Graduate Student Research Seminar

Speaker: (Western)

"TBA"

Time: 13:00 - 14:00

Room: MC 204

Geometry and Combinatorics

Speaker: Yu Li (Notre Dame University)

"Minimal nilpotent orbits and cluster toric varieties"

Time: 15:30 - 16:30

Room: MC 108

I will explain how the coordinate ring of the minimal nilpotent orbit of type A is related to the equivariant cohomology ring of the cluster toric variety of type C via Gröbner degeneration. The main theorem could be viewed as a complementary result of the Hikita conjecture for the minimal nilpotent orbit. This is joint work with Boming Jia.

Geometry and Physics Seminar

Speaker: Masoud Khalkhali (Western)

"From Harmonic Analysis on Trees to p-adic AdS/CFT, Part II"

Time: 12:30 - 13:30

Room: MC 108

This lecture series approaches p-adic AdS/CFT through the geometry and harmonic analysis of trees. Our guiding theme is the analogy between hyperbolic spaces, Riemann surfaces, and their non-Archimedean counterparts. Beginning with homogeneous trees and Cartier’s approach to their harmonic analysis, we will study the Bruhat–Tits trees associated with a non-Archimedean local field. Its boundary at infinity is canonically identified with G/K and the action on the tree extends the Möbius action on this boundary. This provides a geometric starting point for comparing the Archimedean and non-Archimedean settings. The first part develops the basic dictionary: geodesics and horocycles, adjacency and Laplace operators, spherical functions, and boundary integral representations. Particular emphasis will be placed on the relationship between spectral theory in the interior and function theory on the boundary. We will also examine Schottky uniformization in the classical and non-Archimedean settings, relating Riemann surfaces and hyperbolic handlebodies to Mumford curves and the quotient graphs that encode their skeletal geometry. These comparisons will illustrate both the power and the limitations of the tree analogy.

Applied dynamical system seminar

Speaker: Sam Bradford (Western)

"Surface Waves of Ideal Fluids in Annular Cylindrical Domains: Asymptotics, Computation, and Stability"

Time: 13:00 - 14:30

Room: MC 204

For fluids in annular cylindrical time-dependent domains, such as blood vessels and ferrofluid jets, we are interested in studying and computing the waves on the free outer surface. How do they behave under small-wave approximations? What differential equations describe those approximations, and when do they become unstable? How can we use these approximations to compute these surface waves more efficiently than direct computation of the Euler equations?

We begin with a discussion on the derivation and validity of the coupled integro-differential system (the AFM free-surface formulation) that describes surface waves of ideal fluids in cylindrical domains. Under a small-wave approximation and a quasi-monochromatic ansatz, we show that ferrofluid jets are governed, in a certain sense, by the nonlinear Schrödinger equation. We then discuss an operator-splitting scheme as an efficient and accurate method of computing these waves.

Time permitting, we may have a short discussion on the regularity and well-posedness of our surface formulation (something I am struggling with), as well as future directions.

Keywords: Euler equations, Dirichlet to Neumann maps, AFM method, quasi-monochromatic expansions, nonlinear Schrodinger equation, split-step Fourier method.

Graduate Student Research Seminar

Speaker: (Western)

"TBA"

Time: 13:00 - 14:00

Room: MC 204

Geometry and Physics Seminar

Speaker: Masoud Khalkhali (Western)

"From Harmonic Analysis on Trees to p-adic AdS/CFT, Part III"

Time: 12:30 - 13:30

Room: MC 108

Applied dynamical system seminar

Speaker: Bingyang Cui (Western)

"Analysis of a two-species discrete competition model assuming a single-cycle maturation delay"

Time: 13:00 - 14:30

Room: MC 204

We introduce and analyze a discrete-time, two-species competition model that requires one cycle before newborn cohort can contribute to the growth of the population. Our formulation admits all of the outcomes of the classical discrete Lotka-Volterra like competition model:both species die out; one species wins the competition independent of the initial conditions; there is a unique coexistence fixed point that is a saddle and the winning species depends on the initial conditions; or there is a unique coexistence fixed point that is globally asymptotically stable.It also admits two novel bistable regimes. In the first regime, one stable and one unstable interior fixed point coexist with a stable and unstable boundary fixed point. In the second regime, the two boundary fixed points are saddles and there are three interior fixed points; two are attractors and the one in between them is a saddle. Using monotone dynamical systems theory, we show that every forward orbit converges to some fixed point. In the general case, where the two species’ parameters need not coincide, we derive sufficient conditions for extinction, exclusion, and unique coexistence.We also establish sufficient conditions for each of the novel bistable regimes to occur provided we know how many interior fixed points exist. In the symmetric case where certain ratios of the parameters of each species are identical, the first of the bistable regimes is ruled out and we obtain necessary and sufficient conditions for the second case. Finally, we illustrate each regime with phase portraits and bifurcation diagrams.

Graduate Student Research Seminar

Speaker: (Western)

"TBA"

Time: 13:00 - 14:00

Room: MC 204

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"On the Hodge extension of Kähler forms, I"

Time: 15:30 - 16:30

Room: MC 107

I will discuss the question of extending locally defined Kähler forms on a Stein manifold $X$ to globally defined on $X$ Hodge forms (i.e., Kähler forms in the integral cohomology classes). In the first talk I will give a general background on Kähler forms on Stein manifolds and their place in De Rham cohomology.

Applied dynamical system seminar

Speaker: Rujing Zhao (Western)

"TBA"

Time: 13:00 - 14:30

Room: MC 204

Colloquium

Speaker: Omid Amini (Ecole Polytechnique, Paris)

"TBA"

Time: 15:30 - 16:30

Room: MC 108

Graduate Student Research Seminar

Speaker: (Western)

"TBA"

Time: 13:00 - 14:00

Room: MC 204

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"On the Hodge extension of Kähler forms, II"

Time: 15:30 - 16:30

Room: MC 107

I will discuss the question of extending locally defined Kähler forms on a Stein manifold $X$ to globally defined on $X$ Hodge forms (i.e., Kähler forms in the integral cohomology classes). In the second talk I will outline sufficient conditions for such an extension for a wide class of Stein manifolds and discuss potential counterexamples in the general case.

Applied dynamical system seminar

Speaker: Andrea Yuming Li (Western)

"TBA"

Time: 13:00 - 14:30

Room: MC 204

Colloquium

Speaker: Benjamin Landon (University of Toronto)

"TBA"

Time: 15:30 - 16:30

Room: MC 108