9/6/19 Alicia Lamarche, Algebra seminar 1. Can decompose a birational map as a finite sequence of blow-ups and blow-downs on smooth subvarieties. 2. How in general would one prove that X is rational, without constructing a birational map or an isomorphism of function fields? 3. The two definitions of "rational" in AG are apparently implicitly related. How? 9/19/19 Michael Filaseta, Number Theory seminar 1. The irreducible reciprocal factors of f(x)x^n + g(x) can be described independently of n. 2. Sawin, Schusterman, and Stoll describe the notion of a "robust" factorization of a polynomial. *Roughly* this involves minimalities of the heights of the two factors. 3. A trick of Ljunggren is useful in checking irreducibility. If the non-reciprocal part of a polynomial is reducible, then there is a clever trick which will detect this. 9/20/19 Robert Vandermolen, Algebra Seminar 1. Derived categories were introduced as a note keeping device for algebraic homology. 2. Quotients of varieties by group actions can be shown to not be varieties by checking Hausdorffness. (I was a bit confused by this point, would like to understand better.) 3. "Walls and chambers" (in derived categories) are related to the question of which line bundles describe which open sets in a GIT quotient. 9/21-22/19 Palmetto Number Theory Series, Charlotte NC Alex Kontorovich, Conjectures about Fibonacci numbers. 1. The idea: compute (conjecturally) the liminf of the quotient (# prime factors of F_n L_n w/multiplicity)/log n. Expected to be nonzero. 2. Take a "random" set of product of finitely many exponentially growing integers. Should this predict the same result? 3. The "affine sieve" connects this all to algebraic geometry. Would be interesting to better understand how. Alex Dunn, Twisted moments of 1/2-integral weight L-functions 1. These modular forms are well behaved after twisting by Dirichlet characters, although the level goes up. 2. The Mellin transform of a modular form L-function spits out the completed L-function. 3. Question of Hoffstein: Does Lindelof persist (in either t or q aspect) in contexts where GRH is not expected to hold? Kalani Thalagoda, Group presentation of GL(2, O_F) 1. GL(2, O_F), where F is an imaginary quadratic field, has an action on H^3 with a nice tesselation. 2. Group presentations can be described in terms of combinatorial data -- the CW structure of this tesselation. 3. "Combinatorial paths" through the tesselation let you write down relations explicitly. Zack Tripp, Multiplicities of zeroes of zeta functions 1. GRH pair correlation estimate for Dedekind zeta function of an abelian number field. 2. Bound from below the number of distinct zeroes for the same zeta function. 3. Montgomery proved an interesting offbeat variation of the explicit formula. Eun Hye Lee, Multiple Dirichlet series associated to binary cubic forms 1. Li Mei Lim: Studied a MFS associated to the space of ternary quadratic forms, with respect to the action of a parabolic subgroup of SL(3, Z). 2. Look at the action of [1,1;0,1] on binary cubic forms -- four "semi-invariants". 3. Main theorem: construct a 2-variable MDS using two of the semi-invariants, and give a meromorphic continuation to C^2. Sarah Peluse, Bounds for sets with no arithmetic progressions. 1. Multidimensional Szemeredi: let S \subset \Z^d finite. If [A] \subset [N^d] has no homothetic copies of S (i.e. a+bS) then |A|= o_S(N). 2. A finite field result: if a subset of F_p doesn't contain any triples x, x+y, x+y^2, then it is of size at most O(p^(14/15)). 3. Z is much harder than F_p because the algebraic geometry goes away. 4. The explicit formula for the L^4 norm of the Fourier transform in terms of Gowers norms: could this be useful in reverse, e.g. we understand the other side and want to bound the Fourier transforms -- for example in the setting of prehomogeneous vector spaces? Shenhui Liu, GL(3) analogue of Selberg's S(t) result 1. arg zeta(1/2 + it) closely related to N(t). 2. Average of S(t, \chi)^{2n} over \chi is independent of t, up to an error term. 3. Hecke-Maass forms for SL(3, Z) are known to have nice L-functions. Amita Malik - Zeroes of derivatives of completed L-functions 1. On RH, \xi^{(m)} = 0 --> Re(s) = 1/2. 