10:00am **SCGP:** Localization Techniques Program Seminar: Silviu Pufu

**Where:** SCGP 313**When:** Tue, Jan 23 10:00am — 11:00am

**Title:** Coulomb branch operators from supersymmetric localization

**Abstract:** Certain position-dependent linear combinations of Coulomb branch operators in

3d {\cal N} = 4 SCFTs form a topological sector. I will explain how to compute correlation functions of these operators using supersymmetric localization in Abelian gauge theories. As an application, I will combine these results with earlier ones on Higgs branch operators and show tests of 3d mirror symmetry in a few examples.

3d {\cal N} = 4 SCFTs form a topological sector. I will explain how to compute correlation functions of these operators using supersymmetric localization in Abelian gauge theories. As an application, I will combine these results with earlier ones on Higgs branch operators and show tests of 3d mirror symmetry in a few examples.

1:00pm **SCGP:** SCGP Weekly Talk: Maxim Zabzine

**Where:** SCGP 102**When:** Tue, Jan 23 1:00pm — 2:00pm

**Title:** Localization of supersymmetric gauge theories on compact manifolds

**Abstract:** I will try to give an overview of the present situation with the equivariant localization of supersymmetric gauge theories on compact manifolds in different dimensions. I will compare compact and non-compact calculations and compare the calculations in low and higher dimensions. As concrete examples, I will discuss briefly 3D gauge theories and 5D gauge theories and explain the structural problems in their understanding.

1:00pm Graduate Student Seminar: Xujia Chen - Belyi's theorem and dessin d'enfants

**Where:** Math Tower P-131**When:** Wed, Jan 24 1:00pm — 2:00pm

**Title:** Belyi's theorem and dessin d'enfants

**Speaker:** Xujia Chen [Stony Brook University]

**Abstract:** Every compact Riemann surface can be realized as an algebraic curve in P^2 (normalization of the zero locus of some irreducible homogeneous polynomial). Belyi's theorem states that a compact Riemann surface S can be written as an algebraic curve with all coefficients algebraic numbers if and only if there is a branched covering from S to P^1 with at most three branching values. Such a Riemann surface with such a branched covering is called a Belyi pair. Belyi pairs are in one to one correspondence to a certain kind of graphs, dessins d'enfants (``children's drawing''), which are defined purely combinatorially. I will begin from the definition and basic properties of Riemann surfaces. Belyi's theorem will not be proved, but I will explain the general idea and give part of the proof if time permits.

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2:30pm **SCGP:** Lecture Course by Mark Mineev

**Where:** SCGP 313**When:** Wed, Jan 24 2:30pm — 3:30pm

**Title:** "Introduction to integrable processes far from equilibrium: Laplacian growth, DLA, and others"

**Abstract:** Synopsis: This multi-disciplinary course of lectures on integrable growth is a critical account of the field (maybe somewhat subjective).

Numerous unstable non-equilibrium physical processes produce a multitude of rich complex patterns, both compact and (more often) fractal. Shapes of patterns in a long time limit include fingers in porous media, dendritic trees in crystallization, fractals in bacterial colonies and malignant tissues, river networks, and other bio- and geo- systems. The patterns are often universal and reproducible.

Geometry and dynamics of these forms present great challenges, because mathematical treatment of nonlinear, non-equilibrium, dissipative, and unstable dynamical systems is, as a rule, very difficult, if possible at all. While many impressive experimental and computational results were accumulated, powerful analytic methods for these systems were very limited until recently.

Remarkably, many of these growth processes were reduced (after some idealization) to a mathematical formulation, which owns rich, powerful and beautiful integrable structure, and reveals deep connections with other exact disciplines, lying far from non-equilibrium growth.

In physics the examples include quantum gravity, quantum Hall effect, and phase transitions.

In classical mathematics, this structure was found to be deeply interconnected with such fields as the inverse potential problem, classical moments, orthogonal polynomials, complex analysis, and algebraic geometry.

In modern mathematical physics we established tight relations of nonlinear growth to integrable hierarchies and deformations, normal random matrices, stochastic growth, and conformal theory.

