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Since a ground breaking work by Cazenave-Lions in 1982, showing uniqueness (up to symmetries) of variationally constructed solutions to Hamiltonian PDEs has played an indispensable role for verifying their orbital stability. In this talk, we discuss how to obtain the uniqueness of a family of binary star solutions to the Euler-Poisson equations, variationally constructed by McCann in 2006. Main methodology is based on perturbation arguments crucially relying on the exact asymptotic behaviors of solutions.

In the context of Voevodsky’s triangulated category of motives, we will describe a tower of triangulated functors which induce a finite filtration on the Chow groups. For smooth projective varieties, this finite filtration is a good candidate for the (still conjectural) Bloch-Beilinson filtration.

Zoom ID: 352 730 6970, PW: 9999 All time is in Korean Standard Time KST= UTC+9h.

Zoom ID: 352 730 6970, PW: 9999 All time is in Korean Standard Time KST= UTC+9h.

We consider the Cauchy problem of the self-dual Chern-Simons-Schrödinger equation (CSS) under equivariance symmetry. It is $L^2$-critical, has the pseudoconformal symmetry, and admits a soliton $Q$ for each equivariance index $m \geq 0$. An application of the pseudoconformal transform to $Q$ yields an explicit finite-time blow-up solution $S(t)$ which contracts at the pseudoconformal rate $|t|$. In the high equivariance case $m \geq 1$, the pseudoconformal blow-up for smooth finite energy solutions in fact occurs in a codimension one sense; it is stable under a codimension one perturbation, but also exhibits an instability mechanism. In the radial case $m=0$, however, $S(t)$ is no longer a finite energy blow-up solution. Interestingly enough, there are smooth finite energy blow-up solutions, but their blow-up rates differ from the pseudoconformal rate by a power of logarithm. We will explore these interesting blow-up dynamics (with more focus on the latter) via modulation analysis. This talk is based on my joint works with Soonsik Kwon and Sung-Jin Oh.

In a wide class of random constraint satisfaction problems, ideas from statistical physics predict that there is a rich set of phase transitions governed by one-step replica symmetry breaking (1RSB). In particular, it is conjectured that for models in the 1RSB universality class, the solution space condenses into large clusters, just below the satisfiability threshold. We establish this phenomenon for the first time for random regular NAE-SAT in the condensation regime. That is, most of the solutions lie in a bounded number of clusters and the overlap of two independent solutions concentrates on two points. Central to the proof is to calculate the moments of the number of clusters whose size is in an O(1) window. This is joint work with Danny Nam and Allan Sly.

Zoom ID: 832 222 6176 Password: saarc

Zoom ID: 832 222 6176 Password: saarc

Abstract: In combinatorics, Hopf algebras appear naturally when studying various classes of combinatorial objects, such as graphs, matroids, posets or symmetric functions. Given such a class of combinatorial objects, basic information on these objects regarding assembly and disassembly operations are encoded in the algebraic structure of a Hopf algebra. One then hopes to use algebraic identities of a Hopf algebra to return to combinatorial identities of combinatorial objects of interest.
In this talk, I introduce a general class of combinatorial objects, which we call multi-complexes, which simultaneously generalizes graphs, hypergraphs and simplicial and delta complexes. I also introduce a combinatorial Hopf algebra obtained from multi-complexes. Then, I describe the structure of the Hopf algebra of multi-complexes by finding an explicit basis of the space of primitives, which is of combinatorial relevance. If time permits, I will illustrate some potential applications.
This is joint work with Miodrag Iovanov.

Zoom ID: 934 3222 0374 (ibsdimag)

Zoom ID: 934 3222 0374 (ibsdimag)

The exponential random graph model (ERGM) is a version of the Erdos-Renyi graphs, obtained by tilting according to the subgraph counting Hamiltonian. Despite its importance in the theory of random graphs, lots of fundamental questions have remained unanswered owing to the lack of exact solvability. In this talk, I will introduce a series of new concentration of measure results for the ERGM in the entire sub-critical phase, including a Poincare inequality, Gaussian concentration, and a central limit theorem. Joint work with Shirshendu Ganguly.

Zoom ID: 832 222 6176 Password: saarc

Zoom ID: 832 222 6176 Password: saarc

In the theory of turbulence, a famous conjecture of Onsager asserts that the threshold Hölder regularity for the total kinetic energy conservation of (spatially periodic) Euler flows is 1/3. In particular, there are Hölder continuous Euler flows with Hölder exponent less than 1/3 exhibiting strict energy dissipation, as proved recently by Isett. In light of these developments, I'll discuss Hölder continuous Euler flows which not only have energy dissipation but also satisfy a local energy inequality.

Zoom seminar: https://kaist.zoom.us/j/3098650340

Zoom seminar: https://kaist.zoom.us/j/3098650340

A graph $H$ is \emph{common} if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is minimised by the random colouring. Burr and Rosta, extending a famous conjecture by Erd\H{o}s, conjectured that every graph is common. The conjectures by Erd\H{o}s and by Burr and Rosta were disproved by Thomason and by Sidorenko, respectively, in the late 1980s.
Despite its importance, the full classification of common graphs is still a wide open problem and has not seen much progress since the early 1990s. In this lecture, I will present some old and new techniques to prove whether a graph is common or not.

Zoom ID: 862 839 8170 Password: 123450

Zoom ID: 862 839 8170 Password: 123450