2. Conrey: can prove a positive proportion of such are on the critical line. 3. Can get equidistribution for x\gamma_i (mod 1) for any real x, where \gamma_i runs over the imaginary parts of zeroes of zeta or of \zeta^{(m)}. Fatma Cicek, Selberg's central limit theorem 1. Selberg central limit theorems for |log(\zeta(1/2 + it))| and arg zeta(1/2 + it) -- *almost* identical, but with slightly different error terms. 2. log |\zeta'| is easier to understand than arg \zeta'. 3. There are results concerning differences of zeroes between different L(s, chi), and between L(s, chi) and zeta. Shabnam Akhtari, Representations of integers by binary forms. 1. Thue: There are finitely many solutions to F(x, y) = n with F an irred binary n-ic form with n \geq 3. The proof is easy, granting results on Diophantine approximation. 2. Bombieri-Schmidt: At most 215n solutions to F(x,y) = \pm 1 for large degree n. Improved substantially by the speaker. 3. Question (asked by Shabnam): why is 1 important here? Scott Ahlgren: Kloostermann sums, Maass forms, and partitions 1. Character sum identities allow you to go back and forth between different explicit formulas for p(n). 2. There are interesting and apparently nontrivial identities for individual Kloostermann sums. 3. It's conjectured that \sum_{c \leq X} A_c(n) / c \ll (nX)^{\epsilon}. Can one (conjecturally!) do still better? Does the infinite sum converge? Steven Jin, L_1 norms of exponential sums 1. Lower bounds for \int_0^1 | \sum_{n = 1}^N a_n e(nx) | dx have gotten attention (in complete generality). 2. Assuming that a_n is supported on squarefree integers allows one to do much better. 3. Lower bounds can be obtained from upper bounds, and in particular from Linnik's large sieve. Anup Dixit - Classification for L-functions 1. Beurling 1951. A Dirichlet series satisfying certain analytic properties of zeta(s) has to be zeta(s). 2. Theorem: given analytic data including the residue at s=1, can determine the zeta function up to a list of bounded size. 3. What bound would this give on the number of Dedekind zeta functions of number fields of bounded discriminant? Tom Wright, A conditional density for Carmichael numbers 1. Heath-Brown's conjecture on improved Linnik can be used as input to other results (e.g. involving Carmichael numbers) 2. Erdos: to look for Carmichael numbers, look for good LCMs. 3. A trick appearing in the guts of the proof: pick q's such that the q-1 have lots of common factors. Debanjana Kundu, Iwasawa theory for fine Selmer groups 1. Fine Selmer groups are subgroups of classical Selmer groups, with extra conditions at primes over p to make them act more like ordinary class groups. 2. If E/F has rank 0 and finite Sha, then for a density 1 subset of primes, the p-power part of Sel(E/F_cyc) is trviail. 3. The speaker was able to get some equivalences between a elliptic curve Selmer group conjecture and a classical one. 9/26: Joe Foster, Number Theory Seminar -- What do Michael's students do? 1. Cohn: If p is prime, the polynomial which represents it with x=10 is irreducible. 2. In generalizations, if the polynomial has a factor, it is likely to be an integer shift of a cyclotomic polynomial. 3. Recurrence relations play a role in the theory. 9/27: Lea Beneish, Modules for certain subgroups of M_24 1. Gives a natural interpretation of the coefficients of E_2(z) as dimensions of M_24-modules. 2. New proof of Mazur result: can find a form congruent to (1/24)*(E_2(Nz) - E_2(z)) modulo something appropriate. 3. "Virtual modules": like direct sums, but with negative multiplicities allowed. 9/27: Jackson Morrow. Non-Archimedean entire curves in closed subvarieties of semi-abelian varieties 1. Look at holomorphic maps C -> X(C). Interesting to *not* require them to be maps of varieties. 2. Abelian varieties are required to be projective. Just being commutative is not enough. 3. "Semi-norms" allow |x| = 0 even for nonzero x. 10/3: Jeremiah Southwick, The Infinitude of Weak Primes 1. There are infinitely many "weak primes": if you change any digit, get a composite number. 2. For all (a, j) \neq (2, 6), there is a prime p so that the order of a (mod p) is j. 3. Linnik's theorem proves quite useful here. The exact value of the exponent doesn't matter.