Robert Hough
SUNY: Stony Brook University, Mathematics, Faculty Member
Alexander Carney (University of Rochester) Torsion points in families and big adelic line bundles Given a family of abelian varieties A → T over a quasiprojective curve with a point P on the generic fiber, I’ll show that the Néron-Tate... more
Alexander Carney (University of Rochester) Torsion points in families and big adelic line bundles Given a family of abelian varieties A → T over a quasiprojective curve with a point P on the generic fiber, I’ll show that the Néron-Tate height of Pt on each fiber At is a height on T given by a metrized line bundle MP . I’ll then show that this makes a conjecture of Zhang on specializations of small height equivalent to the bigness of MP . Huayang Chen (University of Houston) Expected value of the smallest denominator in a random interval of fixed radius We compute the probability mass function of the random variable which returns the smallest denominator of a reduced fraction in a randomly chosen real interval of radius δ. We prove that the expected value of the smallest denominator is asymptotic, as δ → 0, to (8 √ 2/π2)δ−1/2. Pavel Čoupek (Purdue University) Ramification bounds for mod p étale cohomology via prismatic cohomology
Research Interests:
Enumerating integral orbits in prehomogeneous vector spaces plays an important role in arithmetic statistics. We describe a method of proving subconvexity of the zeta function enumerating the integral orbits, illustrated by proving a... more
Enumerating integral orbits in prehomogeneous vector spaces plays an important role in arithmetic statistics. We describe a method of proving subconvexity of the zeta function enumerating the integral orbits, illustrated by proving a subconvex estimate for the Shintani ζ function enumerating class numbers of binary cubic forms. The method also obtains a subconvex estimate for the zeta function twisted by a Maass cusp form.
Research Interests:
We use the method of Shintani, as developed by Taniguchi and Thorne, to prove the joint cuspidal equidistribution of the shape of quartic fields paired with the shape of its cubic resolvent, when the fields are ordered by discriminant.... more
We use the method of Shintani, as developed by Taniguchi and Thorne, to prove the joint cuspidal equidistribution of the shape of quartic fields paired with the shape of its cubic resolvent, when the fields are ordered by discriminant. Our estimate saves a small power in the corresponding Weyl sums.
Research Interests:
We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues.... more
We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues. We also identify the poles and residues of the zeta function when restricted to irreducible binary cubic forms. This zeta function can be used to prove the equidistribution of the lattice shape of cubic rings.
Research Interests:
Enumerating integral orbits in prehomogeneous vector spaces plays an important role in arithmetic statistics. We describe a method of proving subconvexity of the zeta function enumerating the integral orbits, illustrated by proving a... more
Enumerating integral orbits in prehomogeneous vector spaces plays an important role in arithmetic statistics. We describe a method of proving subconvexity of the zeta function enumerating the integral orbits, illustrated by proving a subconvex estimate for the Shintani $\zeta$ function enumerating class numbers of binary cubic forms. The method also obtains a subconvex estimate for the zeta function twisted by a Maass cusp form.
Research Interests:
We use the method of Shintani, as developed by Taniguchi and Thorne, to prove the joint cuspidal equidistribution of the shape of quartic fields paired with the shape of its cubic resolvent, when the fields are ordered by discriminant.... more
We use the method of Shintani, as developed by Taniguchi and Thorne, to prove the joint cuspidal equidistribution of the shape of quartic fields paired with the shape of its cubic resolvent, when the fields are ordered by discriminant. Our estimate saves a small power in the corresponding Weyl sums.
Research Interests:
We introduce a new method for proving central limit theorems for random walk on nilpotent groups. The method is illustrated in a local central limit theorem on the Heisenberg group, weakening the necessary conditions on the driving... more
We introduce a new method for proving central limit theorems for random walk on nilpotent groups. The method is illustrated in a local central limit theorem on the Heisenberg group, weakening the necessary conditions on the driving measure. As a second illustration, the method is used to study walks on the $n\times n$ uni-upper triangular group with entries taken modulo $p$. The method allows sharp answers to the behavior of individual coordinates: coordinates immediately above the diagonal require order $p^2$ steps for randomness, coordinates on the second diagonal require order $p$ steps; coordinates on the $k$th diagonal require order $p^{\frac{2}{k}}$ steps.
