Georgia Institute of Technology College of Sciences

The Ahlswede–Daykin inequality (also known as the Four Function Theorem) is a classical cornerstone of probability theory and statistical mechanics. Its power lies in its ability to reduce proving correlation and log-concave inequalities to a set of local, usually straightforward, conditions. In this talk, we extend this classical result to the setting of symmetric functions, introducing a new framework tailored for proving Schur positivity. As an application, we give a simple proof of the Schur log-concave inequality for Schur polynomials, a landmark result originally established by Lam–Postnikov–Pylyavskyy (2007). We also establish Schur log-concave inequality for Grothendieck polynomials, thereby resolving a conjecture of Mihalcea.

This talk is a joint work with Hong Chen, Igor Pak, and Daniel Soskin.

Over the last 40 years, a considerable amount of progress has been made in studying the diffeomorphism group of the ball in all dimensions. It is now known that the space of diffeomorphisms of the n-ball that restrict to the identity on the boundary is not contractible for all dimensions n greater than or equal to four whereas in dimensions three and below, it is contractible. Over twenty years ago, Eliashberg asked the same question for the space of contactomorphisms/symplectomorphisms of the ball. To this end, Eliashberg and Mishachev showed it was true in dimension three while Gromov answered it affirmatively for dimension 4. In dimensions one and two, it follows from the corresponding result for diffeomorphisms. In my talk I will outline the proof that in all odd dimensions greater than or equal to 5, the contactomorphism group of the ball with the standard contact structure is not contractible.

In this talk, I will present a classification result for time-periodic oscillation modes of capillary droplet under zero gravity, agitated at low Rayleigh modes (\leq14). To overcome the difficulties of nonlinear Fourier analysis on curved manifold, we develop a toolset paradifferential and para-Fourier integral operators on the sphere based on invariant Littlewood-Paley decomposition. In case of zonal symmetry, an effectively computable symbolic calculus is available. This paradifferential method balances the loss of regularity caused by small denominators, allowing a "fixed point" style proof instead of Nash-Moser/KAM iteration. Furthermore, the bifurcation analysis predicts observable frequency-amplitude relations, which were previously only partially addressed in physical references due to mishandled internal resonances. The bifurcation analysis is computer-assisted due to the hard task of finding all integral points on a given elliptic curve, together with heavy computations involving Legendre polynomials. The talk is based on a working paper co-authored with Haocheng Yang. 

TBA

We discuss the bow lemma or Schur comparison theorem, and in particular its higher dimensional analogs. We first investigate isoperimetry under upper and lower curvature bounds, including the Cartan-Hadamard conjecture. We then discuss Fenchel's theorem for CAT(0) spaces and its higher dimensional versions concerning tight (hyper)surfaces in Cartan-Hadamard manifolds. We also develop a rigidity mechanism for flat hypersurfaces in Alexandrov ambient spaces, which is applied to completely characterize surfaces which optimize total Gauss curvature in 3-folds with upper or lower curvature bounds.

Teams link: https://teams.microsoft.com/meet/2172100941812051?p=0YimD0FGzxQfer6TrD

Zoom Link: https://gatech.zoom.us/j/94382136954?pwd=Pg5gLHppRs4UdyWQPmCtzf8fvZHyAx.1

Meeting ID: 943 8213 6954
Passcode: 621373

Abstract

We consider a one-dimensional discrete Dirac operator in a potential given by a family of i.i.d. random variables modulated by a decreasing envelope. In [1], we showed that these models exhibit a rich phase diagram as a function of the rate of decay of the random potential. The spectrum of the operator

• is absolutely continuous for fast decay,

• is pure point for slow decay,

• presents a spectral transition for critical decay.

We also showed dynamical localization in the sub-critical region by means of the fractional moments method with control on the eigenfunctions, while in the critical regime, transport persists even where the spectrum is pure point.

In [2], we studied spectral statistics. In the fast decay case, the rescaled spectrum of the operator converges to the clock process while for critical decay, it converges to the Sine/Schrödinger process. In this way, we recover the results of Kritchevski, Valkó and Virag established for the Anderson model. Our proof is based on the scaling limit of Prüfer transform associated to the system and uses the monotonicity of the Prüfer phase to deduce the convergence. In the slow decay, the spectral statistics are expected to be given by a Poisson process.

References

[1] O. Bourget, G. R. Moreno Flores, A. Taarabt, One-dimensional Discrete Dirac Operators in a Decaying Random Potential I: Spectrum and Dynamics, Mathematical Physics, Analysis and Geometry 23 (2020).

[2] G. R. Moreno Flores, A. Taarabt, One-dimensional Discrete Dirac Operators in a Decaying Random Potential II: Clock, Schr¨odinger and Sine statistics, Discrete Continuous Dyn. Syst. 46 (2026).

Abstract:

We present several results on exotic open 4-manifolds and knotted surfaces in 4-manifolds. These include obstructive, constructive, and qualitative results. 

First, we discuss obstructions to diffeomorphism using end Floer homology (a direct limit of Heegaard Floer homology), and use this to study exotic R4's. This yields the first completely constructive infinite family of exotic R4's. We find more new phenomena about slice R4's, including the first example of distinct slice R4's made using the same slice disk complement and same Casson handle. As another application, we recover a result of Bizaca-Etnyre on smooth structures of four-manifolds with collarable ends. 

Next, we give a general construction of exotic open 2-handles. We use these to find new exotic smooth structures on open 4-manifolds, which have desirable properties. As an application, we construct examples of knotted proper surfaces in R4 which are topologically standard but smoothly distinct, and remain smoothly distinct after end-sum with exotic planes, making progress on a question of Gompf. As more applications of the 2-handles, we construct a family of topologically slice links, generalizing a theorem of Cha-Powell, and prove all open infinite towers in the sense of disk embedding theory are topologically standard. 

Finally, we give the simplest known decomposition of an exotic R4 into two open sets: two thickened circles, glued along the standard thickened wedge of two circles. This makes progress on a question of Kirby, asking whether any exotic R4 is the union of two R4's. Our proof is obtained by studying cuts in Casson-Gordon fibered slice disk complements, and partially attached Casson handles.

Committee: John Etnyre (advisor), Jennifer Hom, Johnathan Zung, Robert Gompf (UT Austin), Akram Alishahi (UGA).

Fix a graph or uniform hypergraph $F$ and consider the following simple 
process. Start with the complete (hyper)graph $K_n$ on $n$ vertices and 
repeatedly remove the edge set of a copy of $F$, chosen uniformly at 
random among all copies present at that time. Stop when no copy of $F$ 
is left. This is the $F$-removal process, and the key question is when 
it ends, that is, how many edges are left at the end.

This question has proven to be very hard. In 2015, Bohman, Frieze and 
Lubetzky famously proved that the triangle-removal process typically 
ends with $n^{3/2+o(1)}$ edges. Until now, the answer was not known for 
any other $F$.

We determine when the $F$-removal process ends for many choices of $F$. 
This includes cliques, where it confirms a well-known conjecture of 
Bennett and Bohman on the final number of edges. This is joint work with 
Kühn.