This is a reading seminar for two graduate students.) This talk studies the birational geometry of fibered surfaces, which are integral, projective, flat schemes of dimension 2 over a Dedekind scheme. In contrast to smooth projective curves, birational equivalence for surfaces does not imply isomorphism, which leads to the problem of understanding and selecting canonical representatives within a birational class. We first introduce basic tools for birational surface theory, including blowing-ups, contraction, and desingularization. We then explain how intersection theory on regular surfaces is used to analyze these operations and to identify exceptional curves. This perspective naturally leads to minimal surfaces and to applications of contraction criteria in the construction of canonical models.

Abstract: In this seminar, we study the logistic diffusion equation, a reaction–diffusion model, and its equilibria. We first establish existence and regularity of positive solutions to the parabolic problem. We then use the comparison principle to show that, as time tends to infinity, the solution converges to a steady state solving the corresponding elliptic equation. We recall why the existence of solutions to this elliptic problem is not easily obtained by standard variational methods. Finally, we discuss how stability depends on the resource term and how the solution behavior changes with the diffusion rate. References: [1] Cantrell, R.S., Cosner, C. Spatial ecology via reaction-diffusion equation. Wiley series in mathematical and computational biology, John Wiley & Sons Ltd (2003)

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

Abstract: In this talk, we discuss finite-time blow-up dynamics for the nonlinear heat equation (NLH). We explain the notion of finite-time blow-up, introduce Type I and Type II blow-ups, and discuss the difference between these two behaviors. Restricting to radially symmetric solutions, we review known blow-up results and give a heuristic explanation of when only Type I blow-up is possible and when Type II blow-up may occur. Finally, we describe possible Type II blow-up scenarios through their formal mechanisms. Reference: [1] Hiroshi Matano, Frank Merle. On Nonexistence of type II blowup for a supercritical nonlinear heat equation. Communications on Pure and Applied Mathematics, 2004, 57. 1494 - 1541. [2] Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete and Continuous Dynamical Systems, 2021, 41(10): 4847-4885

We will define twisted homology and Reidemeister torsion. These are invariants of smooth manifolds together with a representation. We will show that Reidemeister torsion can be used to classify lens spaces up to diffeomorphism and we will see that Reidemeister torsion can be used to give invariants of knots and links.

We show how specific families of positive definite kernels serve as powerful tools in analyses of iteration algorithms for multiple layer feedforward Neural Network models. Our focus is on particular kernels that adapt well to learning algorithms for data-sets/features which display intrinsic self-similarities at feedforward iterations of scaling.

(This is a reading seminar for two graduate students.) This talk studies the birational geometry of fibered surfaces, which are integral, projective, flat schemes of dimension 2 over a Dedekind scheme. In contrast to smooth projective curves, birational equivalence for surfaces does not imply isomorphism, which leads to the problem of understanding and selecting canonical representatives within a birational class. We first introduce basic tools for birational surface theory, including blowing-ups, contraction, and desingularization. We then explain how intersection theory on regular surfaces is used to analyze these operations and to identify exceptional curves. This perspective naturally leads to minimal surfaces and to applications of contraction criteria in the construction of canonical models.

The subfield of low-dimensional topology colloquially called "3.5-dimensional topology" studies closed 3-manifolds through the eyes of the 4-manifolds that they bound. This talk focusses on Casson's question of which rational homology 3-spheres bound rational homology 4-balls. Since rational homology 3-spheres bounding rational homology 4-balls are a rare phenomenon, we will discuss how to construct examples.