[iasmath-seminars] Mathematics Seminars, Week of March 11, 2019

[Kristina Phillips]

INSTITUTE FOR ADVANCED STUDY

School of Mathematics

Princeton, NJ 08540

 

Mathematics Seminars

Week of March 11, 2019

 

 

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Please Note:

There will only be one Special Year Seminar from 1:00pm-3:00pm on Tuesday, March 12

There will be an additional Analysis Seminar on Friday, March 15 @ 2:00pm

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Monday, March 11

 

Computer Science/Discrete Mathematics Seminar I

Topic:                    Near log-convexity of measured heat in (discrete) time and consequences

Speaker:              Mert Sağlam, University of Washington

Time/Room:       11:00am - 12:00pm/Simonyi Hall 101

Abstract Link:      <http://www.math.ias.edu/seminars/abstract?event=128909> http://www.math.ias.edu/seminars/abstract?event=128909

 

Seminar on Theoretical Machine Learning

Topic:                    A Theoretical Analysis of Contrastive Unsupervised Representation Learning

Speaker:              Orestis Plevrakis, Princeton University

Time/Room:       12:15pm - 1:45pm/White Levy Room

Abstract Link:      <http://www.math.ias.edu/seminars/abstract?event=139490> http://www.math.ias.edu/seminars/abstract?event=139490

 

Members' Seminar

Topic:                    Geometry of 2-dimensional Riemannian disks and spheres.

Speaker:              Regina Rotman, University of Toronto; Member, School of Mathematics

Time/Room:       2:00pm - 3:00pm/Simonyi Hall 101

Abstract Link:      <http://www.math.ias.edu/seminars/abstract?event=129428> http://www.math.ias.edu/seminars/abstract?event=129428

 

Symplectic Dynamics/Geometry Seminar

Topic:                    Equivariant and nonequivariant contact homology

Speaker:              Jo Nelson, Rice University

Time/Room:       3:30pm - 5:00pm/Simonyi Hall 101

Abstract Link:      <http://www.math.ias.edu/seminars/abstract?event=137958> http://www.math.ias.edu/seminars/abstract?event=137958

 

Joint IAS/Princeton University Algebraic Geometry Seminar

Topic:                    A non-archimedean Ax–Lindemann theorem

Speaker:              Antoine Chambert-Loir, Université Paris 7

Time/Room:       5:00pm - 6:00pm/Princeton University, Fine 314

Abstract Link:      <http://www.math.ias.edu/seminars/abstract?event=141995> http://www.math.ias.edu/seminars/abstract?event=141995

 

 

 

Tuesday, March 12

 

Computer Science/Discrete Mathematics Seminar II

Topic:                    Halting problems for sandpiles and abelian networks

Speaker:              Lionel Levine, Cornell University; von Neumann Fellow

Time/Room:       10:30am - 12:30pm/Simonyi Hall 101

Abstract Link:      <http://www.math.ias.edu/seminars/abstract?event=129145> http://www.math.ias.edu/seminars/abstract?event=129145

 

Variational Methods in Geometry Seminar

Topic:                    Macroscopically minimal hypersurfaces

Speaker:              Hannah Alpert, Ohio State University

Time/Room:       1:00pm - 3:00pm/Simonyi Hall 101

Abstract Link:      <http://www.math.ias.edu/seminars/abstract?event=141218> http://www.math.ias.edu/seminars/abstract?event=141218

 

Symplectic Dynamics Working Group

Speaker:              Jo Nelson, Rice University

Time/Room:       1:30pm - 3:00pm/Simonyi Hall Classroom 114

 

 

 

Wednesday, March 13

 

BYOP at Lunch Working Group

Time/Room:       12:30pm - 1:30pm/Dilworth Room

 

Working Group on Geometric Applications of the Langlands Correspondence

Time/Room:       3:30pm - 5:30pm/Simonyi Hall 101

 

Mathematical Conversations

Topic:                    Wiggling and wrinkling

Speaker:              Daniel Álvarez-Gavela, Member, School of Mathematics

Time/Room:       6:00pm - 7:30pm/Dilworth Room

Abstract Link:      <http://www.math.ias.edu/seminars/abstract?event=136648> http://www.math.ias.edu/seminars/abstract?event=136648

