More Like this

1. Liouville closed HT-fields

   Kaplan, Elliot (April 2023, Journal of algebra)

   Let T be a complete, model complete o-minimal theory extending the theory of real closed ordered fields. An HT-field is a model K of T equipped with a T-derivation ∂ such that the underlying ordered differential field of (K,∂) is an H-field. We study HT-fields and their extensions. Our main result is that if T is power bounded, then every HT-field K has either exactly one or exactly two minimal Liouville closed HT-field extensions up to K-isomorphism. The assumption of power boundedness can be relaxed to allow for certain exponential cases, such as T = Th(Ran,exp). 

   more » « less
2. A dichotomy for T$$T$$‐convex fields with a monomial group

   https://doi.org/10.1002/malq.202300017  

   Kaplan, Elliot; Kesting, Christoph (November 2023, Mathematical Logic Quarterly)

   Abstract We prove a dichotomy for o‐minimal fields , expanded by a ‐convex valuation ring (where is the theory of ) and a compatible monomial group. We show that if is power bounded, then this expansion of is model complete (assuming that is), it has a distal theory, and the definable sets are geometrically tame. On the other hand, if defines an exponential function, then the natural numbers are externally definable in our expansion, precluding any sort of model‐theoretic tameness. 

   more » « less
3. On the K-theory of division algebras over local fields

   https://doi.org/10.1007/s00222-019-00909-x  

   Hesselholt, Lars; Larsen, Michael; Lindenstrauss, Ayelet (July 2019, Inventiones mathematicae)

   Let K be a complete discrete valuation field with finite residue field of characteristic p, and let D be a central division algebra over K of finite index d. Thirty years ago, Suslin and Yufryakov showed that for all prime numbers ℓ different from p and integers j≥1 , there exists a "reduced norm" isomorphism of ℓ-adic K-groups Nrd_{D/K}:K_j(D,Z_ℓ)→K_j(K,Z_ℓ) such that d⋅Nrd_{D/K} is equal to the norm homomorphism N_{D/K}. The purpose of this paper is to prove the analogous result for the p-adic K-groups. To do so, we employ the cyclotomic trace map to topological cyclic homology and show that there exists a "reduced trace" equivalence Trd_{A/S}:THH(A|D,Z_p)→THH(S|K,Z_p) between two p-complete cyclotomic spectra associated with D and K, respectively. Interestingly, we show that if p divides d, then it is not possible to choose said equivalence such that, as maps of cyclotomic spectra, d⋅Trd_{A/S} agrees with the trace Tr_{A/S}, although this is possible as maps of spectra with T-action 

   more » « less
4. Pizza and 2-Structures

   https://doi.org/10.1007/s00454-023-00600-2  

   Ehrenborg, Richard; Morel, Sophie; Readdy, Margaret (December 2023, Discrete & Computational Geometry)

   Let $$\mathcal{H}$$ be a Coxeter hyperplane arrangement in $$n$$-dimensional Euclidean space. Assume that the negative of the identity map belongs to the associated Coxeter group $$W$$. Furthermore assume that the arrangement is not of type $$A_1^n$$. Let $$K$$ be a measurable subset of the Euclidean space with finite volume which is stable by the Coxeter group $$W$$ and let $$a$$ be a point such that $$K$$ contains the convex hull of the orbit of the point $$a$$ under the group $$W$$. In a previous article the authors proved the generalized pizza theorem: that the alternating sum over the chambers $$T$$ of $$\mathcal{H}$$ of the volumes of the intersections $$T\cap(K+a)$$ is zero. In this paper we give a dissection proof of this result. In fact, we lift the identity to an abstract dissection group to obtain a similar identity that replaces the volume by any valuation that is invariant under affine isometries. This includes the cases of all intrinsic volumes. Apart from basic geometry, the main ingredient is a theorem of the authors where we relate the alternating sum of the values of certain valuations over the chambers of a Coxeter arrangement to similar alternating sums for simpler subarrangements called $$2$$-structures introduced by Herb to study discrete series characters of real reduced groups. 

   more » « less
5. S-duality in $$ T\overline{T} $$-deformed CFT

   https://doi.org/10.1007/JHEP05(2023)140  

   Benjamin, Nathan; Collier, Scott; Kruthoff, Jorrit; Verlinde, Herman; Zhang, Mengyang (May 2023, Journal of High Energy Physics)

   A bstract $$ T\overline{T} $$ T T ¯ deformed conformal field theories can be reformulated as worldsheet theories of non-critical strings. We use this correspondence to compute and study the $$ T\overline{T} $$ T T ¯ deformed partition sum of a symmetric product CFT. We find that it takes the form of a partition sum of a second quantized string theory with a worldsheet given by the product of the seed CFT and a gaussian sigma model with the two-torus as target space. We show that deformed symmetric product theory admits a natural UV completion that exhibits a strong weak coupling ℤ 2 duality that interchanges the momentum and winding numbers and maps the $$ T\overline{T} $$ T T ¯ -coupling λ to its inverse 1/ λ . The ℤ 2 duality is part of a full O(2, 2, ℤ)-duality group that includes a PSL(2, ℤ) acting on the complexified $$ T\overline{T} $$ T T ¯ coupling. The duality symmetry eliminates the appearance of complex energies at strong coupling for all seed CFTs with central charge c ≤ 6. 

   more » « less