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Optimal bubble cluster problems concern the study of partitions of $\mathbb{R}^n$ into a finite collection of chambers, some with finite volume and some with infinite volume. One looks for local minimizers of interfacial area subject to volume constraints on the finite-volume chambers. The case…
We investigate fluctuation phenomena for the graph distance and the number of cut points associated with random media arising from the range of a random walk. Our results demonstrate a sequence of dimension-dependent phase transitions in the scaling behavior of these fluctuations, leading to…
The dimer model refers to the study of random dimer covers (or perfect matchings) of a bipartite graph. A remarkable feature of these models is the emergence of limit shapes: in large periodic graphs, a random matching concentrates around a deterministic shape. Although general dimer models…
Any reasonable exotic phenomena in simply connected 4-manifolds are unstable. It is an open question if there is a universal upper bound to the number of stabilizations needed. The case of 1 stabilization was proven in works of Lin and Guth-K., but whether we need more than two stabilizations…
In cosmology, a basic explanation of the observed concentration of mass in singular structures is provided by the Zeldovich approximation, which takes the form of free-streaming flow for perturbations of a uniform Einstein-de Sitter universe in co-moving coordinates. The adhesion model…
We will meet to discuss organizational matters (topic of interest, etc.)
definition of crystalline cohomology, functoriality [BO, 5.4-19], and connections [BO, 2.1-2.10].
Lusztig's theory of canonical bases reveals a remarkably rigid and positive algebraic structure on quantum groups and their modules. In symmetric types, it is known that the structure constants for multiplication in the negative part $U^-$, as well as for the action of Chevalley generators $E_i…
Abstract: Three years ago, the best AI models were benchmarked on middle school mathematics. Now, they are regularly solving research math problems (albeit relatively simple ones ... so far). It seems inevitable that AI will redefine the mathematics profession. I will survey the development of…