| Date: | January 20, 2020 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Shamgar Gurevich, U. Wisc. |
| Title: | Harmonic Analysis on GL(n) over Finite Fields. |
| Abstract: | Harmonic Analysis on GL(n) over Finite Fields.
Abstract: There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio:
trace(ρ(g)) / dim(ρ),
for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.
Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank. Rank suggests a new organization of representations based on the very few "small" ones. This stands in contrast to Harish-Chandra's "philosophy of cusp forms", which is (since the 60s) the main organization principle, and is based on the (huge collection) of "large" representations.
This talk will discuss the notion of rank for the group GL(n) over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks.
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| Date: | January 27, 2020 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | A. Harper, TAMU |
| Title: | Border apolarity and matrix multiplication |
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| Date: | February 3, 2020 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | G. Pearlstein, TAMU |
| Title: | Differential Geometry of the Mixed Hodge Metric |
| Abstract: | We will discuss joint work with Chris Peters on canonical metrics attached to the period maps of families of complex algebraic varieties, with the goal of proving rigidity results for such families. |
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| Date: | February 7, 2020 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | Changho Han, University of Georgia |
| Title: | Counting hyperelliptic curves over global fields of bounded height via hyperelliptic fibrations |
| Abstract: | A hyperelliptic curve y^2=f(x) with coefficients in a global field (such as Q) comes equipped with a natural invariant called the height of the discriminant. Then, it is a natural question to find the asymptotic behavior of a function counting the number of reasonably behaving hyperelliptic curves with bounded height B. However, this problem turns out to be wide open if the base field is a number field. By jointly working with Junyong Park, we instead consider this problem for function fields, namely F_q(t) the generic point of a projective line over F_q. In this case, we can reinterpret hyperelliptic curves as hyperelliptic fibrations over P^1, allowing us to use both geometry of algebraic surfaces and arithmetic of the space of maps from P^1 into moduli spaces of hyperelliptic curves. By using this observation, I will illustrate the counting function over F_q(t) (which heuristically expect to look similar to that of number fields) and related results along the way (such as motive/point count of relevant moduli spaces and birational geometry involved). |
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| Date: | February 8, 2020 |
| Time: | 09:00am |
| Location: | BLOC 628 |
| Title: | Hodge Theory, Arithmetic and Moduli II. |
| Abstract: | See: www.math.tamu.edu/~gpearl/hodge-moduli.pdf |
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| Date: | February 9, 2020 |
| Time: | 09:00am |
| Location: | BLOC 628 |
| Title: | Hodge Theory, Arithmetic and Moduli II. |
| Abstract: | See: www.math.tamu.edu/~gpearl/hodge-moduli.pdf |
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| Date: | February 10, 2020 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Yordanka Kovacheva , University of Maryland |
| Title: | Explicit expression of the Suslin-Voevodsky's isomorphism for quasi-projective variety |
| Abstract: | The Suslin-Voevodsky isomorphism relates finite etale covers of
varieties with singular (Suslin) homology of varieties. I would give an
explicit way to calculate this isomorphism in terms of functions on curves
using Artin reciprocity and Kummer/Weil pairing. I will also give a
geometric interpretation of the Weil pairing and relate the Tame and Weil
symbols.
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| Date: | February 14, 2020 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | Vladimir Dragovic, UT Dallas |
| Title: | Periodic ellipsoidal billiards and Chebyshev polynomials on several intervals |
| Abstract: | A comprehensive study of periodic trajectories of the billiards within ellipsoids in the d-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between the periodic billiard trajectories and the extremal polynomials of the Chebyshev type on the systems of d intervals on the real line.
As a byproduct, for d = 2 a new proof of the monotonicity of the rotation number is obtained and the result is generalized for any d. The case study of trajectories of small periods T, d < T < 2d is given. In particular, it is proven that all d-periodic trajectories are contained in a coordinate-hyperplane and that for a given ellipsoid, there is a unique set of caustics which generates d + 1-periodic trajectories. A complete catalog of billiard trajectories with small periods is provided for d = 3. This talk is based on: V. Dragovic, M. Radnovic, Periodic ellipsoidal billiard trajectories and extremal poly-
nomials, arXiv 1804.02515, Comm. Math. Physics. https://doi.org/10.1007/s00220-
019-03552-y, 2019, Vol. 372, p. 183-211.
V. Dragovic, M. Radnovic, Caustics of Poncelet polygons and classical extremal polynomials, arXiv 1812.02907, Regular and Chaotic Dynamics, (2019), Vol. 24, No. 1, p.
1-35. A. Adabrah, V. Dragovic, M. Radnovic, Periodic billiards within conics in the Min-
kowski plane and Akhiezer polynomials, arXiv; 1906.0491, Regular and Chaotic Dy-
namics, No. 5, Vol. 24, 2019, p. 464-501. |
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| Date: | February 17, 2020 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Da Rong Cheng, University of Chicago |
| Title: | Bubble tree convergence of cross product preserving maps |
| Abstract: | We study a class of weakly conformal 3-harmonic maps, called Smith maps, which parametrize associative 3-folds in 7-manifolds equipped with G2-structures. These maps satisfy a first-order system of PDEs generalizing the Cauchy-Riemann equation for J-holomorphic curves, and we are interested in their bubbling phenomena. Specifically, we first prove an epsilon-regularity theorem for Smith maps in W^{1, 3}, and then explain how that combines with conformal invariance to yield bubble trees of Smith maps from sequences of such maps with uniformly bounded 3-energy. When the G2-structure is closed, we show that both 3-energy and homotopy are preserved in the bubble tree limit. The result can be viewed as an associative analogue of the bubble tree convergence theorem for J-holomorphic curves. This is joint work with Spiro Karigiannis and Jesse Madnick. |
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| Date: | February 28, 2020 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | Baris Coskunuzer, UT Dallas |
| Title: | Minimal Surfaces in Hyperbolic 3-Manifolds |
| Abstract: | In this talk, we will discuss the existence of smoothly embedded closed minimal surfaces in infinite volume hyperbolic 3-manifolds. The talk will be non-technical, and accessible to graduate students.
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