# UMASS Boston Mathematics Colloquium Series Spring 2016

## Meeting: 15:00 ROOM: M-02-404

17 February, Normal time 15:00

Dan Cristofaro-Gardiner Harvard
Title: Embedded contact homology and its applications

ABSTRACT:Embedded contact homology (ECH) is a tool for relating the topology of three and four dimensional manifolds to information about their contact and symplectic geometry. I will give an introduction to ECH, and discuss some of its applications; an emphasis will be recent joint work with Hind on using ECH to study high dimensional symplectic embedding problems.

2 March

Fulton Gonzalez Tufts
Title: Mean value Operators, Group Theory, and Integral Geometry

ABSTRACT: An old classical result says that a continuous function $$f$$ on $$\mathbb{R}^n$$ is harmonic if and only if its average value over any sphere equals its value at the center. In this talk, we'll explore various extensions of this principle, focusing primarily on symmetric spaces. We will also explore various integral transforms associated with mean value operators, and consider some applications to wave equations in one and many time variables.

9 March

Siu-Cheong Lau BU
Title: SYZ mirror symmetry and modular forms

ABSTRACT: Mirror symmetry is a deep duality found by string theorists between complex and symplectic geometries. In this talk I will introduce the SYZ constructive approach to mirror symmetry. Applying the mirror construction to certain local Calabi-Yau manifolds which correspond to the affine A-type Dynkin diagram, we shall see a surprising relation between certain symplectic invariants and modular forms. This is a joint work with Atsushi Kanazawa.

23 March

Tomoyuki Kakehi ,Okayama University, Japan
Title: Schroedinger equation on certain compact symmetric spaces.

ABSTRACT: As is well known in quantum physics, Schroedinger equation describes the motion of a particle. For this reason, the Schroedinger equation on $\mathbb{R}^n$ has been extensively studied by a lot of people. On the other hand, nothing is known about the Schroedinger equation on (compact) symmetric spaces. In this colloquium, I will talk about some strange property of the fundamental solution to the Schroedinger equation on certain compact symmetric spaces. Fisrt, I will start with the Schroedinger equation on $S^1$, and then generalize the result on $S^1$ to the case of odd dimensional spheres. Finally, I will mention what happens on general compact symmetric spaces.

30 March

Quang-Nhat Le , Brown
Title: Counting lattice points inside a $$d$$-dimensional polytope via Fourier analysis

ABSTRACT: Given a convex body $$B$$ which is embedded in a Euclidean space $$\mathbb{R}^d$$, we can ask how many lattice points are contained inside $$B$$, i.e. the number of points in the intersection of $$B$$ and the integer lattice $$\mathbb{Z}^d$$. Alternatively, we can count the lattice points inside $$B$$ with weights, which sometimes creates more nicely behaved lattice-point enumerating functions. The theory of lattice-point enumeration in convex bodies is a classical subject that has been studied by Minkowski, Hardy, Littlewood and many others. For polytopes, the work of Ehrhart, Macdonald and McMullen in 1960s and 1970s has revealed many curious properties of various weighted and unweighted lattice-point counts of integer polytopes such as polynomiality and reciprocity laws. The enumerative theory of lattice points inside polytopes have found far-reaching applications in many mathematical areas such as toric varieties, symplectic geometry, number theory, mirror symmetry, etc. The use of Fourier analysis in the theory of lattice-point enumeration has recently enjoyed a renaissance that was pioneered by Barvinok, Brion & Vergne, Diaz & Robins, Randol and others. In this talk, we will investigate Macdonald's solid-angle sum of a polytope, which is a weighted lattice-point count with solid-angle weights. We will employ the Poisson summation formula and other combinatorial techniques to convert the calculation of the solid-angle sum to the computation of the Fourier transform of the polytope. Classically, the theory is concerned with integer dilates of integer and rational polytopes, but our methods are applicable to arbitrary real dilates of any real polytope. This is joint work with Ricardo Diaz and Sinai Robins.

6 April

Egon Schulte Northeastern
Title: Chirality in Polytopes

ABSTRACT: A geometric structure is called chiral if it cannot be superimposed on its mirror images. We study chirality in polytopes, both abstract and geometric. Abstract polytopes are ranked combinatorial structures patterned after convex polytopes and their combinatorics. The most highly symmetric polytopes are regular or chiral. Regular polytopes have maximum reflexive combinatorial or geometric symmetry and their automorphism group or symmetry group is flag-transitive. Chiral polytopes only have maximum rotational combinatorial or geometric symmetry and their automorphism group or symmetry group has two flag-orbits represented by a pair of adjacent flags. Regular polytopes have been well-studied and much work has been done on their classification and their groups. By contrast, relatively little is known about chirality of polytopes. We describe some of the historical developments in this area and report about recent progress.

27 April

Sinai Robins ICERM, Brown University
Title: Covering Euclidean space by translations of a polytope

ABSTRACT: We study the problem of covering Euclidean space $$\mathbb{R}^d$$ by possibly overlapping translates of a convex body $$P$$, such that almost every point is covered exactly $$k$$ times, for a fixed integer $$k$$. Classical tilings by translations (which we call 1-tilings in this context) began with the work of the crystallographer Fedorov and with the work of Minkowski, who founded the Geometry of Numbers. Some 50 years later Venkov and McMullen gave a complete characterization of all convex objects that 1-tile Euclidean space. Today we know that $$k$$-tilings can be tackled by methods from Fourier analysis, though some of their aspects can also be studied using purely combinatorial means. For many of our results, there is both a combinatorial proof and a Harmonic analysis proof. For $$k$$ larger than 1, the collection of convex objects that $$k$$-tile is much wider than the collection of objects that 1-tile; So it's a more diverse subject with plenty (infinite families) of examples even in $$\mathbb{R}^2$$. There is currently no complete knowledge of the polytopes that $$k$$-tile in dimension 3 or larger, and even in 2 dimensions it is still challenging. We will cover both ancient'', as well as very recent, results concerning 1-tilings and other $$k$$-tilings. This is based on some joint work with Nick Gravin, Dmitry Shiryaev, and Mihalis Kolountzakis.

4 May

Yu-Shen Lin Stanford
Title: Counting Riemann Surfaces via Combinatorics

ABSTRACT: One of the central problems in enumerative geometry is counting Riemann surfaces with various conditions. Inspired by the Strominger-Yau-Zaslow conjecture, we will reduce the counting of holomorphic discs in certain Calabi-Yau surfaces to weighted counting of certain graphs, known as tropical dsics. The countings of holomrophic discs would satisfy the Kontsevich-Soibelman wall-crossing formula and are conjeturally related to the explicit formula for Ricci-flat metrics.