Events

Let \(\{a(t):t \in \mathbb{R}\}\) be a diagonalizable subgroup of \(SL(d,\mathbb{R})\) for which the expanded horosphere \(U\) is abelian. By the Birkhoff ergodic theorem, for any point \(x \in SL(d,\mathbb{R})/SL(d,\mathbb{Z})\) and almost every \(u \in U\) the point \(ux\) is Birkhoff generic for the flow \(a(t)\). One may ask whether the same is true when the points in \(U\) are sampled with respect to a measure singular to the Lebesgue measure. In this talk, we discuss work with Omri Solan proving that almost every point on an analytic curve within U is Birkhoff generic when the curve satisfies a non-degeneracy condition.

The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold M with the Grothendieck group of constructible sheaves on M . When M is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of k-vector spaces on M . In this talk, we will characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. Finally, we will give consequences of our result for the stability of the K-theory of persistence modules.

This talk concerns the (generalized) Soler model: a nonlinear (massive) Dirac equation with a nonlinearity taking the form of a space-dependent mass. The equation admits standing wave solutions and they are generally expected to be stable (i.e., small perturbations in the initial conditions stay small) based on numerical simulations. However, contrarily to the nonlinear Schrödinger equation for example, there are very few results in this direction. The results that I will discuss concern the simpler question of spectral stability (and instability), i.e., the absence (or presence) of exponentially growing solutions to the linearized equation around a solitary wave. As in the case of the nonlinear Schrödinger equation, this is equivalent to the presence or absence of "unstable eigenvalues" of a non-self-adjoint operator with a particular block structure. I will highlight the differences and similarities with the Schrödinger case, present some partial results for the one-dimensional case, and discuss open problems. This is joint work with Danko Aldunate, Edgardo Stockmeyer and Hanne van den Bosch.

"Tilting" establishes equivalences between algebraic objects of distinct nature, and therefore has important applications in geometry and number theory. In this talk we will have an overview of the theory, including historical background and some motivating examples.

I will explain some new results showing that the Chow and cohomology rings of moduli spaces of stable curves in relatively low genus and low number of marked points are isomorphic and equal to the tautological ring. These computations involve both concrete geometric techniques in order to explicitly study various strata in the moduli spaces and more abstract techniques relating the computations in the Chow ring to those in cohomology. This part is joint work with Hannah Larson. Next, I will explain a surprising extended application of these results to the vanishing of the eleventh cohomology of moduli spaces of pointed stable curves of genus g at least 2. This part is joint work with Hannah Larson and Sam Payne.

Can one learn a differential operator from pairs of solutions and righthand sides? If so, how many pairs are required? These two questions have received significant research attention in partial differential equation (PDE) learning. Given input-output pairs from an unknown elliptic or parabolic PDE, we will derive a theoretically rigorous scheme for learning the associated Green's function. By exploiting the hierarchical low-rank structure of Green’s functions and randomized linear algebra, we will have a provable learning rate. Along the way, we will develop essential new Green's function theory associated with parabolic PDEs and a more general theory for the randomized singular value decomposition.

Fröhlich and Spencer proved the Berezinskii-Kosterlitz-Thouless transition in 1981, through a relation with delocalisation of height functions. Their delocalisation proof goes through a relation with the Coulomb gas. In recent years it is becoming clear that this phase transition can also be understood in a simpler way through couplings with planar percolation models. This talk presents one such delocalisation proof and outlines some other recent advancements.

Counting points with integer coordinates in geometric objects are challenging and well-studied problems in mathematics and use methods from both number theory and geometry. These problems are typically very hard. In this talk, I will present the problem of counting lattice points in the triangle bounded by the coordinate axes and a line L in the plane. Albeit being a geometric question, certain arithmetic conditions on the slope of L determine the solution to the problem. We will see explicit formulae for rational slope and asymptotic formulae for irrational slope (which behave differently if, e.g., the slope is algebraic). No prior knowledge of number theory is required.

It is well known that martingale transport plans between marginals $\mu\neq\nu$ are never given by Monge maps---with the understanding that the map is over the first marginal $\mu$, or forward in time. Here, we change the perspective, with surprising results. We show that any distributions $\mu,\nu$ in convex order with $\nu$ atomless admit a martingale coupling given by a Monge map over the \emph{second} marginal $\nu$. Namely, we construct a particular coupling called the barcode transport. Much more generally, we prove that such ``backward Monge'' martingale transports are dense in the set of all martingale couplings, paralleling the classical denseness result for Monge transports in the Kantorovich formulation of optimal transport. Various properties and applications are presented, including a refined version of Strassen's theorem and a mimicking theorem where the marginals of a given martingale are reproduced by a ``backward deterministic'' martingale, a remarkable type of process whose current state encodes its whole history. Joint work with Ruodu Wang (Waterloo) and Zhenyuan Zhang (Stanford).

On Calabi--Yau threefolds there are two types of integral invariants, quantum K-invariants and Gopakumar--Vafa invariants. In this talk, I will explain a joint project (with You-Cheng Chou) which aims to show that the quantum K-invariants and Gopakumar invariants are equivalent. At genus zero, this is a conjecture by Jockers--Mayr and Garoufalidis--Scheidegger (for the quintic), and a proof of the JMGS conjecture will be presented.

The moduli spaces of one-dimensional sheaves on CP2 are first studied by Carlos Simpson and Le Potier, and they admit a Hilbert-Chow morphism to a projective base that behaves like a completely integrable system. Following a proposal of Maulik-Toda, one should be able to obtain certain BPS invariants from this morphism. In this talk, we investigate the cohomology ring structure of these moduli spaces. We will derive a minimal set of tautological generators for the cohomology ring, and discuss how they are related, through the key notion of perversity, to the curve counting invariants for local CP2. Based on joint work with Junliang Shen, and with Junliang Shen and Yakov Kononov in progress.