2017 2017/2/13-17
1 1 1.1 Švadlenka Karel................................. 3 1.2............................................ 4 1.3 Hecke Iwahori Whittaker..... 5 1.4.......................................... 6 1.5................................. 7 2 9 2.1 On the ranks of elliptic curves in towers of function fields.................. 11 2.2 Loop homology of some global quotient orbifolds....................... 12 2.3 Couette Lyapunov.................. 13 2.4 L K3.................. 14 2.5 Beilinson............ 15 2.6 SL n -.................................... 16 2.7 von Neumann Factorizable Markov Connes Embedding Problem.......................... 17 2.8............................... 18 2.9 Landauer..................................... 19 2.10.................................... 20 2.11................................ 21
2.12 K-theory for C -algebras and its applications......................... 22 2.13................................... 23 2.14 Witt exponential........................ 24 2.15 Hilbert SL(2)-..................... 25 2.16 W......................................... 26 2.17 Bridgeland.......................... 27 2.18 1.............. 28 2.19 Λ-adic Stark Systems....................................... 29 2.20............................... 30 2.21 Kan s horn-filling for simplices and cubes in HoTT...................... 31 2.22....................... 32 2.23 On generic vanishing for pluricanonical bundles and its applications............. 33 2.24 Depth functions of monomial ideals............................... 34 2.25 A Casselman-Shalika formula for the generalized Shalika model of SO 4n.......... 35 2.26 Fano.......................... 36 2.27 CR Q-prime............................... 37 2.28 Lindelöf............................ 38 2.29 Horofunction boundary........................... 39 2.30 Ricci Myers......................... 40 2.31......... 41
2.32 Stanley g..................................... 42 2.33 3....................... 43 2.34...................... 44 2.35................................... 45 2.36 Thom form in equivariant Čech-de Rham cohomology.................... 46 2.37....................... 47 2.38.................................... 48 2.39...................... 49 2.40.......................... 50 2.41 Representations of Compact Lie Groups and Transversally Elliptic Operators....... 51 2.42 Nahm................ 52
3 Švadlenka Karel, 2017 2 u : Ω R F (u) = 1 + u(x) 2 dx Ω F F 1 thresholding method Wulff 3 [1] Frank Morgan, Geometric Measure Theory, A Biginner s Guide, Academic Press, (2013). [2],, Introduction to Interdisciplinary Mathematics: Phenomena, Modeling and Analysis, (2016). karel@math.kyoto-u.ac.jp
4, 2017 2 ( ) ( ) (sort space) (pressing) (cosmic family) cosmic = universal family universe = Lecture Notes, Group Actions, Representations, and Quotient Families takamura@math.kyoto-u.ac.jp
5 Hecke Iwahori Whittaker, 2017 2 F G G(F ) Borel B(F ) split torus T (F ) unipotent radical N(F ) B(F ) = T (F )N(F ) T (F ) τ : T (F ) C (T (o) ) ˆT (C) (Langlands ) z ˆT (C) τ = τ z G(F ) G(F ) f(g) f(bg) = (δ 1/2 τ)(b)f(g) (b B, g G) I(τ) := Ind G B(τ) = {f : G C loc.const.function f(bg) = δ 1/2 τ(b)f(g)} G (principal series representation) δ : B(F ) R >0 modular quasicharacter τ τ(tn) = τ(t)(t T (F ), n N(F )) N(F ) trivial B(F ), I(τ) Iwahori J I(τ) J Iwahori J o F p q F q cardinality K := G(o) G(F q ) B(F q ) I(τ) J W Casselman Hecke [BBL], [R] Iwahori Whittaker [BBL] B. Brubaker, D. Bump and A. Licata, Whittaker functions and Demazure operators, J. Number Theory, 146, (2015), 41-68. [BN] [NN] [R] D. Bump and M. Nakasuji, Casselman s basis of Iwahori vectors and the Bruhat order, Canadian Journal of Mathematics, Vol. 63, (2011), 1238-1253. M. Nakasuji and H. Naruse, Yang-Baxter basis of Hecke algebra and Casselman s problem (extended abstract), DMTCS proc. BC, (2016), 935 946. M. Reeder, p-adic Whittaker functions and vector bundles on flag manifolds, Comp. Math. 85, (1993), 9 36. nakasuji@sophia.ac.jp