 

 

 

Thursday, March 14

 

Venkatesh Working Group

Time/Room:       10:00am - 12:00pm/Simonyi Hall 101

 

Analysis Seminar

Topic:                    Gradient Gibbs models and homogenization

Speaker:              Scott Armstrong, New York University

Time/Room:       1:00pm - 2:00pm/Simonyi Hall 101

Abstract Link:      <http://www.math.ias.edu/seminars/abstract?event=142510> http://www.math.ias.edu/seminars/abstract?event=142510

 

Working Seminar in Algebraic Number Theory

Topic:                    The map to K2 without symbols

Speaker:              Askhay Venkatesh, Professor, School of Mathematics

Time/Room:       2:15pm - 4:15pm/Simonyi Hall 101

Abstract Link:      <http://www.math.ias.edu/seminars/abstract?event=140055> http://www.math.ias.edu/seminars/abstract?event=140055

 

Joint IAS/Princeton University Number Theory Seminar

Topic:                    TBD

Speaker:              David Hansen, Notre Dame University

Time/Room:       4:30pm - 5:30pm/TBD

 

 

 

Friday, March 15

 

Analysis Seminar

Topic:                    Localization and delocalization for interacting 1D quasiperiodic particles.

Speaker:              Ilya Kachkovskiy, Michigan State University

Time/Room:       2:00pm - 3:00pm/Simonyi Hall 101

Abstract Link:      <http://www.math.ias.edu/seminars/abstract?event=143606> http://www.math.ias.edu/seminars/abstract?event=143606

 

 

 

 

1 Near log-convexity of measured heat in (discrete) time and consequences 
   Mert Sağlam 




We answer a 1982 conjecture of Erd&‌#337;s and Simonovits about the growth of number of $k$-walks in a graph, which incidentally was studied earlier by Blakley and Dixon in 1966. We prove this conjecture in a more general setup than the earlier treatment, furthermore, through a refinement and strengthening of this inequality, we resolve two related open questions in complexity theory: the communication complexity of the $k$-Hamming distance is $\Omega(k \log k)$ and that consequently any property tester for k-linearity requires $\Omega(k \log k)$.

http://www.math.ias.edu/seminars/abstract?event=128909

2 A Theoretical Analysis of Contrastive Unsupervised Representation Learning 
   Orestis Plevrakis 




Recent empirical works have successfully used unlabeled data to learn feature representations that are broadly useful in downstream classification tasks. Several of these methods are reminiscent of the well-known word2vec embedding algorithm: leveraging availability of pairs of semantically “similar” data points and “negative samples,” the learner forces the inner product of representations of similar pairs with each other to be higher on average than with negative samples. The current paper uses the term contrastive learning for such algorithms and presents a theoretical framework for analyzing them by introducing latent classes and hypothesizing that semantically similar points are sampled from the same latent class. This framework allows us to show provable guarantees on the performance of the learned representations on the average classification task that is comprised of a subset of the same set of latent classes. Our generalization bound also shows that learned representations can reduce the (labeled) sample complexity on downstream tasks. We conduct controlled experiments in both the text and image domains to support the theory.

http://www.math.ias.edu/seminars/abstract?event=139490

3 Geometry of 2-dimensional Riemannian disks and spheres. 
   Regina Rotman 




I will discuss some geometric inequalities that hold on Riemannian 2-disks and 2-spheres. 

