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Wooley's and Bourgain--Demeter--Guth's recent resolution of the Vinogradov mean value conjectures achieved a major program towards Waring's problem and dramatically improved previous results. In this talk I will survey modern problems motivated by ergodic theory and additive combinatorics, yet connected to Waring's problem. I will emphasize the arithmetic input which requires more precise information than is required classically. | Wooley's and Bourgain--Demeter--Guth's recent resolution of the Vinogradov mean value conjectures achieved a major program towards Waring's problem and dramatically improved previous results. In this talk I will survey modern problems motivated by ergodic theory and additive combinatorics, yet connected to Waring's problem. I will emphasize the arithmetic input which requires more precise information than is required classically. | ||
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+ | == June 09 == | ||
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+ | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Bobby Grizzard''' | ||
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+ | | bgcolor="#BCD2EE" align="center" | ''Slicing the stars'' | ||
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+ | Masser-Vaaler and Barroero have counted algebraic numbers and algebraic integers, respectively, of a given degree and bounded height, as the height bound tends to infinity. This is done by using results of Chern-Vaaler on counting lattice points in certain "star bodies." By counting carefully studying "slices" of the star bodies, we describe how to count algebraic units, algebraic numbers of a given norm, given trace, given norm and trace, and beyond! We also give improvements on and explicit versions of some of the bounds given by Masser-Vaaler and Barroero. This is joint work with Joseph Gunther (CUNY). | ||
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Revision as of 09:41, 6 June 2016
Return to NTS Spring 2016
Contents
Jan 28
Nigel Boston |
The 2-class tower of Q(√-5460) |
What is the liminf of the root-discriminants of all number fields? It's known (under GRH) to lie between 44.8 and 82.1. I'll explain how trying to tighten this range leads us to ask whether the 2-class tower of Q(√-5460) is finite or not and I'll describe how we find ways to address this question despite repeated combinatorial explosions in the calculation. This is joint work with Jiuya Wang. |
Feb 04
Shamgar Gurevich |
Low Dimensional Representations of Finite Classical Groups |
Group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of representations of the group. In particular, the representations of small dimension tend to contribute the largest terms to these sums, so a systematic knowledge of these small representations could lead to proofs of some of these facts. This talk will discuss a new method for systematically constructing the small representations of finite classical groups. I will explain the method with concrete examples and applications. This is part from a joint project with Roger Howe (Yale). |
Feb 11
Naser Talebi Zadeh |
Optimal Strong Approximation for Quadratic Forms |
Feb 18
Padmavathi Srinivasan |
Conductors and minimal discriminants of hyperelliptic curves with rational Weierstrass points |
Conductors and minimal discriminants are two measures of degeneracy of the singular fiber in a family of hyperelliptic curves. In the case of elliptic curves, the Ogg-Saito formula shows that (the negative of) the Artin conductor equals the minimal discriminant. In the case of genus two curves, equality no longer holds in general, but the two invariants are related by an inequality. We investigate the relation between these two invariants for hyperelliptic curves of arbitrary genus. |
Mar 10
Joseph Gunther |
Integral Points of Bounded Degree in Dynamical Orbits |
What should we mean by a random algebraic number? We'll examine this question in the context of determining the average number of integral points in dynamical orbits on the projective line, where we specifically don't work over a fixed number field. The tools will include variants of the Batyrev-Manin conjecture and a generalization of Siegel's theorem about integral points on curves. |
Mar 17
Jinhyun Park |
Algebraic cycles and crystalline cohomology |
After A. Weil formulated Weil conjectures for Hasse-Weil zeta functions of varieties over finite fields, A. Grothendieck postulated that a reasonable cohomology theory (a good Weil cohomology) and a good understanding of algebraic cycles (e.g. the standard conjectures?) would resolve the Weil conjectures. P. Deligne’s final resolution in 1970s of the Weil conjectures however came through l-adic étale cohomology, and without resorting to the theory of algebraic cycles. In this talk, we try to shed some lights this question again from the point of view of algebraic cycles, with the slogan “Algebraic cycles should know the arithmetic” in mind. More specifically, we discuss how one can describe the de Rham-Witt complexes in terms of algebraic cycles, thus giving a algebraic-cycle theoretic description of crystalline cohomology theory. |
