Published on Department of Mathematics (https://math.yale.edu)

Week of October 20, 2019

October 21, 2019
Group Actions and Dynamics	Topological Complexity of Manifolds of Nonpositive Curvature	Tsachik Gelander - Weizmann Institute	4:15pm -
Geometry, Symmetry and Physics	Affine Springer fibers and small quantum groups	Roman Bezrukavnikov - MIT	4:30pm -

October 22, 2019
Algebra and Number Theory Seminar	Independence of $l$ for Frobenius conjugacy classes attached to abelian varieties	Rong Zhou - Yale University	4:00pm - LOM 205
Geometry & Topology	Tessellations from long geodesics on surfaces	Jenya Sapir - Binghamton	4:15pm - DL 431
YUMS	On the congruent number problem	Yifeng Liu - Yale University	6:00pm - LOM 206

October 23, 2019
Undergraduate Seminar	Putnam seminar		6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215

October 24, 2019
Algebraic and Tropical Geometry	Singularities of generic projection hypersurfaces in arbitrary characteristic	Takumi Murayama - Princeton University	3:00pm - Rosenfeld Hall (109 Grove St)
Analysis	Analysis-Combinatorics Joint Seminar: Sums of random polynomials with independent roots	Sean O'Rourke - University of Colorado, Boulder	3:00pm - AKW 000
Combinatorics Seminar	The size Ramsey number of paths	Deepak Bal - Montclair University	4:00pm - DL 431
Algebraic and Tropical Geometry	Compactification for cluster varieties without frozen variables of finite type	Man-Wai (Mandy) Cheung - Harvard University	4:15pm - LOM 214

October 25, 2019
Algebra and Number Theory Seminar	Entropy, Optimization, and Symmetry	Nisheeth Vishnoi -	11:00am - LOM 107

October 26, 2019
	Virginia Tech Regional Math Contest		9:00am to 11:30am - LOM

Abstracts

Week of October 20, 2019

October 21, 2019
Group Actions and Dynamics	Topological Complexity of Manifolds of Nonpositive Curvature	4:15pm -	We consider complete analytic Riemannian manifolds of bounded nonpositive curvature. These include locally symmetric and, in particular, arithmetic manifolds, the study of which is strongly related to the theory of lattices and arithmetic groups. The complexity of such manifolds is controlled by the volume. This phenomenon can be measured in terms of the growth of topological, geometric, algebraic, arithmetic and representation theoretic invariants, such as Betti numbers and torsion, optimal presentations of Γ=π1(M), the minimal size of a triangulation as well as invariant related to the Plancherel measure associated to L_2(G/Γ). Another line of problems concern with the number of manifolds of a certain type and bounded volume. In the talk I will give an overview of the theory.
Geometry, Symmetry and Physics	Affine Springer fibers and small quantum groups	4:30pm -	“I will describe a work in progress (with Michael McBreen) where we identify a regular block in the category of small quantum group modules with a category that can be described as the category of microlocal sheaves on a certain affine Springer fiber. Here the specific technical incarnation of the “microlocal sheaves” concept is based on presenting the affine Springer fiber as a Hamiltonian reduction of a cotangent bundle. This is heuristically (though at present not technically) related to a conjecture of Shan-Vasserot and myself where this Springer fiber is used to describe the center of the small quantum group; the conjecture was partly motivated by a work Lachowska and Qi on a problem by I. Frenkel. Time permitting I will also describe some ideas from a joint project with Pablo Boixeda Alvarez, Peng Shan and Eric Vasserot aimed at proving that conjecture.

October 22, 2019
Algebra and Number Theory Seminar	Independence of $l$ for Frobenius conjugacy classes attached to abelian varieties	4:00pm - LOM 205	Abstract: Let $A$ be an abelian variety over a number field $E\subset \mathbb{C}$. For $l$ a prime, a result of Deligne implies that upon replacing $E$ by a finite extension, we obtain a representation $\rho_l:\mathrm{Gal}(\overline{E}/E)\rightarrow G(\mathbb{Q}_l)$ where $G$ is the Mumford–Tate group of $A$. For $v\nmid l$ a prime of $E$ where $A$ has good reduction, we show that the conjugacy class of $\rho_l(\mathrm{Frob}_v)$ in $G(\mathbb{Q}_l)$ is defined over $\mathbb{Q}$ and is independent of $l$. This is joint work with Mark Kisin.
Geometry & Topology	Tessellations from long geodesics on surfaces	4:15pm - DL 431	I will talk about joint work of Athreya, Lalley, Wroten and myself. Given a hyperbolic surface S, a typical long geodesic arc will divide the surface into many polygons. We give statistics for the geometry of a typical tessellation. Along the way, we look at how very long geodesic arcs behave in very small balls on the surface.
YUMS	On the congruent number problem	6:00pm - LOM 206	Abstract: A natural integer is a congruent number if it is the area of a right triangle with three rational number sides. It is a problem recorded in an Arab manuscript of the 10th century to find all congruent numbers, which is still open today. In this talk, we will do some experiments on finding congruent numbers, prove that 1 is not a congruent number, and finally speculate a pattern of these numbers. We will also talk about the history and progress of this problem, and its connection with modern number theory.

