• Location: South Engineering 120
• Time: Tuesdays, 11:00-11:50 am
• Organizer: Maria Alfonseca-Cubero

Lane Morrison: Surfaces in 4 Manifolds

Abstract: It is a fact that every second homology class of a closed connected smooth 4 manifold M can be represented by an embedded surface. These classes play a role in determining the topology of M, but determining information about the surfaces which represent these classes can be quite challenging. I plan to discuss properties of these surfaces and give a construction of some smooth surfaces.

Josef Dorfmeister: Why are symplectic 4-manifolds interesting? (Part 2)

Josef Dorfmeister: Why are symplectic 4-manifolds interesting?

Abstract: Low dimensional topology generally refers to manifolds of dimension $\le $ 4.  I will survey what the main questions are in the field and what is known up to dimension 4 and try to illuminate the special role of dimension 4 and what role symplectic manifolds play.  This talk will mostly focus on dimensions 0-3 to provide a contrast to dimension 4.  Depending on how far I get, the second talk will focus on recent work on elliptic surfaces with Tian-Jun Li.

Morgan O'Brien:  The Davis-Choi-Jensen Inequality  (Part 2)

Morgan O'Brien:  The Davis-Choi-Jensen Inequality

Abstract: Jensen's Inequality plays an important role in analysis, as it helps allows one to use convex functions to bound certain integrals by other integrals (for example, in probability theory one finds that E(X^2) is an upper bound of E(X)^2, where E represents the expectation (integral) of a random variable X). In this talk we will look at a few generalizations of this to the C*-algebra setting, first proven by Davis and later improved by Choi, that also show that one can replace the integral of a function with any unital positive map between appropriate function spaces.

Artem Novozhilov: Emergent behavior in flocks (Part 2)

Artem Novozhilov: Emergent behavior in flocks

Abstract: : Among the many mathematical contributions by Steve Smale, there are several that are related to mathematical biology. My plan for the talk is to discuss his (together with Felipe Cucker) paper from 2007, in which a system of ordinary differential equations that model emergent behaviors is formulated and analyzed. 

Azer Akhmedov: Lattices of Lie groups (Part 3)

Azer Akhmedov: Latices of Lie groups (Part 2)

Azer Akhmedov: Lattices of Lie groups

Abstract: A lattice of a Lie group is a discrete subgroup with a finite co-volume with respect to the Haar measure. This is a series of talks (a survey) on lattices of Lie groups and their use in mathematics (geometry/topology and number theory). The talk is aimed at a broad audience.

Boya Liu: Mathematics of Medical Imaging (Part 2)

Boya Liu: Mathematics of Medical Imaging

Abstract: In this talk we will provide an introduction to the field of inverse problems for elliptic PDEs, covering both classical results and recent advancements. We will start with the renowned Calderon problem, which seeks to determine the electrical conductivity of a medium from voltage and current measurements on its boundary. This problem forms the basis for Electrical Impedance Tomography, an imaging modality with applications in seismic and medical imaging. We will present global uniqueness results for this and related problems, addressing both full and partial data cases. We will also outline the mathematical techniques to prove these results and mention a few open problems in the field. Next, we will explore inverse  problems for elliptic PDEs certain classes of Riemannian geometries. We will also address inverse problems for hyperbolic PDEs; in particular, some techniques to solve elliptic inverse problems can be adopted to solve hyperbolic problems. 

Nikita Barabanov: Stability of Recurrent Neural Networks (Part 2)

Nikita Barabanov: Stability of Recurrent Neural Networks

Abstract: We consider the problem of global asymptotic stability of multilayered Recurrent Neural Networks obtained after a learning process. We consider several approaches to solve it. In particular, the relations with Discrete Algebraic Riccati Equations will be discussed. A new approach based on the estimation of the dissipativity domains will be introduced and some results in this area will be presented.

Mariangel Alfonseca: On many questions related to Ulam's floating body problem (Part 2)

Mariangel Alfonseca: On many questions related to Ulam's floating body problem

Abstract: Croft, Falconer and Guy posed a series of questions generalizing Ulam's floating body problem: Given a convex body K in R^3, we consider its plane sections with certain given properties,

(V): Plane sections which cut off a given constant volume,

(A) Plane sections which have a given constant area,

(I) Plane sections which have equal constant principal moments of inertia,

(H) All the planes are at a constant distance from the origin, etc.

In particular, Ulam's floating body problem is equivalent to problem (V,I): If all plane sections of the body K which cut off equal volumes have equal constant moments of inertia, must K be an Euclidean ball?

We discuss the different nature of the problems asking 2 or 3 of the above conditions to hold. Their answers require a variety of techniques from analysis, geometry and differential equations.