Seminars and Colloquia by Series

Construction of multi-soliton solutions for semilinear equations in dimension 3

Series
PDE Seminar
Time
Tuesday, March 4, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Istvan KadarPrinceton University

The existence of multi black hole solutions in General Relativity is one of the expectations from the final state conjecture, the analogue of soliton resolution. In this talk, I will present preliminary works in this direction via a semilinear model, the energy critical wave equation, in dimension 3. In particular, I show 1) an algorithm to construct approximate solutions to the energy critical wave equation that converge to a sum of solitons at an arbitrary polynomial rate in (t-r); 2) a robust method to solve the remaining error terms for the nonlinear equation. The methods apply to energy supercritical problems.

Modeling, analysis, and control of droplet dynamics

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 3, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Hangjie JiNorth Carolina State University

Thin liquid films flowing down vertical fibers spontaneously exhibit complex interfacial dynamics, leading to irregular wavy patterns and traveling liquid droplets. Such droplet dynamics are fundamental components in many engineering applications, including mass and heat exchangers for thermal desalination, as well as water vapor and particle capture. Recent experiments demonstrate that critical flow regime transitions can be triggered by varying inlet geometries and external fields. Similar interacting droplet dynamics have also been observed on hydrophobic substrates, arising from interfacial instabilities in volatile liquid films. In this talk, I will describe lubrication and weighted residual models for falling droplets. The coarsening dynamics of condensing droplets will be discussed using a lubrication model. I will also present our recent results on developing optimal boundary control and mean-field control for droplet dynamics. 

 

2-torsion in instanton Floer homology

Series
Geometry Topology Seminar
Time
Monday, March 3, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Zhenkun LiUniversity of South Florida

Instanton Floer homology, introduced by Floer in the 1980s, has become a power tool in the study of 3-dimensional topology. Its application has led to significant achievements, such as the proof of the Property P conjecture. While instanton Floer homology with complex coefficients is widely studied and conjectured to be isomorphic to the hat version of Heegaard Floer homology, its counterpart with integral coefficients is less understood. In this talk, we will explore the abundance of 2-torsion in instanton Floer homology with integral coefficients and demonstrate how this 2-torsion encodes intriguing topological information about relevant 3-manifolds and knots. This is a joint work with Fan Ye.

 

Strong u-invariant and Period-Index Bounds

Series
Algebra Seminar
Time
Monday, March 3, 2025 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Shilpi MandalEmory University

Please Note: There will be a pre-seminar from 10:55 am to 11:15 am in Skiles 005.

For a central simple algebra $A$ over a field $K$, there are two major invariants, viz., period and index. For a field $K$, the Brauer-$l$-dimension of $K$ for a prime number $l$, is the smallest natural number $d$ such that for every finite field extension $L/K$ and every central simple $L$-algebra $A$ (of period a power of $l$), we have that index($A$) divides period$(A)^d$.

If $K$ is a number field or a local field, then classical results from class field theory tell us that the Brauer-$l$-dimension of $K$ is 1. This invariant is expected to grow under a field extension, bounded by the transcendence degree. Some recent works in this area include that of Harbater-Hartmann-Krashen for $K$ a complete discretely valued field, in the good characteristic case. In the bad characteristic case, for such fields $K$, Parimala-Suresh have given some bounds.

Also, the u-invariant of $K$ is the maximal dimension of anisotropic quadratic forms over $K$. For example, the u-invariant of $\mathbb{C}$ is 1, for $F$ a non-real global or local field the u-invariant of $F$ is 1, 2, 4, or 8, etc.

In this talk, I will present similar bounds for the Brauer-l-dimension and the strong u-invariant of a complete non-Archimedean valued field $K$ with residue field $\kappa$.