6, 2017 2 1 1970., (L- ). Base change lift GL(N).,.,. hiraga@math.kyoto-u.ac.jp
7, 2017 2 [2, Main Conjecture 3.4.3] 0.1 ( ). k X k D k K A S k v S λ v h ϵ > 0 Zariski Z ϵ X C λ v (D, P ) + h(k, P ) < ϵh(a, P ) + C v S P X(k)\Z ϵ X X = P 1 2 P 1 2 X (Siegel ) X 2 ( ; Faltings X D (Schmidt ) X (Faltings ) k Zariski Bombieri Lang abc (Vojta ) Bombieri Lang [1] [3] [1] Joseph H. Silverman, Generalized greatest common divisors, divisibility sequences, and Vojta s conjecture for blowups, Monatsh. Math. 145 (2005), no. 4, 333 350. [2] Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. [3] Yu Yasufuku, Integral points and relative sizes of coordinates of orbits in P N, Math. Z. 279 (2015), no. 3-4, 1121 1141. yasufuku@math.cst.nihon-u.ac.jp
11 On the ranks of elliptic curves in towers of function fields, 2017 2 K 0 y 2 = x 3 + ax + b a, b K, 4a 3 + 27b 2 0 K E E(K) K Q E(K) E(K) E(K) Mordell-Weil E(K) Z r G G r E C(t) C Mordell-Weil Problem 0.1. E C(t) j E / C E(C(t 1 n )) n Mazur ([M]) C(t) F p (t) C(t) ([S]) [M] [S] B.Mazur,Rational points of abelian varieties with values in towers of number fields,invent.math.18(1972),183-266. P.Stiller,The picard numbers of elliptic surfaces with many symmetries,pacific J.Math.128(1987),no.1,157-189. yusuke@math.sci.hokudai.ac.jp
12 Loop homology of some global quotient orbifolds 1 M2, 2017 2 X X S 1 LX = Map(S 1, X) H (LX) 1990 Chas Sullivan Batalin-Vilkovisky Lupercio, Uribe, Xicot encatl M G [M/G] Map(S 1, M G EG) [M/G] M G M G (M, G) continuous action pair (CAP) G G M (M, G, G) continuous action pair with finite subgroup (CAPS) CAPS (M, G, G) ΩG LM LM; (a, l) a l H 0 (ΩG; k) H (LM; k) H (LM; k) (M, G, k) homologically trivial action triple (HAT) (M, G, G, k) homologically trivial action triple with fintie subgroup (HATS) 1.1. (M, G, G) CAP S k G (M, G, G, k) HAT S k H (L[M/G]; k) = H (LM; k) Z(k[G]), (1) Z(k[G]) k[g] 1.2. (M, G, G) CAP S G (M, G, G, k) k HATS 1.3. (M, G, G) CAP S H n (LM; k) = k (M, G, G, k) (G, G) HATS asao@ms.u-tokyo.ac.jp
13 Couette Lyapunov, 2017 2 1 Couette, [1] [2]. Lyapunov., (Minimal Flow Unit, MFU) [3], [4] Couette Lyapunov [5]., MFU, Lyapunov,.,, MFU Lyapunov., Bloch, Lyapunov MFU., Lyapunov Lyapunov. Lyapunov., MFU. [1] A. Prigent, G. Grégoire, H. Chaté & O. Dauchot. Long-wavelength modulation of turbulent shear flows, Physica D 174 (2003) 100-113. [2] Y. Duguet, P. Schlatter & D. S. Henningson. Formation of turbulent patterns near the onset of transition in plane Couette flow, J. Fluid Mech. (2010 a), vol. 650, pp. 119-129. [3] J. Jiménez & P. Moin. The minimal flow unit in near-wall turbulence, J. Fluid Mech. (1991), vol. 225, pp. 213-240. [4] J. M. Hamilton, J. Kim & F. Waleffe. Regeneration mechanisms of near-wall turbulence structure, J. Fluid Mech. (1995), vol. 287, pp. 317-348. [5] M. Inubushi, S. Takehiro & M. Yamada. Regeneration cycle and the covariant Lyapunov vectors in a minimal wall turbulence, Phys. Rev. E., 92, 023022 (2015). toshio@kurims.kyoto-u.ac.jp
14 L K3, 2017 2 1 L Honda-Tate., Honda-Tate K3. Taelman semistable reduction conjecture [1]. Taelman K3 p (p 7 ), semistable reduction,., Weil,. K3, ( ) Weil. ( ), K3 X L L trc (X, T ) 1 + T Q[T ] ( )., det(1 T Frob Fq ; H 2 ét(x Fq, Q l (1))) =, L trc (X/F q, T ) := 22 i=1 γ i / µ (1 γ i T ) (1 γ i T ). (µ 1 n.), ( ) L(T ), K3 X, L trc (X, T ) unconditional. [1] Taelman, L., K3 surfaces over finite fields with given L-function, Algebra Number Theory 10 (2016), no. 5, 1133-1146. kito@math.kyoto-u.ac.jp
