theorem with a couple different ways to state it, and then introduce a new higher dimensional analogue to the theorem that involves planes "reflecting" off of ellipsoids.

Shucheng Yu - BC

Title: A spectral approach to Rogers' formula

Abstract: Rogers' formula is a series of moment formulas for certain counting functions on the space of unimodular lattices. Among them, the second moment formula is of most interest due to its many applications to counting problems. Unfortunately, this second moment formula doesn't hold for rank two lattices. In this talk, I'll describe a Rogers' second moment formula on rank two lattices using spectral methods. If time permits, I'll describe a recent joint work with Dubi Kelmer extending this formula to the space of symplectic lattices.

Boston University, 10/21/2017. Speakers: Brian Hepler, Spencer Leslie, Jackson Walters, Yusheng Luo.

Northeastern University, 12/2/2017. Speakers: Angus McAndrew, Eric Chang, Maria Fox, Emre Sen.