Equilibrium states for star flows and the spectral decomposition conjecture

Series
CDSNS Colloquium
Time
Friday, February 28, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 314
Speaker
Fan YangWake Forest University

In this talk, we will discuss recent progress in the theory of smooth star flows that contain singularities and consider their expansiveness, continuity of the topological pressure, and the existence and uniqueness of equilibrium states. We will prove an ergodic version of the Spectral Decomposition Conjecture: $C^1$ open and densely, every singular star flow has only finitely many ergodic measures of maximal entropy, and only finitely many ergodic equilibrium states for Holder continuous potentials satisfying a mild yet optimal condition. Joint with M.J. Pacifico and J. Yang.

Adaptive Estimation from Indirect Observations

Series
Time
Thursday, February 27, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anatoli JuditskyGrenoble Alpes University

We discuss an approach to estimate aggregation and adaptive estimation based upon (nearly optimal) testing of convex hypotheses. The proposed adaptive routines generalize upon now-classical Goldenshluger-Lepski adaptation schemes, and, in the situation where the observations stem from simple observation schemes (i.e., have Gaussian, discrete and Poisson distribution) and where the set of unknown signals is a finite union of convex and compact sets, attain nearly optimal performance. As an illustration, we consider application of the proposed estimates to the problem of recovery of unknown signal known to belong to a union of ellitopes in Gaussian observation scheme. The corresponding numerical routines can be implemented efficiently when the number of sets in the union is “not very large.” We illustrate “practical performance” of the method in an example of estimation in the single-index model.

 

Local-to-global in thin orbits

Series
School of Mathematics Colloquium
Time
Thursday, February 27, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kate StangeUniversity of Colorado, Boulder

Primitive integral Apollonian circle packings are fractal arrangements of tangent circles with integer curvatures.  The curvatures form an orbit of a 'thin group,' a subgroup of an algebraic group having infinite index in its Zariski closure.  The curvatures that appear must fall into a restricted class of residues modulo 24. The twenty-year-old local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture.  This is joint work with Summer Haag, Clyde Kertzer, and James Rickards.  Time permitting, I will discuss some new results, joint with Rickards, that extend these phenomena to certain settings in the study of continued fractions.

Anosov representations of cubulated hyperbolic groups

Series
Geometry Topology Seminar
Time
Wednesday, February 26, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Theodore WeismanUniversity of Michigan

Please Note: Note the unusual date of a research seminar on Wednesday

An Anosov representation of a hyperbolic group $\Gamma$ is a representation which quasi-isometrically embeds $\Gamma$ into a semisimple Lie group - say, SL(d, R) - in a way which imitates and generalizes the behavior of a convex cocompact group acting on a hyperbolic metric space. It is unknown whether every linear hyperbolic group admits an Anosov representation. In this talk, I will discuss joint work with Sami Douba, Balthazar Flechelles, and Feng Zhu, which shows that every hyperbolic group that acts geometrically on a CAT(0) cube complex admits a 1-Anosov representation into SL(d, R) for some d. Mainly, the proof exploits the relationship between the combinatorial/CAT(0) geometry of right-angled Coxeter groups and the projective geometry of a convex domain in real projective space on which a Coxeter group acts by reflections.

Around the convergence problem in mean field control theory and the associated Hamilton-Jacobi equations

Series
PDE Seminar
Time
Tuesday, February 25, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE: https://gatech.zoom.us/j/95641893035?pwd=rZeIGeDdpL0abXWa4t94JDuRKV9wPa.1
Speaker
Samuel DaudinUniversité Paris Cité

The aim of this talk is to discuss recent advances around the convergence problem in mean field control theory and the study of associated nonlinear PDEs.

We are interested in optimal control problems involving a large number of interacting particles and subject to independent Brownian noises. As the number of particles tends to infinity, the problem simplifies into a McKean-Vlasov type optimal control problem for a typical particle. I will present recent results concerning the quantitative analysis of this convergence. More precisely, I will discuss an approach based on the analysis of associated value functions. These functions are solutions of Hamilton-Jacobi equations in high dimension and the convergence problem translates into a stability problem for the limit equation which is posed on a space of probability measures.

I will also discuss the well-posedness of this limiting equation, the study of which seems to escape the usual techniques for Hamilton-Jacobi equations in infinite dimension.

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