15 Beilinson, 2017 2 1 1 L- Bloch Q K- L- Beilinson L- [B] Beilinson Q Bloch Beilinson 2010 L- [ ] a, b, c 3F 2 e, f z (a) n (b) n (c) n z n := (e) n (f) n n! n=0 (a) n Pochhammer 3 F 2 Rogers Zudilin 27 L- s = 2 L(E 27, 2) 3 F 2 Rogers Zudilin L(E 32, 2) L(E 64, 2) 3 F 2 Bloch [B] A.A. Beilinson, Higher regulators and values of L-functions, J. Sov. Math. 30, 1985, p.2036-2070. [I] R. Ito, The Beilinson conjectures for CM elliptic curves via hypergeometric functions, The Ramanujan Journal, Springer. (to appear) thaya9ma@icloud.com
16 SL n -, 2017 2 P SL 2 - P SL 2 - G G- G- SL 2 - SL n - SL 2 SL n - SL 2 - [H] [L] - [FG] SL n C- SL 3 C- [FG] V. Fock and A. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. No. 103 (2006), 1-211 [H] N. Hitchin, Lie groups and Teichmüller space, Topology 31 (1992), no.3, 449-473. [L] F. Labourie, Anosov flows,surface groups and curves in projective space, Invent. Math. 165(2006), no.1, 51-114. y-inagaki@cr.math.sci.osaka-u.ac.jp
17 von Neumann Factorizable Markov Connes Embedding Problem M2, 2017 2 (M, ϕ) (N, ψ) von Neumann. T : M N (ϕ, ψ)-markov T ( i.e. T (1 M ) = 1 N ) ( i.e. n N, T id n ) ( i.e. ψ T = ϕ) modular ( i.e. σ ψ t T = T σ ϕ t, t R). (ϕ, ψ)-markov factorizable von Neumann (L, τ) (ϕ, τ)-markov *- α (ψ, τ)-markov *- β T = β α. (M, ϕ) = (N, ψ) = (M n (C), 1 n T r) T factorizable (L, τ) M n(c) L unitary u T x = (id n τ L )(u (x 1 L )u), x M n (C) 2011 Haagerup Musat ([1]). T M n (C) N. factorizable ϕ-markov FM(M, ϕ) CB-norm. Markov factorizable Markov CB- : τ n -Markov T : M n (C) M n (C) τ k -Markov S : M k (C) M k (C) d CB (T, FM(M n (C))), d CB (S, FM(M k (C))) d CB (T S, FM(M n (C) M k (C))) d CB (T, FM(M n (C))) + d CB (S, FM(M k (C))) + d CB (T, FM(M n (C)))d CB (S, FM(M k (C))). Markov factorizable. Aut(M n (C)) (convex hull). n 2 conv(aut(m n (C))) FM(M n (C)) ( n = 2 ). n 3 FM(M n (C) \ conv(aut(m n (C))). S k : M k (C) M k (C) S k (x) = τ k (x)1 k τ k -Markov. completely deporlization channel. τ N := n i=1 α iτ ki (α i Q +, n i=1 α i = 1) von Neumann N = n i=1 M k i (C) τ n -Markov T M n (C) N k 2 T S k conv(aut(m n (C) M k (C))).. A.Connes Connes Embedding Problem(CEP).. ([2]) [1] Haagerup, Uffe; Musat, Magdalena Factorization and dilation problems for completely positive maps on von Neumann algebras. Comm. Math. Phys. 303 (2011), no. 2, 555-594. [2] Haagerup, Uffe; Musat, Magdalena An asymptotic property of factorizable completely positive maps and the Connes embedding problem. Comm. Math. Phys. 338 (2015), no. 2, 721-752. yuuki114@math.kyoto-u.ac.jp
18, 2017 2 2 Euler C Euler ω D u iv = ω dw 1 dw 2 2πi z w = ω 4π p.v. log (z w) dw (1) D p.v. Cauchy (1) D (1) (1) Gâteaux d 0.1 (). z w C 1 φ (z, z, w, w) φ z w log f Jordan C (f) F F (f) := p.v. φ (z, z, w, w) dw. C(f) F δf Gâteaux [ ] df (f; δf) = p.v. dφ (z, w; δz, δz) dw + 2i φ w Re (δw δz)( i dw). ( ) C δz := δf f 1 (z) δw := δf f 1 (w) Pierrehumbert Crowdy uda@math.kyoto-u.ac.jp C D ( )
19 Landauer 1, 2017 2 1 Landauer Landauer Landauer s limit S, R Hilbert S T Hilbert H S,H R H S H R Hilbert (trace 1 S T ρ S,ρ T ρ S,ρ T ρ T S(ρ S ),S(ρ S ) ρ S,ρ S Q(ρ T ),Q(ρ T ) ρ T,ρ T S = S(ρ S ) S(ρ S ), Q = Q(ρ T ) Q(ρ T ) Q T S T Landauer s limit [RW] David Reeb, Michael M. Wolf, An improved Landauer Principle with finite-size corrections, New J. Phys. 16,103011, (2014) ejima@ms.u-tokyo.ac.jp
20, 2017 2 S B f : S B f K f K 2 f χ f = degf O(K f ) λ f = K 2 f /χ f f g 4(g 1) g λ f 12 4(g 1)/g 3 3 8/3 d λ d := 6(d 3) d 2 K 2 f λ dχ f 3 4 3 [Re] 4 Durfee λ f [Re] M. Reid, Problems on pencils of small genus, Preprint (1990). m-enokizono@cr.math.sci.osaka-u.ac.jp
21, 2017 2 p F G G Langlands G(F ) G L L Langlands G H Langlands Arthur LLC [Art13] H(F ) G(F ) 0.1. G F H π H H(F ) π G(F ) (1) G = GL 2n H = SO 2n+1 π (2) G = Res E/F GL N H = U E/F (N) π E/F 2 H (3) G = GL 2n+1 H = Sp 2n π GL 2n (F ) π ω π π ω π Langlands [Art13] J. Arthur, The endoscopic classification of representations: Orthogonal and symplectic groups, American Mathematical Society Colloquium Publications, vol. 61, American Mathematical Society, Providence, RI, 2013. masaooi@ms.u-tokyo.ac.jp