This mathematical structure made possible to see many outstanding problems in a new light and solve several long-standing challenges in pattern formation and in mathematical physics.

I will provide brief history with key experiments and paradigms in the field, make short surveys of mathematics mentioned above, expose the integrable structure hidden behind the interface dynamics, and will present major results up to date in this rapidly growing field.

Finally I will list and discuss outstanding unresolved challenges, most notably derivation of the fractal spectrum for a diffusion-limited aggregation (DLA), which is still out of analytic reach since 1981, when it was discovered.

Numerous unstable non-equilibrium physical processes produce a multitude of rich complex patterns, both compact and (more often) fractal. Shapes of patterns in a long time limit include fingers in porous media, dendritic trees in crystallization, fractals in bacterial colonies and malignant tissues, river networks, and other bio- and geo- systems. The patterns are often universal and reproducible.

Geometry and dynamics of these forms present great challenges, because mathematical treatment of nonlinear, non-equilibrium, dissipative, and unstable dynamical systems is, as a rule, very difficult, if possible at all. While many impressive experimental and computational results were accumulated, powerful analytic methods for these systems were very limited until recently.

Remarkably, many of these growth processes were reduced (after some idealization) to a mathematical formulation, which owns rich, powerful and beautiful integrable structure, and reveals deep connections with other exact disciplines, lying far from non-equilibrium growth.

In physics the examples include quantum gravity, quantum Hall effect, and phase transitions.

In classical mathematics, this structure was found to be deeply interconnected with such fields as the inverse potential problem, classical moments, orthogonal polynomials, complex analysis, and algebraic geometry.

In modern mathematical physics we established tight relations of nonlinear growth to integrable hierarchies and deformations, normal random matrices, stochastic growth, and conformal theory.

This mathematical structure made possible to see many outstanding problems in a new light and solve several long-standing challenges in pattern formation and in mathematical physics.

I will provide brief history with key experiments and paradigms in the field, make short surveys of mathematics mentioned above, expose the integrable structure hidden behind the interface dynamics, and will present major results up to date in this rapidly growing field.

Finally I will list and discuss outstanding unresolved challenges, most notably derivation of the fractal spectrum for a diffusion-limited aggregation (DLA), which is still out of analytic reach since 1981, when it was discovered.

2:30pm Analysis Seminar: Leonid Kovalev - Metrically removable sets

**Where:** Frey 305**When:** Thu, Jan 25 2:30pm — 3:30pm

**Title:** Metrically removable sets

**Speaker:** Leonid Kovalev [Syracuse University ]

**Abstract:** A compact subset K of Euclidean space R^n is called metrically removable if any two points a,b of its complement can be joined by a curve that is disjoint from K and has length arbitrarily close to |a-b|. Every set of zero (n-1)-dimensional measure is metrically removable, but not conversely. Metrically removable sets can even have positive n-dimensional measure.

I will describe some properties of metrically removable sets and outline a proof of the following fact: totally disconnected sets of finite (n-1)-dimensional measure are metrically removable. This answers a question raised by Hakobyan and Herron in 2008.

Joint work with Sergei Kalmykov and Tapio Rajala.

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I will describe some properties of metrically removable sets and outline a proof of the following fact: totally disconnected sets of finite (n-1)-dimensional measure are metrically removable. This answers a question raised by Hakobyan and Herron in 2008.

Joint work with Sergei Kalmykov and Tapio Rajala.

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4:00pm Colloquium: Frank Thorne - TBA

**Where:** Math Tower P-131**When:** Thu, Jan 25 4:00pm — 5:00pm

**Title:** TBA

**Speaker:** Frank Thorne [University of South Carolina]

**Abstract:** TBA

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11:00pm Analysis Seminar: Frank Thorne - TBA

**Where:** P-131**When:** Fri, Jan 26 11:00pm — Sat, Jan 27 12:00am

**Title:** TBA

**Speaker:** Frank Thorne [South Carolina]

**Abstract:** TBA

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