October 23, 2019
Undergraduate Seminar	Putnam seminar	6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215 , 6:30pm - LOM 214/215	Weekly event for interested students to practice competition math problems in a relaxed setting. Meetings are held every Wednesday from 6:30pm to 8 in LOM 215. All are welcome to attend. Contact Pat Devlin to find out more.

October 24, 2019
Algebraic and Tropical Geometry	Singularities of generic projection hypersurfaces in arbitrary characteristic	3:00pm - Rosenfeld Hall (109 Grove St)	Classically, it is known that every algebraic variety is birational to a hypersurface in some projective or affine space. Using generic linear projections, Doherty proved that over the complex numbers, this hypersurface can be taken to have at worst semi-log canonical singularities in dimensions up to five. This extends classically known results for curves and surfaces. We present a positive-characteristic analogue of Doherty's theorem, by showing that in the same dimensions, generic projection hypersurfaces in positive characteristic are F-pure. This proves cases of a conjecture of Bombieri, Andreotti, and Holm. To show our result, we study F-injective singularities, which are the analogue of Du Bois singularities in positive characteristic, and their behavior under flat morphisms. This work is joint with Rankeya Datta.
Analysis	Analysis-Combinatorics Joint Seminar: Sums of random polynomials with independent roots	3:00pm - AKW 000	Abstract: We consider the sum of independent random polynomials as their degrees tend to infinity. Namely, let $p$ and $q$ be two independent random polynomials of degree $n$, whose roots are chosen independently in the complex plane. We compute the limiting distribution for the zeros of the sum $p+q$ as $n$ tends to infinity by analyzing its logarithmic potential. We will also discuss a generalization for more than two polynomials. These results can be viewed as describing a version of the free additive convolution from free probability theory for zeros of polynomials. The talk is based on joint work with Tulasi Ram Reddy.
Combinatorics Seminar	The size Ramsey number of paths	4:00pm - DL 431	Given a graph $H$, let $\hat{R}(H)$ be the minimum $m$ such that there exists a graph $G$ with $m$ edges such that in every $2$-coloring of the edges $G$, there is a monochromatic copy of $H$. To prove that $\hat{R}(P_n)>m$, one must show that every graph on $m$ edges can be $2$-colored such that every monochromatic path has order less than $n$. We show that $\hat{R}{P_n} > (3.75 - o(1))n$ thereby improving the previous best-known lower bound of $(2.5 - o(1))n$ due to Dudek and Pralat. We also discuss some results concerning the $r$-color version of the problem. This is joint work with Louis DeBiasio.
Algebraic and Tropical Geometry	Compactification for cluster varieties without frozen variables of finite type	4:15pm - LOM 214	Cluster varieties are blow up of toric varieties. They come in pairs (A,X), with A and X built from dual tori. Compactifications of A, studied by Gross, Hacking, Keel, and Kontsevich, generalize the polytope construction of toric varieties while the compactifications of X, studied by Fock and Goncharov, generalize the fan construction. The conjecture is that the A and the X cluster varieties are mirrors to each other. Together with Tim Magee, we have shown that there exists a positive polytope for the type A cluster varieties which give us a hint to the Batyrev-Borisov construction.

October 25, 2019
Algebra and Number Theory Seminar	Entropy, Optimization, and Symmetry	11:00am - LOM 107	Abstract: The principle of maximum entropy states that among all possible ways to write a given point as a convex combination of the domain, pick the one that maximizes the entropy of the probability distribution corresponding to the convex combination. This talk will start by reviewing the computational and structural properties of max-entropy distributions over discrete domains, and then move on to defining and studying them over continuous manifolds with applications to quantum mechanics, semidefinite programming, and statistics. The latter part draws from mathematical physics, in particular from the theory of Lie groups. Based on joint works with Mohit Singh, Damian Straszak, and Jonathan Leake.

October 26, 2019
	Virginia Tech Regional Math Contest	9:00am to 11:30am - LOM	The annual VTRMC event, which is an informal 2.5 hour-long undergraduate math contest. Contact Pat Devlin to register or find out more. Interested students would also benefit from the weekly Putnam seminar, held every Wednesday night. More information available at the contest website: http://intranet.math.vt.edu/people/plinnell/Vtregional/