22 K-theory for C -algebras and its applications, 2017 2 1 1970 C - K-, [1]. K- C - 2 K 0,K 1,K 0 C -,K 1. AF- C - K 0 - [2]. C -,K-. K-, K-. [1] B. Blackadar, K-THEORY FOR OPERATOR ALGEBRAS, MSRI Publications,Volume 5, second edition (1998), 9 12. [2] M. Rørdam, F. Larsen, and N. J. LAUSTSEN, An Introduction to K-Theory for C -Algebras, LMS Student Texts 49 (2000), 109 130. kohta@ms.u-tokyo.ac.jp
23 M2, 2017 2 E Q E Q 1 E Q Q Galois G Q = Gal( Q/Q) E Weierstrass l G Q E l E[l] = (Z/lZ) 2 E modulo l ϕ E,l : G Q Aut(E[l]) = GL 2 (F l ) ϕ E,l ϕ E,l (G Q ) Q E[l] Q(E[l]) Q Galois ϕ E,l l E l Q(E[l])/Q Galois GL 2 (F l ) E Serre [3] Mazur [2] 11 Mazur E 2,3,5,7,11,13,17,19,37,43,67,163 ϕ E,l (G Q ) GL 2 (F l ) Cartan Duke [1] [1] [1] Duke, W., Elliptic curves with no exceptional primes (English, French summary), C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 8, 813-818. [2] Mazur, B., Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129-162. [3] Serre, J.-P., Propriétés galoisiennes des points d ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259-331. odawara@math.kyoto-u.ac.jp
24 Witt exponential 2, 2017 2 p. y = P (T ) y exp(p (T )) = n=0 P (T )n /n!., P (T ) = T. R, 1/n! 0, t R exp(t)., Q p, p 1/n!., Q p exp(p (T ))., Rodolphe Richard Q p exp(p (T )) Witt. A W (A) Witt. K/Q p ultrametric extention, R. Artin-Hasse exponential E p (T ) := exp(t + 1 p T p + 1 p T p2 + ), AH : W (K) 1 + T K[[T ]] a = (a 2 0, a 1, ) W (K) AH(a) = E p (a n T pn ) n=0. [R], Witt AH, Q p exponential.. 0.1. d 0, ζ 1 p d+1., P (T ) := D i=1 a it i K[T ]. 0 i D, ζ d i := ζ pi D, d i = log p i. P (T ) := a 1 ζ d1 1 T + a 2 ζ d2 1 T 2 + + a D ζ dd 1 T D, exp( P (T )) = 1 + i=1 ãit i., exp(p (T )) ρ, { } ρ = max λ R max ã i λ i 1 1 i D., ζ Lubin-Tate., exp(x+x p /p) exponential. [R] Rodolphe, Richard, Des π-exponentielles I : Vecteurs de Witt annulés par Frobénius et Algorithme de (leur) rayon de convergence, Rendiconti del Seminario Matematico della Universita di Padova, 133 (2015), 125-158. kcb.c314271@gmail.com
25 Hilbert SL(2)- 2, 2017 2 G, X G-. Hilbert Hilb G h (X), G- X Z, C[Z] G- ( Hilbert h ). Alexeev Brion Hilbert, G G-Hilbert, Hilbert h : Univ G h (X) X Hilb G h (X) γ X /G γ Hilbert-Chow, [Z] Hilb G h (X) Z /G. γ ( Hilb G h (X) ), Hilbert-Chow γ X /G?., γ SL(2)-.,. Popov [P], SL(2)- (l, m) {Q (0, 1]} N (, E l,m ), Kraft Panyushev, Gaĭfullin, Batyrev, Haddad. [BH] E l,m C 5, Hilbert, E l,m., E l,m. 0.1. E l,m. Hilbert-Chow γ E l,m, E l,m. [BH] [P] Batyrev, Victor, Haddad, Fatima, On the geometry of SL(2)-equivariant flips, Mosc. Math. J., 8, (2008), no. 4, 621 646, 846. Popov, V. L., Quasihomogeneous affine algebraic varieties of the group SL(2), Izv. Akad. Nauk SSSR Ser. Mat, 37, (1973), 792 832. 0ce3880636b282e@fuji.waseda.jp
26 W, 2017 2 1 g C Lie, ĝ = g[t, t 1 ] CK (untwisted ) Lie. k C, ĝ V k (g) = U(ĝ) U(g[t] CK) C k. [F] generic k, M k n +,λ λ h, k ĝ 0 V k (g) ˆρ M k+h n +,0 l i=1 S i l i=1 M k+h n +, α i /(k+h ) (1). l = rank g. f g, f sl 2 -triple {e, h, f}, 1 2 h g 1 2 Z-. m + = j 1 g j Drinfeld-Sokolov W W k (g, f) := H 0 DS,f ( ˆm +, V k (g)). H 0 DS,f ( ˆm +,?) (1) 0 W k (g, f) M k+h r +,0 F (g 1 2 ) l i=1 Q i l i=1 M k+h r +, α i/(k+h ) F (g 1 2 ) (2). r + = n + g 0, F (g 1 2 ) g 1 2. 1.1. generic k, (2) W Q i. W W k (g, f) generic k. [G1] W paraboric. [F] E. Frenkel. Wakimoto modules, opers and the center at the critical level. Advances in Math., 195:297 404, 2005. [G1] N. Genra. Screening operators for W-algebras. arxiv:1606.00966. [G2] N. Genra. Wakimoto representations for W-algebras. in preparation. gnr@kurims.kyoto-u.ac.jp
