Seminars Sorted by Series

Abstract: This will be an overview of spacetime aspects of mean curvature includingtrapped surfaces, marginally outer trapped surfaces, and the role that they playin the spacetime positive energy theorem and Penrose inequality. We willsurvey the...

Abstract: The Allen-Cahn equation behaves as a desingularization of the area functional. This allows for a PDE approach to the construction of minimal hypersurfaces in closed Riemannian manifolds. After presenting and overview of the subject, I will...

Abstract: The Clifford torus is the simplest nontotally geodesic minimal surface in S^3. It is a product surface, it is helicoidal, and it is a solution obtained by separation of variables. We will show that there are more minimal submanifolds with...

Abstract: Consider an area minimizing integral current $T$ which has a smooth boundary $\Gamma$ of multiplicity $1$ in some smooth Riemannian ambient manifold $\Sigma$. If the current has codimension $1$ a famous work of Hardt and Simon gives full...

Abstract: In this talk we show that given any regular cone with entropy less than that of round cylinder, all smooth self-expanding solutions of the mean curvature flow that are asymptotic to the cone are in the same isotopy class. This is joint...

Abstract: Let M be a manifold with bounded Ricci curvature and which is noncollapsed, i.e. |Rc|v>0. Together with Wenshuai Jiang we prove that there then exists the apriori L^2 curvature estimate \int |Rm|^2 C and L^1 hessian estimate \int |\nabla...

Abstract: I will survey the recent progress on the existence problem for minimal hypersurfaces and then point for some new directions. This is joint work with Fernando Marques.

Abstract: We give some existence and non-existence results for minimal annuli in H2xR whose data at infinity are given.

Abstract: I will first concentrate on doubling gluing constructions for minimal surfaces, including a recent construction for free boundary minimal surfaces in the unit ball (with D. Wiygul: arXiv:1711.00818). I will then discuss the Linearized...

Abstract: In 1963 Brill and Lindquist asked where one might find the apparent horizons of charged black holes in geometrostatic manifolds that arise as time symmetric solutions to vacuum Einstein Maxwell constraint equations. These manifolds are...

Abstract: We define a relative entropy for two expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same smooth cone at infinity. Adapting work of White and using recent results of Bernstein and Bernstein-Wang, we show that...

Abstract: We will explain how to prove properness of a complete embedded minimal surface in Euclidean three-space, provided that the surface has finite genus and countably many limit ends (and possibly compact boundary). This is joint work with...

Abstract: We will discuss a new family of free boundary minimal Möbius bands in $B^4$ that are critical for higher Steklov eigenvalues among $S^1$-invariant metrics.

Abstract: I will go over some recent work that I have been involved in on surface geometry in complete locally homogeneous 3-manifolds X. In joint work with Mira, Perez and Ros, we have been able to finish a long term project related to the Hopf...

Abstract: I will discuss the following question: is Langlands functoriality given by algebraic cycles? After a survey of some examples of interest, the talk will focus mostly on one case, namely that of inner forms GL(2) over a totally real field...

Abstract: I will report on ongoing computations, with G. Dispinescu and W. Niziol, of the p-adic etale cohomology of the Drinfield tower, and applications to the p-adic local Langlands correspondence.

Abstract: I will introduce the mod p derived spherical Hecke algebra of a p-adic group, and discuss its structure via a derived version of the Satake homomorphism. Then, I will survey some speculations about its action on the cohomology of...

Abstract: We will describe new algorithms to compute an explicit finite simplicial model for compact, congruence locally symmetric spaces and Hecke actions thereon. Joint work with Aurel Page.

Abstract: In usual Hida theory, one constructs a module over weight space whose specialitzation to sufficiently regular weight is a space over classical ordinary modular forms. The goal of higher Hida theory is to construct a perfect complex of...

Abstract: We prove an ordinary modularity lifting theorem for certain non-regular 4-dimensional symplectic representations over totally real fields. The argument uses both higher Hida theory and the Calegari-Geraghty version of the Taylor-Wiles...

Abstract: We will discuss joint work with Calegari, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne that proves potential automorphy of symmetric powers of rank two compatible systems of weight zero. As a consequence, we deduce the...

Abstract: Wiles' work on modularity of elliptic curves over the rationals, used as a starting point that odd, irreducible represenations $G_Q \rightarrow GL_2 (F_3)$ arise from cohomological cusp forms (i.e. new forms of weight $K \geq 2$).In...

Abstract: I will discuss a recent conjecture formulated in an ongoing project with Jan Vonk relating the intersection numbers of one-dimensional topological cycles on certain Shimura curves to the arithmetic intersections of associated real...

Abstract: Euler systems are compatible families of cohomology classes for a global Galois represenation, which plan an important role in studying Selmer groups. I will outline the construction of a new Euler system, for the Galois representation...

Abstract: Given a p-adic reductive group G and its (pro-p) Iwahori-Hecke algebra H, we are interested in the link between the category of smooth representations of G and the category of H-modules. When the field of coefficients has characteristic...

Abstract: In his ladmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. We use deformation theory of pseudorepresentations to study...

Abstract: In his classical work, Mazur considers the Eisenstein ideal $I$ of the Hecke algebra $\mathbb{T}$ acting on cusp forms of weight $2$ and level $\Gamma_0(N)$ where $N$ is prime. When $p$ is an Eisenstein prime, i.e. $p$ divides the...

Abstract: Zywina showed that after passing to a suitable field extnesion, every abelian surface $A$ with real multiplication over some number field has geometrically simple reduction modulo $\frak{p}$ for a density one set of primes $\frak{p}$. One...

Abstract: An important input into modularity lifting theorems is an understanding of the geometry of Galois deformation rings, especially local deformation rings with p-adic Hodge theory conditions at $\ell=p$. Outside of a few cases (ordinary...

Abstract: A rather notorious mistake occurs p. 217 in the proof of Lemma 6.3 of the book "Simple algebras, base change, and the advanced theory of the trace formula" (1989) by J. Arthur, L. Clozel. E. Lapid and J. Rogawski (1998) proposed a proof of...

Abstract: We prove that the weight 4 Beilinson's regulator map can be expressed via the classical n-logarithms, $n \leq 4$. This plus Borel's theorem implies Zagier's conjecture, relating the value of the Dedekind zeta functions at $s=4$ and the...

Deep learning has led to rapid progress being made in the field of machine learning and artificial intelligence, leading to dramatically improved solutions of many challenging problems such as image understanding, speech recognition, and control...

Few-shot classification, the task of adapting a classifier to unseen classes given a small labeled dataset, is an important step on the path toward human-like machine learning. I will present some of the key advances in this area, and will then...

Genomics has revolutionized biology, enabling the interrogation of whole transcriptomes, genome-wide binding sites for proteins, and many other molecular processes. However, individual genomic assays measure elements that interact in vivo as...

Existing generative models are typically based on explicit representations of probability distributions (e.g., autoregressive or VAEs) or implicit sampling procedures (e.g., GANs). We propose an alternative approach based on modeling directly the...

In this talk, I would like to share some of my reflections on the progress made in the field of interpretable machine learning. We will reflect on where we are going as a field, and what are the things that we need to be aware of to make progress...