Research Interests:
Sandpile dynamics are considered on graphs constructed from periodic plane and space tilings by assigning a growing piece of the tiling either torus or open boundary conditions. A method of computing the Green's function and spectral... more
Sandpile dynamics are considered on graphs constructed from periodic plane and space tilings by assigning a growing piece of the tiling either torus or open boundary conditions. A method of computing the Green's function and spectral gap is given for various tilings and a cut-off phenomenon in the mixing is demonstrated under general conditions. It is shown that the boundary condition does not affect the mixing time for planar tilings, but that the mixing time is altered for the $\mathrm{D4}$ lattice in dimension 4. For all sufficiently large $d$, the boundary condition does not affect the mixing of the cubic lattice $\mathbb{Z}^d$.
Research Interests:
We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues.... more
We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues. We also identify the poles and residues of the zeta function when restricted to irreducible binary cubic forms. This zeta function can be used to prove the equidistribution of the lattice shape of cubic rings.
Research Interests:
In their previous work, the authors studied the abelian sandpile model on graphs constructed from a growing piece of a plane or space tiling, given periodic or open boundary conditions, and identified spectral parameters which govern the... more
In their previous work, the authors studied the abelian sandpile model on graphs constructed from a growing piece of a plane or space tiling, given periodic or open boundary conditions, and identified spectral parameters which govern the asymptotic spectral gap and asymptotic mixing time. This paper gives a general method of determining the spectral parameters either computationally or asymptotically, and determines the spectral parameters in specific examples.
Research Interests:
Research Interests:
Research Interests:
A generalized `$15$ puzzle' consists of an $n \times n$ numbered grid, with one missing number. A move in the game switches the position of the empty square with the position of one of its neighbors. We solve Diaconis' `15 puzzle... more
A generalized `$15$ puzzle' consists of an $n \times n$ numbered grid, with one missing number. A move in the game switches the position of the empty square with the position of one of its neighbors. We solve Diaconis' `15 puzzle problem' by proving that the asymptotic total variation mixing time of the board is at least order $ n^4 $ when the board is given periodic boundary conditions and when random moves are made. We demonstrate that for any $f(n) \to \infty$ with $n$, the number of fixed points after $n^4 f(n)$ moves converges to a Poisson distribution of parameter 1. The order of total variation mixing time for this convergence is $n^4$ without cut-off. We also prove an upper bound of order $n^{4 }\log n$ for the total variation mixing time.
Research Interests:
We consider an analogue of the Kac random walk on the special orthogonal group, in which at each step a random rotation is performed in a randomly chosen 2-plane of R^N. We obtain sharp asymptotics for the rates of convergence in total... more
We consider an analogue of the Kac random walk on the special orthogonal group, in which at each step a random rotation is performed in a randomly chosen 2-plane of R^N. We obtain sharp asymptotics for the rates of convergence in total variance distance, establishing a cutoff phenomenon; in the special case where the rotation angle is fixed this confirms a conjecture of Rosenthal. Under mild conditions we also establish a cutoff for convergence of the walk to stationarity under the L^2 norm. Depending on the distribution of the randomly chosen angle of rotation, several surprising features emerge. For instance, it is sometimes the case that the mixing times differ in the total variation and L^2 norms. Our analysis hinges on a new method of estimating the characters of the orthogonal group, using a contour formula and saddle point estimates.
Research Interests:
Research Interests:
We consider the size of large character sums, proving new lower bounds for Δ(N,q)=sup χ≠χ0 mod q∣∑ n<Nχ(n)∣ in almost all ranges of N. The proofs use the resonance method and saddle point analysis.
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Research Interests:
Research Interests:
Research Interests:
We consider an analogue of the Kac random walk on the special orthogonal group, in which at each step a random rotation is performed in a randomly chosen 2-plane of R^N. We obtain sharp asymptotics for the rates of convergence in total... more
We consider an analogue of the Kac random walk on the special orthogonal group, in which at each step a random rotation is performed in a randomly chosen 2-plane of R^N. We obtain sharp asymptotics for the rates of convergence in total variance distance, establishing a cutoff phenomenon; in the special case where the rotation angle is fixed this confirms a conjecture of Rosenthal. Under mild conditions we also establish a cutoff for convergence of the walk to stationarity under the L^2 norm. Depending on the distribution of the randomly chosen angle of rotation, several surprising features emerge. For instance, it is sometimes the case that the mixing times differ in the total variation and L^2 norms. Our analysis hinges on a new method of estimating the characters of the orthogonal group, using a contour formula and saddle point estimates.