27 Bridgeland, 2017 2, Douglus Π-, Bridgeland ([Bri07]).,,.,,.,, : X, D b (X) X. D b (X), Bridgeland. D b (X) σ, D b (X), σ-( ),, σ-., : 0.1. X, D b (X)., X, X, D b (X) ([Tod13], [Tod14]).,,. 0.2. X, C X ( 2)., C A 1 - Y., D b (X) σ, Y σ-. 0.3. Y, Y Y., D b (Y) σ, Y σ-. [Bri07] T. Bridgeland. Stability conditions on triangulated categories. Ann. of Math. (2), 166(2):317 345, 2007. [Tod13] Y. Toda. Stability conditions and extremal contractions. Math. Ann., 357(2):631 685, 2013. [Tod14] Y. Toda. Stability conditions and birational geometry of projective surfaces. Compos. Math., 150(10):1755 1788, 2014. koseki@ms.u-tokyo.ac.jp
28 1, 2017 2 Kontsevich,,., DG., 2 T 2 1 Ť 2, T 2,, a n Q (n a )L a n n, a E(L a )., n E(L a n ). E(L a n ) DG, E(L a n ) C(ψ) E(L b m ) ψ T E(L a n )., T, C(ψ) ψ 0, dimext 1 (E(L b ), E( m )) = 1., Atiyah L a n, µ Ť 2 = C/2π(Z τz) (τ H)., µ Ť 2 E(L a n ) [E(L a n ) µ], C(ψ) m + n, a + b, Atiyah µ Ť 2 [C(ψ)] = [E(L a+b ) µ ]., m+n µ Ť 2,, L a L b, L a+b. m m+n,. n, K. Kobayashi, On exact triangles consisting of stable vector bundles on tori. mathdg/1610.02821. afka9031@chiba-u.jp
29 Λ-adic Stark Systems, 2017 2 Kolyvagin Selmer Euler. Kolyvagin Heegner Euler Kolyvagin cohomology, Selmer., Rubin Euler 2. p. K, R p Noether. T R R- Gal(K/K)-, K. F T Selmer ( ), χ(f) Z., T F. Mazur Rubin Kolyvagin F Kolyvagin., R χ(f) = 1 F Kolyvagin 1, F Kolyvagin F Selmer., Euler Selmer Kolyvagin. χ(f) > 1.. R, χ(f) > 0., Mazur-Rubin F Stark. Stark Rubin-Stark. Mazur Rubin F Stark 1, F Stark F Selmer., χ(f) = 1 F Stark F Kolyvagin, Stark Kolyvagin. Stark, Stark., Stark. Stark sakamoto@ms.u-tokto.ac.jp
30, 2017 2 1 n A B p(n A) = p(n B). s t N, x N {0} f(x) f(x + 2) = f(x)f(x + 1) f(0) = s f(1) = t. 1.1. s t s 1 f(x) p(n f(x) f(x) f(x + 1) ) = p(n {s, t} ). (1) 1. s t s 1 n t 2. s = t s = 1 n [R] George E.Andrews, Kimmo Eriksson,, pae sato@cc.nara-wu.ac.jp
31 Kan s horn-filling for simplices and cubes in HoTT, 2017 2 (HoTT) HoTT HoTT HoTT HoTT Awodey-Warren Voevodsky Voevodsky HoTT HoTT (Univalent foundation) HoTT (globular) (simplicial) (cubical) ( ) Licata Brunerie 2 2 HoTT HoTT- Kan fibrant horn-filling gksato@ms.u-tokyo.ac.jp
32, 2017 2 k 0 G k k U (k ) π 1 (U) U A.Grothendieck k Grothendieck U U π 1 (U) G k k 0 U (ie, k 1 ) π 1 (U) U U π 1 (U) ([1]Exposé 12) k π 1 (U) k U U π 1 (U) [1] Grothendieck, A., Revêtemental étales et groupe fondamental, SGA1, Lecture Notes in Mathematics 224, Springer-Verlag, 1971
33 On generic vanishing for pluricanonical bundles and its applications, 2017 2 Generic vanishing theory X F ( ω X ) L Pic 0 (X) H i (X, F L) F i cohomology support locus V i (F) = {α Pic 0 (X) H i (X, F α) 0} Pic 0 (X) Zariski V i (F) (i 0) Generic vanishing theory Generic vanishing theory X f : X A X Albanese 0.1 (Hacon [Hac]). i, j 0 codim V i (R j f ω X ) i 0.2 (Simpson [Sim]). i 0 V i (ω X ) m(k X + ) [Hac] C. D. Hacon, A derived category approach to generic vanishing, J. Reine Angew. Math. 575 (2004), 173 187. [Sim] C. Simpson, Subspaces of moduli spaces of rank one local systems, Ann. Sci. E.N.S. (4) 26 (1993), no. 3, 361 401. tshibata@math.kyoto-u.ac.jp
34 Depth functions of monomial ideals, 2017 2 (R, M) M 1 HM i (R) R M i R depth depthr = min{i; H i M(R) 0} depth function. R = K[x 1,, x n ] I I depth function f : Z 0 \ {0} Z 0 f(k) = depthr/i k depth function [1] [2] 0.1. f : Z 0 \ {0} Z 0 depth function 0.2. f : Z 0 \ {0} Z 0 f(k) f(k + 1) 1 for all k 1; a = f(1) b = lim k f(k) f 1 (a) f 1 (a 1) f 1 (b + 1) depth function [1] M. Brodmann, The asymptotic nature of the analytic spread, Math. Proc. Cambridge Philos. Soc. 86 (1979), 35 39. [2] J. Herzog and T. Hibi, The depth of powers of an ideal, J. Alg. 291 (2005), 534 550. t-suzuki@ist.osaka-u.ac.jp 1