Apr 01
Jacob Tsimerman |
Coming soon... |
This talk will now occur in the Algebraic Geometry Seminar, which occurs Friday Apr 01 at 2:25 PM in B113. |
Feb 04
Shamgar Gurevich |
Low Dimensional Representations of Finite Classical Groups |
Group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of representations of the group. In particular, the representations of small dimension tend to contribute the largest terms to these sums, so a systematic knowledge of these small representations could lead to proofs of some of these facts. This talk will discuss a new method for systematically constructing the small representations of finite classical groups. I will explain the method with concrete examples and applications. This is part from a joint project with Roger Howe (Yale). |
Apr 07
Jose Rodriguez |
Numerically computing Galois groups for applications |
The Galois/monodromy group of a family of equations (or of a geometric problem) is a subtle invariant that encodes the structure of the solutions. In this talk, we will use numerical algebraic geometry to compute Galois groups. Our algorithm computes a witness set for the critical points of our family of equations. With this witness set, we use homotopy continuation to construct a generating set for the Galois group. Examples from classical algebraic geometry, kinematics, and formation shape control will be presented to illustrate the method. A background in algebraic geometry or numerical analysis will not be assumed. Joint work with Jonathan Haeunstein and Frank Sottile. |
Apr 14
Eyal Goren |
Unitary Shimura varieties in positive characteristic |
I will report on joint work with Ehud DeShalit (Hebrew University). Unitary Shimura varieties are moduli spaces of abelian varieties with an action of a quadratic imaginary field. They form an interesting class of Shimura varieties and have been studied intensively as a test ground for the Langlands conjectures on automorphicity of L-functions, in the context of the local Langlands correspondence and in the context of Kudla's program. I will offer a rather detailed picture of such varieties associated to the unitary group GU(2,1), the so-called Picard modular surfaces. Many of the results extend to the case of signature (n,1), n>0. |
Apr 21
Raphael von Känel |
Integral points on moduli schemes |
We present explicit finiteness results for integral points on certain moduli schemes of abelian varieties of GL(2)-type. Parts of the results were obtained jointly with Benjamin Matschke or with Arno Kret. We also explain the strategy of proof which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity results. |
May 05
Mirela Çiperiani |
Divisibility questions for genus one curve |
Genus one curves with a fixed Jacobian can be viewed as elements of the Weil-Châtelet group. We will discuss divisibility questions within this group. This leads us to analyzing the divisibility properties of the Tate-Shafarevich group. There are two related questions: 1. Are the elements of the Tate-Shafarevich group divisible within the Weil-Châtelet group? (Cassels' question) 2. How does the Tate-Shafarevich group intersect the maximal divisible subgroup of the Weil-Châtelet group? (Bashmakov's question) We will discuss our answers to these questions. This is joint work with Jakob Stix. |
May 12
Kevin Hughes |
Beyond Waring's problem |
Wooley's and Bourgain--Demeter--Guth's recent resolution of the Vinogradov mean value conjectures achieved a major program towards Waring's problem and dramatically improved previous results. In this talk I will survey modern problems motivated by ergodic theory and additive combinatorics, yet connected to Waring's problem. I will emphasize the arithmetic input which requires more precise information than is required classically. |
June 09
Bobby Grizzard |
Slicing the stars |
Masser-Vaaler and Barroero have counted algebraic numbers and algebraic integers, respectively, of a given degree and bounded height, as the height bound tends to infinity. This is done by using results of Chern-Vaaler on counting lattice points in certain "star bodies." By counting carefully studying "slices" of the star bodies, we describe how to count algebraic units, algebraic numbers of a given norm, given trace, given norm and trace, and beyond! We also give improvements on and explicit versions of some of the bounds given by Masser-Vaaler and Barroero. This is joint work with Joseph Gunther (CUNY). |