Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Department Colloquium - Laura DeMarco (Northwestern University)

Laura DeMarco, Northwestern University

Friday, March 2nd, 2018

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Laura DeMarco (Northwestern University)

Title: Complex dynamics and arithmetic equidistribution

Abstract: About 5 years ago, Matt Baker and I formulated a conjecture about the dynamics of rational maps on P1, connecting geometry and arithmetic in the moduli space of such maps. My goal is to present recent progress on the conjecture, illustrating some of the main ideas appearing in proofs of special cases. One important special case includes a result about torsion points on elliptic curves, and I hope to discuss how this case can be related to dynamical stability and the Mandelbrot set.

Laura DeMarco (Northwestern University): Laura DeMarco received her Ph.D in Mathematics from Harvard University in 2002 under the supervision of Curtis McMullen. From 2002 to 2007, she was at he University of Chicago (as L.E. Dickinson Instructor from 2002 to 2005, and Assistant Professor from 2005 to 2007). From 2007 to 2014 she was at the University of Illinois at Chicago (as Assistant Professor from 2007 to 2009, Associate Professor from 2009 to 2012, and Professor from 2012 to 2014). In 2014, Prof DeMarco joined Northwestern University. Her awards include the NSF Postdoctoral Fellowship at the University of Chicago (2003-2006), the Sloan Foundation Research Fellowship (2008-2010), the NSF Career Award (2008-2013), the Simons Foundation Fellowship (2015-2016), and the Ruth LyttleSatter Prize (2017). In 2012, she became Fellow of the American Mathematical Society. Laura DeMarco is an Invited Speaker at the International Congress of Mathematicians in Rio de Janeiro in 2018. Her research interest include dynamical systems, complex analysis, and arithmetic geometry. She mainly focuses on the dynamics of rational maps on P1 and their moduli spaces.

Number Theory - Francois Seguin (Queen's University)

Wednesday, February 28th, 2018

Time: 2:15 p.m.  Place: Jeffery Hall 319

Speaker: François Séguin (Queen's University)

Title: Prime divisors of sparse values of cyclotomic polynomials.

Abstract: We will be presenting a result about the largest prime divisors of cyclotomic polynomials evaluated at a specific integer. We will also see how this ties in to problems we previously encountered.

Curves Seminar - Mike Roth (Queen's University)

Wednesday, February 28th, 2018

Time: 3:30 p.m.  Place: Jeffery Hall 319

Speaker: Mike Roth (Queen's University)

Title: Zak’s theorems on tangencies II

Abstract: We will prove a result due to F. Zak on the secant variety to a smooth variety (under a certain codimension condition). This result proves a conjecture of Hartshorne on linear normality of subvarieties of small codimension.

Department Colloquium - Catherine Pfaff (UC-Santa Barbara)

Catherine Pfaff, UC-Santa Barbara

Friday, February 16th, 2018

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Catherine Pfaff (University of California-Santa Barbara)

Title: A Nielsen-Thurston Inspired Story of Iterating Free Group Automorphisms and Efficiently Deforming Graphs

Abstract: While many fundamental contributions to the study of outer automorphisms of free groups date back to the early 20th century, the real explosion of activity in the eld came with two much more recent developments: the denition by Culler and Vogtmann of the deformation space of metric graphs on a surface, namely Outer Space, and the development by Bestvina, Feighn, and Handel of a train track theory for outer automorphisms of free groups. The explosion was a result of a new ability to study free group outer automorphisms using generalizations of techniques developed to study surface homeomorphisms (mapping classes) via their action on the deformation space of metrics on the surface (Teichmuller space). In our talk, we focus specically on a Nielsen-Thurston inspired story jointly studying: 1) outer automorphism conjugacy class invariants obtained by iteratively applying the automorphisms and 2) geodesics in Culler-Vogtmann Outer Space.

Catherine Pfaff, (University of California-Santa Barbara): Catherine Pfa obtained her Ph.D. in Mathematics from Rutgers University in 2012 under the supervision of Lee Mosher. Dr. Pfa was Postdoctoral Research Fellow at the Universite d'Aix-Marseille (2013-2014) and at the Universitat Bielefeld (2014-2015). Since 2015, she is Ky Fan Visiting Assistant Professor at the University of California, Santa Barbara. Catherine Pfa's research focuses on geometric group theory and geometric topology. In particular, she studies the outer automorphism group of the free group and Outer Space from a mapping class group perspective.

Department Colloquium - Qiang Zeng (Northwestern University)

Qiang Zeng, Northwestern University

Wednesday, February 14th, 2018

Time: 3:30 p.m.  Place: Jeffery Hall 234

Speaker: Qiang Zeng, Northwestern University

Title: Replica Symmetry Breaking for Mean Field Spin Glass Models

Abstract: In statistical physics, the study of spin glasses was initialized to describe the low temperature state of a class of magnetic alloys in the 1960s. Since then spin glasses have become a paradigm for highly complex disordered systems. Mean eld spin glass models were introduced as an approximation of the physical short range models in the 1970s. The typical mean eld models include the Sherrington- Kirkpatrick (SK) model, the (Ising) mix p-spin model and the spherical mixed p-spin model. Starting in 1979, the physicist Giorgio Parisi wrote a series of ground breaking papers introducing the idea of replica symmetry breaking (RSB), which allowed him to predict a solution for the SK model by breaking the symmetry of replicas innitely many times at low temperature. This is known as full-step replica symmetry breaking (FRSB). In this talk, we will show that Parisi's FRSB prediction holds at zero temperature for the more general mixed p-spin model. As a consequence, at positive temperature the level of RSB will diverge as the temperature goes to zero. On the other hand, we will show that there exist two-step RSB spherical mixed spin glass models at zero temperature, which are the rst examples beyond the replica symmetric, one-step RSB and FRSB phases. This talk is based on joint works with Antonio Aunger (Northwestern University) and Wei-Kuo Chen (University of Minnesota).