35 A Casselman-Shalika formula for the generalized Shalika model of SO 4n M2, 2017 2 Whittaker Casselman-Shalika[1] Whittaker explicit formula (Casselman-Shalika formula) L- Whittaker 1 explicit fomula 1 Shalika Shalika Casselman-Shalika formula [1] W. Casselman and J. Shalika, The unramified principal series of p-adic groups I, II, Composito Math. [2] Y. Sakellaridis, A Casselman-Shalika formula for the Shalika model of GL n, Canad. J. Math 58, (2006), No. 5, p1095-1120. msuzuki@mah.kyoto-u.ac.jp
36 Fano 2, 2017 2 G = (V (G), E(G)) X( (G)) I V (G) G I B(G) = {I V (G) G I I } G graphical building set G N B(G) nested set 1. I, J N I J, J I, I J = 2. I, J N I J = I J / B(G) 3. V (G) N B(G) nested sets N (B(G)) V (G) = {1,..., n+1} e n+1 = e 1 e n R n I V (G) e I = i I e i N N (B(G)) R 0 N = I N R 0e I R 0 N N 1 (G) = {R 0 N N N (B(G))} R n n X( (G)) G G 1,..., G m G X( (G)) = X( (G 1 )) X( (G m )) X( (G)) G 0.1 ([S]). G 1. X( (G)) Fano G 3 2. X( (G)) Fano G G I V (G ) G I 4 0.1 1. Fano 2. X( (G)) G [S] Y. Suyama, Toric Fano varieties associated to finite simple graphs, Tohoku Math. J., to appear; arxiv:1604.08440. d15san0w03@st.osaka-cu.ac.jp
37 CR Q-prime 1, 2017 2 CR CR C m CR Levi CR Fefferman [F74] C m CR CR CR Case-Yang [CY13] Hirachi [H14] Q-prime Q-prime CR Q-prime CR. 3 Q-prime Sasakian η-einstein CR Sasakian η-einstein CR Q-prime [CY13] [F74] Jeffrey Case and Paul Yang. A Paneitz-type operator for CR pluriharmonic functions. Bull. Inst. Math. Acad. Sin. (N.S.), 8(3):285 322, 2013. Charles Fefferman. The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math., 26:1 65, 1974. [H14] Kengo Hirachi. Q-prime curvature on CR manifolds. Differential Geom. Appl., 33(suppl.):213 245, 2014. ytake@ms.u-tokyo.ac.jp
38 Lindelöf, 2017 2 1 Minkowski. Rogers Swinnerton-Dyer [2]. K O m (a 1,..., a m ) K m (a 1,..., a m ) a 1 + + a m = O. N(a i ) x. N(a i ) x m V m (x, K). V m (x, K). K = Q Lindelöf ( 1.1). 1.1. K Lindelöf ε > 0 { V m (x, K) = cm O(x m 1/2+ε ) if m 3, ζ K (m) xm + O(x 3/2+ε log x) if m = 2. c K. K 2 [K : Q] Sittinger [1]. 3 [K : Q]. Lindelöf K m. [1] B. D. Sittinger. The probability that random algebraic integers are relatively r-prime. Journal of Number Theory, 130(1): 164 171, 2010. [2] K. Rogers. and H. P. F. Swinnerton-Dyer. The geometry of numbers over algebraic number fields. Trans. Amer. Math. Soc. 88, 227 242, 1958. takeda-w@math.kyoto-u.ac.jp
39 Horofunction boundary, 2017 2 Gromov [1] Horofunction boundary. X:X C(X) b X x X ϕ x C(X), ϕ x (y) = d(x, y) d(x, b) i i(x) i(x) horofunction i(x) \ i(x) horofunction boundary T R 0 γ : T X lim t γ(t) horofunction ( γ geodesic ray ) geodesic ray Busemann point Busemann point Walsh[2] Develin[3] horofunction boundary Walsh[2] [1] M.Gromov. Hyperbolic manifolds, groups and actions, Ann.Math.Studies, 97:183-215, Princeton University Press, Princeton, 1981. [2] C.Walsh. The action of a nilpotent group on its horofunction boundary has finite orbits, Groups, Geometry, and Dynamics,5(1):189-206, 2011. [3] M.Develin. Cayley compactifications of Abelian groups, Annals of Combinatorics,6 (3-4):295-312, 2002. kenshi-t@math.kyoto-u.ac.jp
40 Ricci Myers, 2017 2 Riemann Riemann. S. B. Myers [2] Ricci, Riemann. Ricci Riemann. A (S. B. Myers [2]). Let (M, g) be an n-dimensional complete Riemannian manifold. If there exists some positive constant λ > 0 such that Ric g λg, then (M, g) is compact with finite fundamental group. Moreover, the diameter of (M, g) satisfies n 1 diam(m, g) π λ. A. J. Cheeger, M. Gromov, M. Taylor Riemann Ricci, Myers. B (J. Cheeger, M. Gromov and M. Taylor [1]). Let (M, g) be an n-dimensional complete Riemannian manifold. Suppose that there exist some point p M and positive constants r 0 > 0 and v > 0 such that the Ricci curvature satisfies Ric g (x) (n 1) ( 1 4 + v2) r 2 (x) for all x M satisfying r(x) r 0, where r(x) denotes the distance between x and p. Then (M, g) must be compact. Moreover, the diameter of (M, g) from p satisfies ( π ) diam p (M, g) r 0 exp. v Ricci, Ricci Ricci, B. [1] J. Cheeger, M. Gromov and M. Taylor, J. Differential Geom. 17 (1982), 15 53. [2] S. B. Myers, Duke Math. J. 8 (1941), 401 404. h-tadano@cr.math.sci.osaka-u.ac.jp
