Švadlenka Karel Hecke Iwahori Whittaker

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2 Švadlenka Karel Hecke Iwahori Whittaker On the ranks of elliptic curves in towers of function fields Loop homology of some global quotient orbifolds Couette Lyapunov L K Beilinson SL n von Neumann Factorizable Markov Connes Embedding Problem Landauer

3 2.12 K-theory for C -algebras and its applications Witt exponential Hilbert SL(2) W Bridgeland Λ-adic Stark Systems Kan s horn-filling for simplices and cubes in HoTT On generic vanishing for pluricanonical bundles and its applications Depth functions of monomial ideals A Casselman-Shalika formula for the generalized Shalika model of SO 4n Fano CR Q-prime Lindelöf Horofunction boundary Ricci Myers

4 2.32 Stanley g Thom form in equivariant Čech-de Rham cohomology Representations of Compact Lie Groups and Transversally Elliptic Operators Nahm

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7 3 Švadlenka Karel, 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

8 4, ( ) ( ) (sort space) (pressing) (cosmic family) cosmic = universal family universe = Lecture Notes, Group Actions, Representations, and Quotient Families takamura@math.kyoto-u.ac.jp

9 5 Hecke Iwahori Whittaker, 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), [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), M. Nakasuji and H. Naruse, Yang-Baxter basis of Hecke algebra and Casselman s problem (extended abstract), DMTCS proc. BC, (2016), M. Reeder, p-adic Whittaker functions and vector bundles on flag manifolds, Comp. Math. 85, (1993), nakasuji@sophia.ac.jp

10 6, , (L- ). Base change lift GL(N).,.,. hiraga@math.kyoto-u.ac.jp

11 7, [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, [2] Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, [3] Yu Yasufuku, Integral points and relative sizes of coordinates of orbits in P N, Math. Z. 279 (2015), no. 3-4, yasufuku@math.cst.nihon-u.ac.jp

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15 11 On the ranks of elliptic curves in towers of function fields, K 0 y 2 = x 3 + ax + b a, b K, 4a b 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), P.Stiller,The picard numbers of elliptic surfaces with many symmetries,pacific J.Math.128(1987),no.1, yusuke@math.sci.hokudai.ac.jp

16 12 Loop homology of some global quotient orbifolds 1 M2, 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

17 13 Couette Lyapunov, 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) [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 [3] J. Jiménez & P. Moin. The minimal flow unit in near-wall turbulence, J. Fluid Mech. (1991), vol. 225, pp [4] J. M. Hamilton, J. Kim & F. Waleffe. Regeneration mechanisms of near-wall turbulence structure, J. Fluid Mech. (1995), vol. 287, pp [5] M. Inubushi, S. Takehiro & M. Yamada. Regeneration cycle and the covariant Lyapunov vectors in a minimal wall turbulence, Phys. Rev. E., 92, (2015). toshio@kurims.kyoto-u.ac.jp

18 14 L K3, 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, kito@math.kyoto-u.ac.jp

19 15 Beilinson, 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 [I] R. Ito, The Beilinson conjectures for CM elliptic curves via hypergeometric functions, The Ramanujan Journal, Springer. (to appear) thaya9ma@icloud.com

20 16 SL n -, 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), [H] N. Hitchin, Lie groups and Teichmüller space, Topology 31 (1992), no.3, [L] F. Labourie, Anosov flows,surface groups and curves in projective space, Invent. Math. 165(2006), no.1, y-inagaki@cr.math.sci.osaka-u.ac.jp

21 17 von Neumann Factorizable Markov Connes Embedding Problem M2, (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, [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, yuuki114@math.kyoto-u.ac.jp

22 18, 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 ( )

23 19 Landauer 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

24 20, 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 [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

25 21, 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, masaooi@ms.u-tokyo.ac.jp

26 22 K-theory for C -algebras and its applications, 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), [2] M. Rørdam, F. Larsen, and N. J. LAUSTSEN, An Introduction to K-Theory for C -Algebras, LMS Student Texts 49 (2000), kohta@ms.u-tokyo.ac.jp

27 23 M2, 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, [2] Mazur, B., Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, [3] Serre, J.-P., Propriétés galoisiennes des points d ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, odawara@math.kyoto-u.ac.jp

28 24 Witt exponential 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 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 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), kcb.c314271@gmail.com

29 25 Hilbert SL(2)- 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 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, , 846. Popov, V. L., Quasihomogeneous affine algebraic varieties of the group SL(2), Izv. Akad. Nauk SSSR Ser. Mat, 37, (1973), ce b282e@fuji.waseda.jp

30 26 W, 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 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: , [G1] N. Genra. Screening operators for W-algebras. arxiv: [G2] N. Genra. Wakimoto representations for W-algebras. in preparation. gnr@kurims.kyoto-u.ac.jp

31 27 Bridgeland, , 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]).,, X, C X ( 2)., C A 1 - Y., D b (X) σ, Y σ Y, Y Y., D b (Y) σ, Y σ-. [Bri07] T. Bridgeland. Stability conditions on triangulated categories. Ann. of Math. (2), 166(2): , [Tod13] Y. Toda. Stability conditions and extremal contractions. Math. Ann., 357(2): , [Tod14] Y. Toda. Stability conditions and birational geometry of projective surfaces. Compos. Math., 150(10): , koseki@ms.u-tokyo.ac.jp