Qiang Zeng (Northwestern University): Qiang Zeng obtained his Ph.D. in Mathematics from the University of Illinois at Urbana-Champaign in 2014 under the supervision of Marius Junge and Renming Song. From 2014 to 2015 he was a Postdoctoral Fellow at Harvard University. In 2015, Dr. Song was a Postdoctoral Fellow at the Mathematical Sciences Research Institute in Berkeley, California. Since 2016, he is Boas Assistant Professor at Northwestern University in Evanston, Illinois. Qiang Zeng works at the interfaces of probability, functional analysis and mathematical physics. His main topic of study is noncommutative probability and spin glasses.

Probability Seminar - Jamie Mingo (Queen's University)

Tuesday, February 13th, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Jamie Mingo (Queen's University)

Title: The Infinitesimal Law of the GOE, Part II

Abstract:  If $X_N$ is the $N \times N$ Gaussian Orthogonal Enemble (GOE) of random matrices, we can expand $\mathrm{E}(\mathrm{tr}(X_N^n))$ as a polynomial in $1/N$, often called a genus expansion. Following the celebrated formula of Harer and Zagier for the GUE, Ledoux (2009) found a five term recurrence for the coefficients of $\mathrm{E}(\mathrm{tr}(X_N^n))$. We show that the coefficient of $1/N$ counts the number of non-crossing annular pairings of a certain type.

Our method is quite elementary. A similar formula holds for the Wishart ensemble. This identification is related to the theory of infinitesimal freeness of Belinschi and Shlyakhtenko.

Free Probability and Random Matrices Seminar Webpage:

Department Colloquium - Brad Rodgers (University of Michigan)

Brad Rodgers, University of Michigan

Monday, February 12th, 2018

Time: 4:30 p.m.  Place: Jeffery Hall 234

Speaker: Brad Rodgers, University of Michigan

Title: Some Applications of Random Matrix Theory to Analytic Number Theory

Abstract: In this talk I'll survey some of the ways that ideas originating from the study of random matrices have had an impact on analytic number theory. I hope to discuss in particular: 1) the statistical spacing of zeros of the Riemann zeta function, and what this spacing has to say about arithmetic, 2) a resolution of conjectures of Saari and Montgomery about the distribution of Rudin-Shapiro polynomials, using a connection to random walks on compact groups, and 3) recent work on the de Bruijn-Newman constant; de Bruijn showed that the Riemann hypothesis is equivalent to the claim that this constant is less than or equal to 0, and I will describe recent work showing the constant is greater than or equal to 0, conrming a conjecture of Newman. This includes joint work with J. Keating, E. Roditty-Gershon, and Z. Rudnick; and with T. Tao.

Brad Rodgers (University of Michigan): Brad Rodgers obtained his Ph.D. in Mathematics from the University of California, Los Angeles in 2013 under the supervision of Terence Tao. From 2013 to 2015 he held a postdoctoral position at the Institut fur Mathematik at the Universitat Zurich. Since 2015, he is a Postdoc Assistant Professor at the University of Michigan. Dr. Rodgers's awards include the AMS-Simons Travel Grant (2013-2016) and a NSF research grant (2017-2020). His research interests include random matrix theory, analytic number theory. In particular, he focuses on the interaction of these disciplines with analysis, probability, and combinatorics.

Department Colloquium - Daniel Le (University of Toronto)

Daniel Le, University of Toronto

Friday, February 9th, 2018

Time: 3:30 p.m.  Place: Jeffery Hall 234

Speaker: Daniel Le, University of Toronto

Title: The geometry of Galois representations

Abstract: The arithmetic of number fields can be profitably studied through the representation theory of their absolute Galois groups. These representations exhibit a number of elegant and surprising phenomena, most famously the quadratic reciprocity law. Many of these phenomena are explained by the modularity conjecture of Langlands that all Galois representations come from modular forms. Startling progress towards this conjecture began with Taylor and Wiles's study of Galois deformation spaces. We give a construction of local models for some Galois deformation spaces coming from geometric representation theory, and describe some applications to modularity conjectures and congruences between modular forms. Much of what we discuss is joint work with Bao Le Hung, Brandon Levin, and Stefano Morra.

Curves Seminar - Daniel Erman (Wisconsin-Madison)

Wednesday, February 7th, 2018

Time: 3:30 p.m.  Place: Jeffery Hall 319

Speaker: Daniel Erman (Wisconsin-Madison)

Title: Big polynomial rings and Stillman’s conjecture

Abstract: Ananyan–Hochster's recent proof of Stillman's conjecture is based on a key principle: if f_1,.., f_r are sufficiently general forms in a polynomial ring, then as the number of variables tends to infinity, they will behave increasingly like independent variables. We show that this principle becomes a theorem if ones passes to a limit of polynomial rings, using either the inverse limit or the ultraproduct. This yields the surprising fact that these limiting rings are themselves polynomial rings (in uncountably many variables). It also yields two new proofs of Stillman's conjecture. This is joint work with Steven Sam and Andrew Snowden.