TY - CONF
AB - We study edge asymptotics of poissonized Plancherel-type measures on skew Young diagrams (integer partitions). These measures can be seen as generalizations of those studied by Baik--Deift--Johansson and Baik--Rains in resolving Ulam's problem on longest increasing subsequences of random permutations and the last passage percolation (corner growth) discrete versions thereof. Moreover they interpolate between said measures and the uniform measure on partitions. In the new KPZ-like 1/3 exponent edge scaling limit with logarithmic corrections, we find new probability distributions generalizing the classical Tracy--Widom GUE, GOE and GSE distributions from the theory of random matrices.
AU - Betea, Dan
AU - Bouttier, Jérémie
AU - Nejjar, Peter
AU - Vuletíc, Mirjana
ID - 8175
T2 - Proceedings on the 31st International Conference on Formal Power Series and Algebraic Combinatorics
TI - New edge asymptotics of skew Young diagrams via free boundaries
ER -
TY - GEN
AB - Suppose that $n\neq p^k$ and $n\neq 2p^k$ for all $k$ and all primes $p$. We prove that for any Hausdorff compactum $X$ with a free action of the symmetric group $\mathfrak S_n$ there exists an $\mathfrak S_n$-equivariant map $X \to
{\mathbb R}^n$ whose image avoids the diagonal $\{(x,x\dots,x)\in {\mathbb R}^n|x\in {\mathbb R}\}$.
Previously, the special cases of this statement for certain $X$ were usually proved using the equivartiant obstruction theory. Such calculations are difficult and may become infeasible past the first (primary) obstruction. We
take a different approach which allows us to prove the vanishing of all obstructions simultaneously. The essential step in the proof is classifying the possible degrees of $\mathfrak S_n$-equivariant maps from the boundary
$\partial\Delta^{n-1}$ of $(n-1)$-simplex to itself. Existence of equivariant maps between spaces is important for many questions arising from discrete mathematics and geometry, such as Kneser's conjecture, the Square Peg conjecture, the Splitting Necklace problem, and the Topological Tverberg conjecture, etc. We demonstrate the utility of our result applying it to one such question, a specific instance of envy-free division problem.
AU - Avvakumov, Sergey
AU - Kudrya, Sergey
ID - 8182
T2 - arXiv
TI - Vanishing of all equivariant obstructions and the mapping degree
ER -
TY - GEN
AB - Denote by ∆N the N-dimensional simplex. A map f : ∆N → Rd is an almost r-embedding if fσ1∩. . .∩fσr = ∅ whenever σ1, . . . , σr are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if r is not a prime power and d ≥ 2r + 1, then there is an almost r-embedding ∆(d+1)(r−1) → Rd. This was improved by Blagojevi´c–Frick–Ziegler using a simple construction of higher-dimensional counterexamples by taking k-fold join power of lower-dimensional ones. We improve this further (for d large compared to r): If r is not a prime power and N := (d+ 1)r−r l
d + 2 r + 1 m−2, then there is an almost r-embedding ∆N → Rd. For the r-fold van Kampen–Flores conjecture we also produce counterexamples which are stronger than previously known. Our proof is based on generalizations of the Mabillard–Wagner theorem on construction of almost r-embeddings from equivariant maps, and of the Ozaydin theorem on existence of equivariant maps.
AU - Avvakumov, Sergey
AU - Karasev, R.
AU - Skopenkov, A.
ID - 8184
T2 - arXiv
TI - Stronger counterexamples to the topological Tverberg conjecture
ER -
TY - GEN
AB - In this paper we study envy-free division problems. The classical approach to some of such problems, used by David Gale, reduces to considering continuous maps of a simplex to itself and finding sufficient conditions when this map hits the center of the simplex. The mere continuity is not sufficient for such a conclusion, the usual assumption (for example, in the Knaster--Kuratowski--Mazurkiewicz and the Gale theorem) is a certain boundary condition.
We follow Erel Segal-Halevi, Fr\'ed\'eric Meunier, and Shira Zerbib, and replace the boundary condition by another assumption, which has the economic meaning of possibility for a player to prefer an empty part in the segment
partition problem. We solve the problem positively when $n$, the number of players that divide the segment, is a prime power, and we provide counterexamples for every $n$ which is not a prime power. We also provide counterexamples relevant to a wider class of fair or envy-free partition problems when $n$ is odd and not a prime power.
AU - Avvakumov, Sergey
AU - Karasev, Roman
ID - 8185
T2 - arXiv
TI - Envy-free division using mapping degree
ER -
TY - JOUR
AU - Ilieva, Kristina M.
AU - Fazekas-Singer, Judit
AU - Bax, Heather J.
AU - Crescioli, Silvia
AU - Montero‐Morales, Laura
AU - Mele, Silvia
AU - Sow, Heng Sheng
AU - Stavraka, Chara
AU - Josephs, Debra H.
AU - Spicer, James F.
AU - Steinkellner, Herta
AU - Jensen‐Jarolim, Erika
AU - Tutt, Andrew N. J.
AU - Karagiannis, Sophia N.
ID - 8227
IS - 10
JF - Allergy
SN - 0105-4538
TI - AllergoOncology: Expression platform development and functional profiling of an anti‐HER2 IgE antibody
VL - 74
ER -