Seminars archive

We work over an algebraically closed field k of characteristic p greater than 0. In 1994, Serre showed that if semi-simple representations V_i of a group \Gamma are such that \sum ( dim(V_i) - 1 ) less than p, then their tensor product is semi-simple. In the late nineties, Serre generalized this theorem comprehensively to the case where \Gamma is a subgroup of G(k), for G a reductive group, and answered the question of “complete reducibility” of \Gamma in G (Seminaire Bourbaki, 2003). In 2014, Deligne generalized the results of Serre (of 1994) to the case when the V_i are semi-simple representations of a group scheme \mathfrak{G}. In my talk I will present the recent work of mine with Deligne and Parameswaran where we consider the case when \mathfrak{G} is a subgroup scheme of a reductive group G and generalize the results of Serre and Deligne. A key result is a structure theorem on “doubly saturated” subgroup schemes \mathfrak{G} of reductive groups G. As an application, we obtain an analogue of classical Luna's etale slice theorem in positive characteristics.

A celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. Over the real numbers, it is a lovely observation of Finashin–Kharlamov and Okonek–Teleman that while the number of real lines depends on the surface, a certain signed count of lines is always 3. We extend this count to an arbitrary field k using an Euler number in A1-homotopy theory. The resulting count is valued in the Grothendieck-Witt group of non-degenerate symmetric bilinear forms. This is joint work with Jesse Kass.

Accelerated optimization methods have gained wide applicability within the machine learning and optimization communities. They are known for leading to optimal convergence rates among schemes that use only use gradient (first order) information in the convex case. In the non-convex case, they appear to provide robustness to shallow local minima. The intuitive idea is that by considering a particle with mass that moves in an energy landscape, the particle will gain momentum and surpass shallow local minimum and settle in in more wider, deeper local extrema in the energy landscape. Although these techniques have been widely used, it was only within the last few years that theoretical attempts have been made to understand them and put them in a mathematical framework. Recent work has shown that accelerated methods may be formulated with variational principles, although in finite dimensions. Motivated by the success of accelerated methods in finite dimensional problems, we formulate optimization problems on infinite dimensional manifolds of diffeomorphisms using a generalization of this approach. The talk will mainly be about the mathematical formulation and some simple examples to illustrate the advantages of this approach. We note very large speed-ups in optical flow computation compared with standard approaches, and robustness to local minimum. Finally, we outline considerations for generalizing this approach to video data and applications in motion-based object segmentation, which require one to optimize diffeomorphisms not just defined on the image domain, but evolving regions of interest that encompass the domain of each of the objects in the scene.\\ \\Bio: Ganesh Sundaramoorthi received the PhD in Electrical and Computer Engineering from Georgia Institute of Technology, Atlanta, USA, and BS in Computer Engineering and BS Mathematics from the same institution in 2003. He was then a postdoctoral researcher in the Computer Science Department at the University of California, Los Angeles between 2008 and 2010. In 2011, he was appointed Assistant Professor of Electrical Engineering and Assistant Professor of Applied Mathematics and Computational Science at King Abdullah University of Science and Technology (KAUST). His research interests include computer vision and its mathematical foundations with recent interest in shape and motion analysis, video analysis, invariant representations for visual tasks, and applications. He was an area chair for IEEE ICCV 2017 and IEEE CVPR 2018.

Studying congruences between modular forms is a prosperous avenue in number theory. One approach to obtaining congruences involves computations on the Jacobian (Mazur, Ribet, ...). For instance, Mazur uses the Jacobian to determine when there is a weight 2 cusp form of prime level congruent to an Eisenstein series, which has various applications. We will explore another approach to obtaining congruences of modular forms using the arithmetic of quaternion algebras and the Jacquet-Langlands correspondence. This will lead to (1) generalizations work of Mazur and Ribet on weight 2 Eisenstein congruences, and (2) a phenomenon of many mod 2 congruences between weight k cusp forms.

A configuration is a finite set of points with no three collinear. Two configurations have the same order type if there exists a bijection between these two configurations that preserves the orientation of every ordered triple. A configuration A contains a copy of a configuration B some subset of A has the same order type of B and we denote by B \subset A. For a configuration B and an integer m, the extremal number ex(m,B)= max {|A| : B is not a subset of A, A \subset [m]^2} is the maximum size of a subset of the grid $[m]^2$ without a copy of $B$. We discuss some bounds on this function for general B.

Objectives. To assess whether the geometrical and hemodynamic reappraisal of the Ishimaru’s Aortic Arch Map according to Aortic Arch Classification in Type I, II, and III, may provide valuable information regarding the suitability for thoracic endovascular aortic repair (TEVAR), and the risk of aortic dissection. Methods. Anonymized thoracic computed tomography scans of healthy aortas were reviewed, and stratified according to the Aortic Arch Classification. Twenty patients of each Type of Arch were selected. Further processing allowed calculation of angulation and tortuosity of each proximal landing zones. Data were described indicating both proximal landing zone and Type of Arch (e.g. 0/I). Also, among these 60 CT angiography scans, 15 were selected, 5 per Type of Arch, for further analysis. Computational fluid dynamics were performed to compute displacement forces, exerted by pulsatile blood flow on the aortic wall in the defined landing areas. Equivalent surface tractions were computed dividing the displacement forces magnitude of each proximal landing zone by the corresponding area. The three-dimensional orientation (x,y,z) of displacement forces was described as an upward (z direction), and a sideways component (x-y plane).

Prasad and Rapinchuk investigated the isopectrality problem for certain Riemannian varieties by analysing in what extent a semisimple algebraic group defined over a number field is determined by its maximal tori. We shall report advances on this topic by Chernousov/Rapinchuk/Rapinchuk, Bayer-Fluckiger/Lee/Parimala and others by discussing the case of non-archimedean fields and local-global principles.