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Vasily Dolgushev, Temple University
In the second talk of this series, I will give more examples of operads. I will also talk about one of the central objects of this series, the operad of parenthesized braids PaB. This is an operad in the category of groupoid and it is "assembled from" Artin's braid groups. This operad was introduced by Dmitry Tamarkin in 1998 and a very similar object was introduced by Dror Bar-Natan in 1996.
Wissam Raji, American University of Beirut
Modular forms are analytic functions defined on the upper half-plane with a specific transformation law under elements of the full modular group $\mathrm{SL}_2(\mathbb{Z})$. In this talk, we give different motivations to the theory and then give an explicit introduction about the main definitions in the theory of modular forms. Interesting series called $L$-series, constructed using the Fourier coefficients of modular forms have important connections to elliptic curves. We show that, on average, the $L$-functions of cuspidal Hilbert modular forms (a generalization of classical modular forms) with sufficiently large weight $k$ do not vanish on the line segments $ \Im(s)=t_0, \ \Re(s) \in (\frac{k-1}{2},\frac{k}{2}-\epsilon)\cup (\frac{k}{2}+\epsilon,\frac{k+1}{2})$.
Izabella Stuhl, Pennsylvania State University
Do hard disks in the plane admit a unique Gibbs measure at high density? This is one of the outstanding open problems of statistical mechanics, and it seems natural to approach it by requiring the centers to lie in a fine lattice; equivalently, we may fix the lattice, but let the Euclidean diameter $D$ of the hard disks tend to infinity. Unlike most models in statistical physics, we find non-universality and connections to number theory, with different new phenomena arising in the triangular lattice $\mathbb{A}_2$, the square lattice $\mathbb{Z}^2$ and the hexagonal tiling $\mathbb{H}_2$.
In particular, number-theoretic properties of the exclusion diameter $D$ turn out to be important. We analyze high-density hard-core Gibbs measures via Pirogov-Sinai theory. The first step is to identify periodic ground states, i.e., maximal density disk configurations which cannot be locally 'improved'. A key finding is that only certain 'dominant' ground states, which we determine, generate nearby Gibbs measures. Another important ingredient is the Peierls bound separating ground states from other admissible configurations.
Answers are provided in terms of Eisenstein primes for $\mathbb{A}_2$ and norm equations in the ring $\mathbb{Z}[\sqrt{3}]$ for $\mathbb{Z}^2$. The number of high-density hard-core Gibbs measures grows indefinitely with $D$ but non-monotonically. In $\mathbb{Z}^2$ we analyze the phenomenon of 'sliding' and show it is rare.
This is a joint work with A. Mazel and Y. Suhov.
Tarik Aougab, Haverford College
Given two finite degree regular covers (not necessarily of the same degree) Y, Z of a surface S, suppose that for any closed curve gamma on S, gamma lifts to a simple closed curve on Y if and only if it does to Z. We prove that Y and Z must be equivalent covers. The proof uses some Teichmuller theory and the curve complex. This represents joint work with Max Lahn, Marissa Loving, and Sunny Yang Xiao.
Pak-Wing Fok, University of Delaware
Dan Margalit, Georgia Tech
PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn About 100 years ago, Artin showed that any homomorphism from the braid group \(B_n\) to the symmetric group \(S_n\) is either cyclic or conjugate to the standard homomorphism. Much more recently, Castel showed that any endomorphism of \(B_n\) is either cyclic or conjugate to (a transvection of) the identity map. With Lei Chen and Kevin Kordek, we extend Castel's result by showing that any homomorphism from \(B_n\) to \(B_{2n}\) is either cyclic or conjugate to (a transvection of) one of the standard maps.
In the morning background talk (9:30am in room 527) I will review braid groups, mapping class groups, canonical reduction systems, and totally symmetric sets.
Kyle Hayden, Columbia University
PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn
From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from the knot Floer homology concordance invariant nu, which we prove is an invariant of a simple 4-manifold associated to a knot, called the knot trace. As a corollary to the existence of exotic Mazur manifolds, we produce integer homology 3-spheres admitting two distinct \(S^1 \times S^2\) surgeries, resolving a question from Problem 1.16 in Kirby's list. We also resolve a related question about the knot concordance invariants tau and epsilon. This is joint work with Tom Mark and Lisa Piccirillo.
In the morning background talk (at 11:00am), we'll review the key constructive ingredients (Dehn surgery, handlebody structures, and cork twists), and provide some extra historical context for the results described above. In the afternoon talk, I'll explain our main results, present examples demonstrating our constructive results, and discuss the key ideas in the proofs. In particular, I'll focus on how the main arguments take tools from smooth 3- and 4-dimensional topology, hyperbolic geometry, and Heegaard Floer homology and play them off one another.
There are no conferences this week.