41, 2017 2 S X, Y Isom S (X, Y ) Isom π1 (S)(π 1 (X), π 1 (Y ))/ Ker(π 1 (Y ) π 1 (S)) p [2] [1] [1] S. Mochizuki, The local pro-p anabelian geometry of curves, Invent. Math. 138 (1999), pp. 319-423. [2] Y. Hoshi and S. Mochizuki, Topics surrounding the combinatorial anabelian geometry of hyperbolic curves I: Inertia groups and profinite Dehn twists, Galois-Teichmüller Theory and Arithmetic Geometry, Adv. Stud. Pure Math. 63, Math. Soc. Japan, 2012, pp. 659-811. stsuji@kurims.kyoto-u.ac.jp
42 Stanley g, 2017 2 P P. d P i f i (P) f(p) = (f 1 (P), f 0 (P),, f d 1 (P)) P f. f h h(p) = (h 0 (P), h 1 (P),, h d (P)). Stanley [1] h. 0.1. (g [1]) h = (h 0, h 1,, h d ) Z d+1 h d P h 3. 1. h j = h d j (0 j d) 2. 1 = h 0 h 1 h d/2 3. h i+1 h i (h i h i 1 ) <i> (1 i d/2 1 ) Stanley 3 P X P Poincare Hard Lefschetz Macaulay. Swartz ([2]). [1] R. P. Stanley. The number of faces of a simplicial convex polytope. Adv. in Math., 35(3):236-238, 1980. [2] E. Swartz: g-elements of matroid complexes, Journal of Comb. Theory Ser. B, Vol. 88, (2003), pp. 369?375.. tnagaoka@math.kyoto-u.ac.jp
43 3, 2017 2 U, V D V \ D = U, (V, D) U. C 3 (V, D), V 1,,, Peternell Schneider (cf. [F])., V. (cf. [K]). 0.1. V 2 3, D 1 D 2. U := V \ (D 1 D 2 ), (V, D 1 D 2, U). K V + D 1 + D 2 nef,. 0.1 (V, D 1 D 2, U)., V. 0.2. (V, D 1 D 2, U) 0.1., K V + D 1 + D 2 = 0., V 14., 14, Fano V, 0.1 (V, D 1 D 2, U). 14. V imprimitive, Fano W C : 1. W = P 3, C 4 1. 2. W P 4 2, C 4. 3. W 5 3 del Pezzo, C 3. V primitive : (i) V P 2 P 2, (1, 2). (ii) V = P 1 P 2. (iii) V 2 Veronese., D 1 D 2 U. [F] [K] Furushima, M.: The complete classification of compactifications of C 3 which are projective manifolds with the second Betti number one. Math. Ann. 297, 627 662 (1993) Kishimoto, T.: Compactifications of contractible affine 3-folds into smooth Fano 3-folds with B 2 = 2. Math. Z. 251, 783 820 (2005) nagaoka@am.ms.u-tokyo.ac.jp
44, 2017 2 n M = {m 1,..., m n } n W = {w 1,..., w n } (n ) m M w W m 1 Gale-Shapley[1] i, j m i w j X ij w j m i Y ij [0, 1] σ S n (m i w σ(i) ) ( ) Xiσ(i) + Y iσ(i) 1 i n 2 n Pittel[2] Random Assignment [1] D. Gale and L. S. Shapley, College admissions and the stability of marriage. American Mathematical Monthly, 69, 9-15, 1962. [2] B. Pittel, On likely solution of a stable marriage problem. The Annals of Applied Probability, 2, 358-401, 1992. paa nagare@cc.nara-wu.ac.jp
45, 2017 2 1 1990 Shokurov Reid 3 X X X (X, ) 1.1 ( ). X (X, ) K X + (X, ) K X + ([BCHM]) 1.2 ( ). X (X, ) K X + K X + R- E 2011 [B] Birkar 1.3 (cf. [B]). 1.2 n 1.1 n 1.4. 1.2 n X = 0 1.2 n 1.1 n [B] C. Birkar, On existence of log minimal models II, J. Reine Angew Math. 658 (2011), 99 113. [BCHM] C. Birkar, P. Cascini, C. D. Hacon, J. M c Kernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405 468. hkenta@math.kyoto-u.ac.jp
46 Thom form in equivariant Čech-de Rham cohomology (Ko FUJISAWA) 1, 2017 2 Atiyah-Bott-Berline-Vergne [1] Thom Mathai-Quillen [2] Thom Čech-deRham Čech-de Rham [3] Thom Chern form Mathai-Quillen [1] Berline, Nicole, Ezra Getzler, and Michele Vergne. Heat kernels and Dirac operators. Springer Science & Business Media, 2003. [2] Mathai, Varghese, and Daniel Quillen. Superconnections, Thom classes, and equivariant differential forms. Topology 25.1 (1986): 85-110. [3] Suwa, T. Indices of vector fields and residues of holomorphic singular foliations. Hermann, Paris (1998). fujisawa1219@math.sci.hokudai.ac.jp