32 28 1, 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/ afka9031@chiba-u.jp

33 29 Λ-adic Stark Systems, 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

34 30, 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 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

35 31 Kan s horn-filling for simplices and cubes in HoTT, (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

36 32, 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

37 33 On generic vanishing for pluricanonical bundles and its applications, 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), [Sim] C. Simpson, Subspaces of moduli spaces of rank one local systems, Ann. Sci. E.N.S. (4) 26 (1993), no. 3, tshibata@math.kyoto-u.ac.jp

38 34 Depth functions of monomial ideals, (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), [2] J. Herzog and T. Hibi, The depth of powers of an ideal, J. Alg. 291 (2005), t-suzuki@ist.osaka-u.ac.jp 1

39 35 A Casselman-Shalika formula for the generalized Shalika model of SO 4n M2, 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, p msuzuki@mah.kyoto-u.ac.jp

40 36 Fano 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 Fano 2. X( (G)) G [S] Y. Suyama, Toric Fano varieties associated to finite simple graphs, Tohoku Math. J., to appear; arxiv: d15san0w03@st.osaka-cu.ac.jp

41 37 CR Q-prime 1, 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): , Charles Fefferman. The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math., 26:1 65, [H14] Kengo Hirachi. Q-prime curvature on CR manifolds. Differential Geom. Appl., 33(suppl.): , ytake@ms.u-tokyo.ac.jp

42 38 Lindelöf, Minkowski. Rogers Swinnerton-Dyer [2]. K O m (a 1,..., a m ) K m (a 1,..., a m ) a 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) 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): , [2] K. Rogers. and H. P. F. Swinnerton-Dyer. The geometry of numbers over algebraic number fields. Trans. Amer. Math. Soc. 88, , takeda-w@math.kyoto-u.ac.jp

43 39 Horofunction boundary, 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: , Princeton University Press, Princeton, [2] C.Walsh. The action of a nilpotent group on its horofunction boundary has finite orbits, Groups, Geometry, and Dynamics,5(1): , [3] M.Develin. Cayley compactifications of Abelian groups, Annals of Combinatorics,6 (3-4): , kenshi-t@math.kyoto-u.ac.jp

44 40 Ricci Myers, 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) ( 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), [2] S. B. Myers, Duke Math. J. 8 (1941), h-tadano@cr.math.sci.osaka-u.ac.jp

45 41, 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 [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 stsuji@kurims.kyoto-u.ac.jp

46 42 Stanley g, 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 (g [1]) h = (h 0, h 1,, h d ) Z d+1 h d P h 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): , [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

47 43 3, U, V D V \ D = U, (V, D) U. C 3 (V, D), V 1,,, Peternell Schneider (cf. [F])., V. (cf. [K]) 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 (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 W P 4 2, C 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, (1993) Kishimoto, T.: Compactifications of contractible affine 3-folds into smooth Fano 3-folds with B 2 = 2. Math. Z. 251, (2005) nagaoka@am.ms.u-tokyo.ac.jp

48 44, 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, [2] B. Pittel, On likely solution of a stable marriage problem. The Annals of Applied Probability, 2, , paa nagare@cc.nara-wu.ac.jp

49 45, 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 n X = n 1.1 n [B] C. Birkar, On existence of log minimal models II, J. Reine Angew Math. 658 (2011), [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, hkenta@math.kyoto-u.ac.jp

50 46 Thom form in equivariant Čech-de Rham cohomology (Ko FUJISAWA) 1, 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, [2] Mathai, Varghese, and Daniel Quillen. Superconnections, Thom classes, and equivariant differential forms. Topology 25.1 (1986): [3] Suwa, T. Indices of vector fields and residues of holomorphic singular foliations. Hermann, Paris (1998). fujisawa1219@math.sci.hokudai.ac.jp

51 47, 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): , [K15] A. Kleshchev. Affine highest weight categories and affine quasi-hereditary algebras. Proc. Lond. Math. Soc. (3), 110(4): , [F16] R. Fujita. Tilting modules of affine quasi-hereditary algebras. arxiv: rfujita@math.kyoto-u.ac.jp

52 48, ,.,..,.,.,.,.. ([1]). L.,,.. ([3]),..,., n = 1, n = 2 [2] 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 [2] Ameya Pitale, Abhishek Saha, and Ralf Schmidt. Lowest weight modules of Sp 4 (R) and nearly holomorphic Siegel modular forms (expanded version). arxiv: [3] G.Shimura. Arithmeticity in the theory of automorphic forms, volume 82 of Mathematical survers and Monograghs. American Mathematical Society, Providence, RI, horinaga@math.kyoto-u.ac.jp

53 49, 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), [2] Matsuzawa, Y., On upper bounds of arithmetic degrees, preprint, 2016, arxiv: v2. [3] Matsuzawa, Y., Sano, K., Shibata, T., Arithmetic degrees and dynamical degrees of endomorphisms on surfaces, preprint, 2017, arxiv: v1. myohsuke@ms.u-tokyo.ac.jp

54 50, 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), ryo-mkm@math.kyoto-u.ac.jp

55 51 Representations of Compact Lie Groups and Transversally Elliptic Operators, 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), [2] M. A. Shubin, Spectral properties and the spectrum distribution function of a transversally elliptic operators, P lenum P ublishing Corporation, (1984), mmasahiro0408@gmail.com

56 52 Nahm M2, (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) 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