For example, I will prove that on any Riemannian 2-sphere there M exist at least three simple periodic geodesics of length at most 20d, where d is the diameter of M, (joint with A. Nabutovsky, Y. Liokumovich). This is a quantitative version of the well-known Lyusternik and Shnirelman theorem.

http://www.math.ias.edu/seminars/abstract?event=129428

4 Equivariant and nonequivariant contact homology 
   Jo Nelson 




I will discuss joint work with Hutchings which constructs nonequivariant and a family floer equivariant version of contact homology. Both theories are generated by two copies of each Reeb orbit over Z and capture interesting torsion information. I will then explain how one can recover the original cylindrical theory proposed by Eliashberg-Givental-Hofer via our construction.

http://www.math.ias.edu/seminars/abstract?event=137958

5 A non-archimedean Ax–Lindemann theorem 
   Antoine Chambert-Loir 




A significant step in the Pila–Zannier approach to the André–Oort conjecture is a geometric transcendence result for the uniformization map of modular curves. I will discuss joint work with François Loeser. We prove an analogue of this result in non-archimedean geometry, namely for the uniformization of Mumford curves whose associated fundamental groups are non-abelian Schottky subgroups of PGL(2,ℚₚ) contained in PGL(2,ℚ). In particular, we characterize bi-algebraic irreducible subvarieties of the uniformization.

http://www.math.ias.edu/seminars/abstract?event=141995

6 Halting problems for sandpiles and abelian networks 
   Lionel Levine 




Will this procedure be finite or infinite? If finite, how long can it last? Bjorner, Lovasz, and Shor asked these questions in 1991 about the following procedure, which goes by the name “abelian sandpile”: Given a configuration of chips on the vertices of a finite directed graph, choose (however you like) a vertex with at least as many chips as out-neighbors, and send one chip from that vertex to each of its out-neighbors. Repeat, until there is no such vertex. 

The first part of the talk will be a little tour of “algebraic directed graph theory”, whose main player is the graph Laplacian considered as an *integer* matrix. I’ll tell you about the curious class of coEulerian graphs, for which sandpile halting is easy to decide even though it may require exponentially many steps. A recent theorem of Nguyen and Wood, confirming a conjecture of Koplewitz, shows that coEulerian graphs are not rare: The directed random graph G(n,p) is coEulerian with limiting probability $1/(\zeta(2)\zeta(3)\zeta(4)\cdots )$ where $\zeta$ is the Riemann zeta function. So (in case you were wondering) about 43.6% of all simple directed graphs are coEulerian. 

The second part of the talk will focus on computation in abelian networks. These are automata networks, generalizing the abelian sandpile, for which the order of execution does not affect the final output. I’ll state some “local-to-global principles” of the form: If every automaton has property X, then the whole network has X. 

The usual Boolean gates are not abelian, but there is a set of simple "abelian logic gates” that suffice to compute any abelian function. These include an infinite family, indexed by prime numbers $p$, computing $x \mapsto \lfloor x/p \rfloor$. For directed acyclic circuits, no finite set of abelian gates suffices. But for circuits allowing a limited type of feedback, a specific set of five gates suffices. 

Is abelian computation weaker than Turing computation? If time permits, I’ll tell you what I know about this maddening question! 

If time still permits, I’ll return to the abelian sandpile to tell you about some of the “critical exponents” physicists would like to calculate. Discrete Fourier techniques may be useful here. 

I will not answer any questions about hats. 

Joint work with Ben Bond, Matt Farrell, Alexander Holroyd, and Peter Winkler.

http://www.math.ias.edu/seminars/abstract?event=129145

7 Macroscopically minimal hypersurfaces 
   Hannah Alpert 




A decades-old application of the second variation formula proves that if the scalar curvature of a closed 3--manifold is bounded below by that of the product of the hyperbolic plane with the line, then every 2--sided stable minimal surface has area at least that of the hyperbolic surface of the same genus. We can prove a coarser analogue of this statement, taking the appropriate notions of macroscopic scalar curvature and macroscopic minimizing hypersurface from Guth's 2010 proof of the systolic inequality for the n--dimensional torus. The appropriate analogue of hyperbolic area in this setting turns out to be the Gromov simplicial norm. Joint work with Kei Funano.

http://www.math.ias.edu/seminars/abstract?event=141218

8 Wiggling and wrinkling 
   Daniel Álvarez-Gavela 




The idea of corrugation goes back to Whitney, who proved that homotopy classes of immersed curves in the plane are classified by their rotation number. Generalizing this result, Smale and Hirsch proved that the space of immersions of a manifold X into a manifold Y is (weakly) homotopy equivalent to the space of injective bundle maps from TX to TY, whenever dim(X)

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