47, 2017 2 Lie g Bernstein-Gelfand-Gelfand O g O Lie (quasi-hereditary algebra) Artin Ringel[R91] Artin (tilting module) Kleshchev[K15] ( ) Hecke H n ( Hecke ) g C[z] g- BGG O BGG Õ. Artin Ringel 2 [R91] C. M. Ringel. The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences. Math. Z., 208(2):209 223, 1991. [K15] A. Kleshchev. Affine highest weight categories and affine quasi-hereditary algebras. Proc. Lond. Math. Soc. (3), 110(4):841 882, 2015. [F16] R. Fujita. Tilting modules of affine quasi-hereditary algebras. arxiv:1610.02621. rfujita@math.kyoto-u.ac.jp
48, 2017 2 100,.,..,.,.,.,.. ([1]). L.,,.. ([3]),..,., n = 1, n = 2 [2]. 0.1. n. ρ GL n (C). f ρ, f, GL n (C) ρ 0 ρ = ρ 0 det. det GL n (C). [1] E. Freitag. Eine Verschwindungssatz für automorphe Formen zur Siegelschen Modulgruppe. Math. Zeitschrift 165, (1979), p. 11 18. [2] Ameya Pitale, Abhishek Saha, and Ralf Schmidt. Lowest weight modules of Sp 4 (R) and nearly holomorphic Siegel modular forms (expanded version). arxiv:1501.00524. [3] G.Shimura. Arithmeticity in the theory of automorphic forms, volume 82 of Mathematical survers and Monograghs. American Mathematical Society, Providence, RI, 2000. horinaga@math.kyoto-u.ac.jp
49, 2017 2 Q X f : X X f ( ) δ f ( ) X P f f P α f (P ) ( f well-definedness cf. (1)) δ f f α f (P ) P 0.1 (Kawaguchi-Silverman). X Weil h X 1 (1) α f (P ) = lim n h X (f n (P )) 1/n (2) P {f n (P ) n 0} X Zariski α f (P ) = δ f. α f (P ) = lim sup n h X (f n (P )) 1/n δ f [2]. f : X X [3] ( f (1) [1] ) f : X X α f (P ) = δ f P P [3] () [1] Kawaguchi, S., Silverman, J. H., Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties, Trans. Amer. Math. Soc. 368 (2016), 5009 5035. [2] Matsuzawa, Y., On upper bounds of arithmetic degrees, preprint, 2016, arxiv:1606.00598v2. [3] Matsuzawa, Y., Sano, K., Shibata, T., Arithmetic degrees and dynamical degrees of endomorphisms on surfaces, preprint, 2017, arxiv:1701.04369v1. myohsuke@ms.u-tokyo.ac.jp
50, 2017 2 Tate y 2 = x(x 1)(x λ) (λ 0, 1) λ = 1 or λ 1 < 1 p p Bradley [1] Tate, John, Rigid analytic spaces. Invent. Math. 12(1971), 257 289. ryo-mkm@math.kyoto-u.ac.jp
51 Representations of Compact Lie Groups and Transversally Elliptic Operators, 2017 2 G Lie M G E M G C (E) E G C (E) (g ϕ)(x) := g[ϕ(g 1 x)](g G, x X, ϕ C (E)) P : C (E) C (E) 2 transversally elliptic 1) P G 2) P σ(p ) TG M\0 ( T G M := {v T M Y g v(ỹπ(v)) = 0} g G, Ỹ Y M ) M, E G Riemann G Hermite C (E), L 2 ϕ, ψ L 2 := M ϕ(x), ψ(x) Ex dm(x) G (C (E), L 2) 1) P λ P λ := {ϕ C (E) P ϕ = λϕ} G G P λ transversally elliptic operator P λ Shubin [2] Lie G Schwartz P, Sobolev transversally elliptic operator [1] J. Roe, Elliptic operators, topology and asymptotic methods, second edition, Chapman Hall/CRC Research, (1998), 95-117 [2] M. A. Shubin, Spectral properties and the spectrum distribution function of a transversally elliptic operators, P lenum P ublishing Corporation, (1984), 406-422 mmasahiro0408@gmail.com
52 Nahm M2, 2017 2 (X, g X ) 4. X Hermite (V, h, A) F (A) L 2, ASD F (A) = F (A), (V, h, A) X L 2 -., (Y, g Y ) 3, Z Y Y., Y \ Z Hermite ( ˆV, ĥ, Â) Hermite End( ˆV ) ˆΦ ( ˆV, ĥ, Â, ˆΦ) Bogomolny F (Â) = Â (ˆΦ), p Z weight k Z rank( ˆV ), ˆΦ k, ( ˆV, ĥ, Â, ˆΦ) Z Dirac Y., 3 T 3 := R 3 /Λ R T 3 L 2 - (V, h, A), T 3 ˆT 3 Dirac ( ˆV, ĥ, Â, ˆΦ) Nahm., T 3 S 1 T 2, (V, h, A) ( ˆV, ĥ, Â, ˆΦ). 1 1. R T 3 L 2 - (V, h, A) Nahm (Γ ±, N ± ) = (Γ i,±, N i,± ) i=1,2,3., (Γ, N) = (Γ i, N i ) i=1,2,3 Nahm, Γ = (Γ i ) Hermite, N = (N i ) Γ j Hermite, N i = sgn(ijk)[n j, N k ] (, (ijk) (123) ). 2. (Γ ± ) Sing(A) ˆT 3, (V, h, A) ˆT 3 \Sing(A) ( ˆV, ĥ, Â, ˆΦ)., ( ˆV, ĥ, Â, ˆΦ) Sing(A) Dirac. 3. T 3 S 1 = R/Z T 2 S 1 T 2. (V, h, A) rank(v ) > 1 L 2 -, ( ˆV, ĥ, Â, ˆΦ) weight k Z rank( ˆV ), (N ± ) su(2). yoshino@kurims.kyoto-u.ac.jp